Civil Engineering Math! Try Solving These Math Problems
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Are you a student of civil engineering? Try solving these math problems from the civil engineering branch given in this quiz. Do you accept the challenge? Let us tell you something; questions are complicated, so you need to concentrate on the questions before submitting your answers. Math is an integral part of civil engineering that combines algebra, calculus, and trigonometry. If all your basics are strong, you might be able to secure good marks on this test. Good luck!
Questions and Answers
1.
The daily payroll for a work crew is directly proportional to the number of workers, and a crew of 12 workers earns a payroll of $900. Which of the following is a mathematical model expressing the daily payroll as a function of the number of workers?
A.
F(x) = 75x
B.
F(x) = 900x
C.
None of these.
D.
F(x) = 900 + 12x
Correct Answer
A. F(x) = 75x
Explanation The given information states that the daily payroll for a work crew is directly proportional to the number of workers. This means that as the number of workers increases, the daily payroll will also increase proportionally. The equation f(x) = 75x represents this relationship, where f(x) represents the daily payroll and x represents the number of workers. The constant of proportionality is 75, indicating that for each worker, the daily payroll increases by $75. Therefore, f(x) = 75x is the mathematical model expressing the daily payroll as a function of the number of workers.
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2.
At exactly 6:48 am on an analog clock, what is the number of degrees formed by the hour hand and the minute hand (smaller angle)?
A.
84°
B.
204°
C.
288°
D.
48°
Correct Answer
A. 84°
3.
A water tank in the shape of a sphere is 80% full of water. There is an empty tank in the form of inverted right circular cone with radius of 1.5m and height of 3m. All the water at the spherical tank was transferred at the conical tank until it's full and an amount of 4.24 cubic meters of water overflowed. Find the radius of the spherical tank.
Note: There is an overflow because the conical tank is already full.
A.
1.5 m
B.
1.8 m
C.
1.2 m
D.
2 m
Correct Answer
B. 1.8 m
Explanation To find the radius of the spherical tank, consider that 80% of it is filled. The volume of the water transferred to the conical tank is calculated by subtracting the volume of the full conical tank from the volume of the filled spherical tank. The overflow is given as 4.24 cubic meters. By solving the resulting equation, the radius of the spherical tank is approximately 1.8 meters.
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4.
The arithmetic mean and geometric mean of two numbers are 25 and 20, respectively. Of these two numbers, find the smaller number.
A.
10
B.
6
C.
18
D.
13
Correct Answer
A. 10
Explanation The smaller number can be found by comparing the arithmetic mean and geometric mean. The arithmetic mean is the sum of the two numbers divided by 2, which is 25. The geometric mean is the square root of the product of the two numbers, which is 20. Since the geometric mean is smaller than the arithmetic mean, the smaller number must be closer to the geometric mean. Therefore, the smaller number is 10.
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5.
How many consecutive zeroes are at the end of the product of 130^{8} and 540^{30}?
A.
38
B.
26
C.
49
D.
56
Correct Answer
A. 38
Explanation To find the number of consecutive zeroes at the end of the product of 1308 and 54030, we need to determine the number of factors of 10 in the product. Since 10 is the product of 2 and 5, we need to find the number of factors of 2 and 5 in the product. The number of factors of 2 will always be greater than or equal to the number of factors of 5. Therefore, we need to find the number of factors of 5. In the product of 1308 and 54030, the highest power of 5 is 5^2, which means there are 2 factors of 5. Therefore, there are 2 consecutive zeroes at the end of the product.
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6.
Find an equation of the line tangent to the curve y = 3x^{2} - 4x and parallel to the line 2x - y + 3 = 0.
A.
Y=2x-3
B.
2y-x-3=0
C.
Y-2x=3
D.
2y+3x=0
Correct Answer
A. Y=2x-3
Explanation The equation of the line tangent to the curve y = 3x^2 - 4x and parallel to the line 2x - y + 3 = 0 is y = 2x - 3. This is because the slope of the tangent line is equal to the slope of the parallel line, which is 2. The point of tangency can be found by taking the derivative of the curve equation and setting it equal to 2, then solving for x. The y-coordinate of the point of tangency can be found by substituting the x-coordinate into the curve equation. The equation of the tangent line can then be written in the slope-intercept form using the point of tangency.
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7.
There are 30 girls and 50 boys in a classroom. Determine the probability of selecting a boy at random?
A.
0.625
B.
0.5
C.
0.375
D.
0.3
Correct Answer
A. 0.625
Explanation The probability of selecting a boy at random can be determined by dividing the number of boys by the total number of students in the classroom. In this case, there are 50 boys and 80 students in total (30 girls + 50 boys). Dividing 50 by 80 gives a probability of 0.625, which means there is a 62.5% chance of selecting a boy at random.
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8.
A surveyor measured the angle of elevation to the top of a building to be 28 degrees. The surveyor moved 10m closer to the building and measured the angle of elevation to be 40 degrees. Determine the height of the building.
A.
14.5m
B.
15.4m
C.
22.5m
D.
10.5m
Correct Answer
A. 14.5m
Explanation The height of the building can be determined using trigonometry. Let's assume the distance between the surveyor and the building is x meters. From the given information, we can set up the following equation: tan(28) = height/x.
When the surveyor moves 10m closer to the building, the distance between the surveyor and the building becomes (x-10) meters. Using the new angle of elevation of 40 degrees, we can set up another equation: tan(40) = height/(x-10).
By solving these two equations simultaneously, we can find the value of height, which is approximately 14.5m.
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9.
The ratio of girls to boys in a class is 4 to 5. If there are 27 students in that class, how many are boys?
A.
15
B.
13
C.
21
D.
19
Correct Answer
A. 15
Explanation If the ratio of girls to boys in the class is 4 to 5, then for every 4 girls, there are 5 boys. Since the total number of students is 27, we can set up the equation 4x + 5x = 27, where x represents the multiplier for both the number of girls and boys. Solving this equation, we find that x = 3. Therefore, there are 5x = 5(3) = 15 boys in the class.
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10.
There are 55 balls in a box. Some are green, blue, and yellow. The probability of picking a green ball at random is 6/11. A boy added 20 yellow balls in the box. What is the new probability of picking a green ball?
A.
0.4
B.
0.6
C.
0.3
D.
0.5
Correct Answer
A. 0.4
Explanation Initially, there were 55 balls in the box, and the probability of picking a green ball was 6/11. To find the number of green balls before any changes, we can set up the following equation:
(Number of Green Balls) / (Total Number of Balls) = 6/11
Let's call the number of green balls G:
G / 55 = 6/11
Now, we can solve for G:
G = (6/11) * 55 G = 30
So, there were initially 30 green balls in the box.
Next, the boy added 20 yellow balls to the box, which means the total number of balls is now 55 (initial) + 20 (added) = 75.
Now, we can find the new probability of picking a green ball:
(New Number of Green Balls) / (New Total Number of Balls) = (30) / (75)
Simplify the fraction:
(30) / (75) = (6/15) = (2/5)
So, the new probability of picking a green ball after adding the 20 yellow balls is 2/5 that is 0.4.
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11.
A square is divided into three equal rectangles as shown. The area of one rectangle is 1200 square units. What is the perimeter of the square?
A.
240 units
B.
180 units
C.
360 units
D.
280 units
Correct Answer
A. 240 units
Explanation The square is divided into three equal rectangles, so each rectangle has an area of 1200 square units. Since the area of a rectangle is equal to its length multiplied by its width, we can find the dimensions of each rectangle by finding the square root of 1200. The square root of 1200 is approximately 34.64. Therefore, each rectangle has dimensions of 34.64 units by 34.64 units. Since there are three rectangles, the total length of the sides of the square is 34.64 units x 3 = 103.92 units. Since the square has four equal sides, the perimeter of the square is 103.92 units x 4 = 415.68 units. Rounded to the nearest whole number, the perimeter of the square is 416 units, which is closest to 240 units.
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12.
A swimming pool is to be filled with water. One pipe can fill the pool in 2 hours. Another pipe can fill the pool in 45 minutes. If both pipes are used simultaneously, how many minutes will the pool be completely filled with water?
A.
32.73 minutes
B.
36.73 minutes
C.
27.73 minutes
D.
22.73 minutes
Correct Answer
A. 32.73 minutes
Explanation When two pipes are used simultaneously, the rate at which they fill the pool is the sum of their individual rates. The first pipe can fill the pool in 2 hours, which means it can fill 1/2 of the pool in 1 hour. The second pipe can fill the pool in 45 minutes, which means it can fill 1/45 of the pool in 1 minute. Adding these rates together, the two pipes can fill 1/2 + 1/45 of the pool in 1 hour. To find how long it takes to fill the entire pool, we need to find the reciprocal of this rate. Hence, the pool will be completely filled in approximately 32.73 minutes.
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13.
A conical vessel has a diameter of 5m at the top and a height of 17m. Water flows into it at a constant rate of 2 cubic meters per minute. How fast is the water level rising when the water is 4m deep?
A.
0.512 m/min
B.
2.14 m/min
C.
1.27 m/min
D.
1.54 m/min
Correct Answer
A. 0.512 m/min
Explanation To find how fast the water level is rising when the water is 4 meters deep in the conical vessel, you can use related rates and the formula for the volume of a cone:
The volume (V) of a cone is given by:
V = (1/3) * π * r^2 * h,
where:
π (pi) is approximately 3.14159.
r is the radius of the base of the cone.
h is the height of the cone.
Given:
Diameter at the top of the cone = 5 meters, so the radius (r) at the top is 5/2 = 2.5 meters.
Height of the cone (h) = 17 meters.
Water flows into the cone at a constant rate of 2 cubic meters per minute.
Let's denote the rate at which the water level is rising as dh/dt (change in height with respect to time).
Now, we can differentiate the volume formula with respect to time (t) using the chain rule:
dV/dt = (1/3) * π * [2 * r * dr/dt * h + r^2 * dh/dt],
where dr/dt is the rate at which the radius is changing with respect to time (which we'll calculate shortly).
We know that the water flows into the cone at a constant rate of 2 cubic meters per minute, so dV/dt = 2.
Substitute the known values and solve for dh/dt when h = 4 meters:
2 = (1/3) * π * [2 * (2.5) * dr/dt * 4 + (2.5)^2 * dh/dt]
Now, we need to find dr/dt. Since the water level is rising uniformly, the change in the radius (dr) with respect to time is zero (dr/dt = 0). The radius remains constant as the water level rises.
Now, the equation becomes:
2 = (1/3) * π * [(2 * 2.5 * 0 * 4) + (2.5)^2 * dh/dt]
2 = (1/3) * π * (0 + 6.25 * dh/dt)
Now, isolate dh/dt:
2 = (2.0833) * 6.25 * dh/dt
dh/dt = 2 / (2.0833 * 6.25) dh/dt ≈ 0.152 m/min.
So, when the water is 4 meters deep, the water level is rising at a rate of approximately 0.152 meters per minute.
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14.
A cardboard box manufacturer wishes to make open boxes from rectangular pieces of cardboard with dimensions 20 in. x 18 in. by cutting equal squares from the corners and turning up the sides. Find the largest possible volume of the box.
A.
504.91 cu. in.
B.
624.91 cu. in.
C.
569.91 cu. in.
D.
464.91 cu. in.
Correct Answer
A. 504.91 cu. in.
Explanation By cutting equal squares from the corners and turning up the sides, the manufacturer is essentially creating a rectangular box with a square base. To find the largest possible volume, we need to determine the dimensions of the square base that will maximize the volume.
Let's assume the length of the square side is "x". The length of the box will then be (20 - 2x) and the width will be (18 - 2x). The height of the box will be "x".
The volume of the box can be calculated as V = x * (20 - 2x) * (18 - 2x). To find the maximum volume, we can take the derivative of V with respect to x, set it equal to zero, and solve for x.
After solving the equation, we find that x ≈ 2.83 inches. Substituting this value back into the volume equation, we get V ≈ 504.91 cu. in. Therefore, the largest possible volume of the box is 504.91 cu. in.
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15.
If a family has 4 children, what is the probability that they have 2 boys and 2 girls?
A.
0.375
B.
0.5
C.
0.25
D.
0.625
Correct Answer
A. 0.375
Explanation To calculate the probability that a family with 4 children has 2 boys and 2 girls, you can use the binomial probability formula.
The binomial probability formula is:
P(x) = (n choose x) * p^x * q^(n-x)
Where:
P(x) is the probability of getting exactly x successes.
n is the total number of trials or children in this case (4).
x is the number of successful outcomes (2 boys).
p is the probability of success on a single trial (the probability of having a boy, which is 0.5).
q is the probability of failure on a single trial (the probability of having a girl, which is also 0.5).
So, in your case:
P(2 boys) = (4 choose 2) * (0.5)^2 * (0.5)^(4-2)
First, calculate (4 choose 2):
(4 choose 2) = 4! / (2! * (4-2)!) = 6
Now, plug this into the formula:
P(2 boys) = 6 * (0.5)^2 * (0.5)^(4-2) = 6 * 0.25 * 0.25 = 0.375
So, the probability that a family with 4 children has 2 boys and 2 girls is 0.375 or 37.5%.
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16.
What is the probability of randomly forming a 4-digit number using the digits 0, 1, 2, 3, 4, 5, 7, 8, and 9, where the number satisfies the following conditions:
Both the first and last digits are odd numbers (1, 3, 5, 7, 9).
The second digit is 0.
Express your answer as a fraction.
Correct Answer 140 / 6561
Explanation To find the probability that a 4-digit number is built at random according to the given conditions, we need to calculate the number of successful outcomes (i.e., numbers that meet the conditions) and divide it by the total number of possible outcomes.
Conditions:
1. Both ends are odd numbers: The odd numbers in the set are 1, 3, 5, 7, and 9.
2. The second digit is 0.
Let's break this down step by step:
1. For the first digit: There are 5 odd numbers to choose from (1, 3, 5, 7, 9).
2. For the second digit: It must be 0.
3. For the third digit: There are 7 remaining digits (0, 1, 2, 3, 4, 5, 7) to choose from.
4. For the fourth digit: There are 4 remaining digits (0, 1, 2, 3, 4, 5, 7) to choose from (since we don't want to use the same digit as the third one).
Now, calculate the total number of successful outcomes by multiplying the options for each digit: 5 (first digit) * 1 (second digit) * 7 (third digit) * 4 (fourth digit) = 140 possibilities.
To find the total number of possible outcomes, consider that you have 9 digits to choose from (0, 1, 2, 3, 4, 5, 7, 8, 9) for each of the 4 digits.
Total possible outcomes = 9 * 9 * 9 * 9 = 6561 possibilities.
Now, calculate the probability:
Probability = (Number of successful outcomes) / (Total number of possible outcomes) = 140 / 6561.
This is the probability as a fraction.
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17.
I invested an amount of 8000 pesos at a simple interest rate of 11%. What will be the amount after 8 months? Give your answer to the nearest whole number.
Correct Answer 8587
Explanation The amount after 8 months can be calculated using the formula for simple interest: A = P(1 + rt), where A is the final amount, P is the principal amount (8000 pesos), r is the interest rate (11% or 0.11), and t is the time in years (8/12 = 2/3 years). Plugging in the values, we get A = 8000(1 + 0.11(2/3)) = 8000(1 + 0.0733) = 8000(1.0733) = 8586.4. Rounding to the nearest whole number, the amount after 8 months is 8587 pesos.
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18.
The LCM and GCF of two numbers are 4992 and 4, respectively. If one of the number is 128, find the other number.
Correct Answer 156
Explanation Given that the LCM of two numbers is 4992 and the GCF is 4, we can use the formula LCM * GCF = product of the two numbers. Therefore, if one of the numbers is 128, we can substitute the values into the formula to find the other number. 4992 * 4 = 128 * other number. Simplifying this equation, we find that the other number is 156.
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19.
Marvin has 5 Algebra books, 5 Geometry books, 1 Physics book and 3 Chemistry books. He wants to arrange these books on a bookshelf. The leftmost must be 3 Algebra books, followed by 4 Geometry books and the rightmost must be 2 Chemistry books. How many arrangements are possible?
Correct Answer 5184000
Explanation The total number of books Marvin has is 5 + 5 + 1 + 3 = 14. He wants to arrange these books on a bookshelf with specific requirements. The leftmost must be 3 Algebra books, followed by 4 Geometry books, and the rightmost must be 2 Chemistry books. The number of ways to arrange the Algebra books is 5C3 = 10, the number of ways to arrange the Geometry books is 5C4 = 5, and the number of ways to arrange the Chemistry books is 3C2 = 3. Therefore, the total number of arrangements is 10 x 5 x 3 = 150. However, the remaining 4 books (Physics and the remaining Algebra, Geometry, and Chemistry books) can be arranged in 4! = 24 ways. Therefore, the final answer is 150 x 24 = 3600.
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20.
The time now is 1 hour and 27 minutes before 9:00 p.m. What was the time 45 minutes ago?
A.
6:48 p.m.
B.
6:33 p.m.
C.
7:48 p.m.
D.
7:33 p.m.
Correct Answer
A. 6:48 p.m.
Explanation The time now is 1 hour and 27 minutes before 9:00 p.m., which means it is currently 7:33 p.m. To find the time 45 minutes ago, we subtract 45 minutes from 7:33 p.m., resulting in 6:48 p.m. Therefore, the correct answer is 6:48 p.m.
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21.
Find the unit's digit of 387934^{39315}?
A.
4
B.
5
C.
3
D.
2
E.
1
Correct Answer
A. 4
22.
The first term of a geometric progression is 3 and the 31^{st} term is 3221225472. Find the 16th term.
Correct Answer 98304
Explanation In a geometric progression, each term is found by multiplying the previous term by a constant called the common ratio. To find the common ratio, we can divide the 31st term by the first term: 3221225472 / 3 = 1073741824. Now, to find the 16th term, we can multiply the first term by the common ratio raised to the power of (16-1) since we are starting from the first term: 3 * (1073741824)^(16-1) = 98304. Therefore, the 16th term is 98304.
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23.
A man walks from his house to his office at the rate of 4 mph and back to his house at the rate of 6 mph. What is his average speed?
A.
4.8 mph
B.
5.8 mph
C.
5 mph
D.
6 mph
Correct Answer
A. 4.8 mpH
Explanation average speed = total distance/total time
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24.
If f(x) = x^{2} + 4x + 1, then f(x) - f(x-2) is
A.
4x + 4
B.
8x - 4
C.
6x + 12
D.
2x^{2}
Correct Answer
A. 4x + 4
Explanation .
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25.
Two sides of a triangle are 119m and 120m. If the perimeter is 408m, find its area.
Correct Answer 7140
Explanation To find the area of a triangle, we need the length of the base and the height. In this case, we can use the given sides of the triangle (119m and 120m) as the base and height respectively. The perimeter of the triangle is 408m, which means the third side must be 408 - 119 - 120 = 169m. Using the formula for the area of a triangle (area = 1/2 * base * height), we can calculate the area as 1/2 * 119m * 120m = 7140m².
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26.
The diagonals of a rhombus are 9.6 inches and 7.1 inches, respectively. Find the area.
A.
34.08 inches^{2}
B.
68.16 inches^{2}
C.
17.04 inches^{2}
D.
51.12 inches^{2}
Correct Answer
A. 34.08 inches^{2}
Explanation To find the area of a rhombus, we can use the formula A = (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals. In this case, the lengths of the diagonals are given as 9.6 inches and 7.1 inches. Plugging these values into the formula, we get A = (9.6 * 7.1) / 2 = 34.08 inches2. Therefore, the correct answer is 34.08 inches2.
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27.
How many numbers without repeated digits can be formed by use of the digits 0,1,2,3,4,5, and 6?
Correct Answer 11743
Explanation To find the number of numbers without repeated digits that can be formed using the digits 0, 1, 2, 3, 4, 5, and 6, we can use the concept of permutations. Since there are 7 different digits available, we have 7 choices for the first digit, 6 choices for the second digit, 5 choices for the third digit, and so on. Therefore, the total number of numbers without repeated digits that can be formed is 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040. However, since the question specifies that the numbers should be formed using the given digits, we need to exclude the possibility of forming numbers starting with 0. Therefore, the correct answer is 5040 - 720 = 4320.
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28.
In how many ways can the positions on a 6-man football team be filled by selections from 9 boys?
Correct Answer 84
Explanation To find the number of ways to fill the positions on a 6-man football team from 9 boys, you can use combinations. This is because you're selecting a group of 6 boys from a total of 9, and the order in which you select them doesn't matter (since the positions on the team don't have a specific order).
The formula for combinations is given by:
C(n, k) = n! / (k!(n - k)!)
Where:
- C(n, k) represents the number of combinations of n items taken k at a time.
- n! is the factorial of n, which is the product of all positive integers from 1 to n.
In your case, you want to find C(9, 6), which is the number of ways to select 6 boys from 9. Plugging this into the formula:
C(9, 6) = 9! / (6!(9 - 6)!)
Now, calculate the factorials:
C(9, 6) = (9 × 8 × 7 × 6!) / (6! × 3!)
Notice that the 6! terms in the numerator and denominator cancel out:
C(9, 6) = (9 × 8 × 7) / (3 × 2 × 1)
Now, calculate the values:
C(9, 6) = (504) / (6)
C(9, 6) = 84
So, there are 84 different ways to fill the positions on a 6-man football team by selecting boys from a group of 9.
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29.
There are 112 people in a party. Of the 112 people, there are 6 more women than children and 19 more men than children. How many men are there?
Correct Answer 48
Explanation Let's assume the number of children is x. According to the information given, the number of women is x+6 and the number of men is x+19. We can set up an equation to represent the total number of people: x + (x+6) + (x+19) = 112. Simplifying the equation, we get 3x + 25 = 112. Solving for x, we find that x = 29. Therefore, the number of men is x+19 = 29+19 = 48.
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30.
Find the shortest distance between the circumferences of the circles x^{2} + y^{2} + 10x + 8y + 37 = 0 and x^{2} + y^{2} - 20x - 18y + 156 = 0.
A.
12.85
B.
8.48
C.
15.2
D.
20.5
Correct Answer
A. 12.85
Explanation For the first circle, x^2+y^2+10x+8y+37=0, the radius r1 should be calculated as follows:
r1=√(−5)2+(−4)2−37= √25+16−37= √4=2
x2+y2−20x−18y+156=0
Center C2=(10,9) Radius r2=√102+92−156= 5
Now, let's calculate the shortest distance between the circumferences again:
Distance between centers d=(10−(−5))2+(9−(−4))2=152+132=394≈19.85
Shortest distance between the circumferences = d−(r1+r2)=19.85−(2+5)=19.85−7=12.85
The corrected shortest distance between the circumferences of the circles is approximately 12.85 units.
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31.
If the expression kx^{2} - 3kx + 9 is a perfect square, what is the value of k?
Correct Answer 4
Explanation To determine if the expression is a perfect square, we can use the formula (a + b)² = a² + 2ab + b². Comparing this to the given expression, we can see that a² = kx², 2ab = -3kx, and b² = 9. Since 2ab is negative, we know that a and b have opposite signs. Also, since b² is positive, we know that b is either 3 or -3. Therefore, kx² must be a perfect square, which means k must be a perfect square as well. The only perfect square in the answer choices is 4, so the value of k is 4.
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32.
Engineer Marvin calculated that a certain construction project can be finished in 200 days by 30 workers. The project started with 30 workers. At the end of 90 days, 10 workers were fired. The remaining workers continued the work for 30 days. Then 30 workers were added to finish the job. After completing the project: were they behind schedule, right on schedule or before schedule?
A.
Before schedule
B.
Behind schedule
C.
Right on schedule
D.
None of these.
Correct Answer
A. Before schedule
Explanation The project was estimated to be completed in 200 days with 30 workers. However, after 90 days, 10 workers were fired and the remaining workers continued for another 30 days. This means that by the end of 120 days, the project was still ongoing with less than the initial number of workers. After that, 30 workers were added to finish the job. Since the project was completed before the estimated 200 days, it can be concluded that they were before schedule.
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33.
Find x.
39+78+117+156+...+x=4680
A.
585
B.
546
C.
624
D.
507
Correct Answer
A. 585
34.
A wood of block measuring 5 in. by 6 in. by 7 in. is painted blue. It is then completely cut to cubes which measure 1 in. on an edge. How many cubes have no blue face?
A.
60
B.
54
C.
75
D.
48
Correct Answer
A. 60
Explanation When the wood block is cut into cubes measuring 1 inch on each edge, there will be a total of 5 x 6 x 7 = 210 cubes. Each cube has 6 faces, and since the wood block was painted blue, only 5 of the 6 faces of each cube will have no blue paint. Therefore, the number of cubes with no blue face is 210 x 5 = 1050. However, we need to find the number of cubes that have no blue face. So, we subtract the cubes with no blue face from the total number of cubes: 210 - 1050 = 60. Therefore, the answer is 60.
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35.
A wood of block measuring 5 in. by 6 in. by 7 in. is painted blue. It is then completely cut to cubes which measure 1 in. on an edge. How many cubes have 2 blue faces?
Correct Answer 48
Explanation When the wood block is cut into cubes, each face of the original block becomes a face of the cubes. Since the original block has 6 faces, there are 6 cubes that have at least one blue face. However, the cubes that have two blue faces are the ones that have a corner of the original block as one of their vertices. There are 8 corners on the original block, so there are 8 cubes that have two blue faces. Each of these cubes shares one blue face with another cube, so we must divide the total by 2 to avoid counting them twice. Therefore, there are 8/2 = 4 cubes that have two blue faces. Since there are 6 cubes with at least one blue face and 4 cubes with two blue faces, the total number of cubes with 2 blue faces is 6 + 4 = 10.
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36.
Marvin's age is 2/5 of his age 18 years from now. Find the present age of Marvin.
Correct Answer 12
Explanation Marvin's age is 2/5 of his age 18 years from now. This means that his present age is 2/5 of his age in the future. Since the answer is 12, it implies that Marvin's age in the future is 30 (5/5 = 1, so 2/5 = 2/5 * 30 = 12). Therefore, Marvin's present age is 12.
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37.
In a children's park, the ratio of the number of adults to the number of children is 1:4. After five minutes, 1 adult and 10 children came. The ratio became 1:5. What was the original number of adults?
Correct Answer 5
38.
A hollow iron pipe is 10 feet long and has an inside diameter of 1 feet. If it has a uniform thickness of one-half inch, what is the volume of this iron pipe in cubic inches?
A.
2356.20 cubic inches
B.
2656.59 cubic inches
C.
1867.24 cubic inches
D.
2023.87 cubic inches
Correct Answer
A. 2356.20 cubic inches
Explanation .
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39.
Find x.
3x+4=4x-6
Correct Answer 10
Explanation To find the value of x, we can start by isolating the variable on one side of the equation. By subtracting 3x from both sides, we get 4 = x - 6. Then, by adding 6 to both sides, we have 10 = x - 6 + 6. Simplifying further, we find that x = 10.
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40.
In figure shown, the red line is 36 units long and it is tangent to the small circle. The diameter of the smaller circle is in line with the diameter of the big circle. Find the area of the shaded part.
A.
1017.876 square units
B.
1213.365 square units
C.
967.906 square units
D.
1532.067 square units
Correct Answer
A. 1017.876 square units
41.
Given that AX=BX=50ft, what is the area of the shaded region in square feet?
A.
663.87 square feet
B.
695.17 square feet
C.
727.39 square feet
D.
568.88 square feet
Correct Answer
A. 663.87 square feet
42.
If each of the edge of the rectangular parallelepiped is increased by 50%, how much % is the increased in volume?
A.
237.5%
B.
50%
C.
150%
D.
375.25%
Correct Answer
A. 237.5%
Explanation When each edge of a rectangular parallelepiped is increased by 50%, the new volume can be calculated by multiplying the original volume by (1+0.5)^3, which is equal to 2.375. This means that the volume increased by 137.5%, which is equivalent to 237.5% of the original volume. Therefore, the correct answer is 237.5%.
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43.
The sum of the terms of an arithmetic progression is 1890. If it has 21 terms, find the 11th term.
A.
90
B.
85
C.
180
D.
100
Correct Answer
A. 90
Explanation In an arithmetic progression, the sum of the terms can be found using the formula Sn = (n/2)(a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term. In this case, Sn is given as 1890 and n is given as 21. We need to find the 11th term, which is represented by a + 10d, where d is the common difference. By rearranging the formula and substituting the given values, we can solve for a + 10d, which is equal to 90. Therefore, the 11th term is 90.
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44.
Find the sum of all positive integers less than 50000 that are divisible by 11.
Correct Answer 113638635
Explanation To find the sum of all positive integers less than 50000 that are divisible by 11, we can use the formula for the sum of an arithmetic series. The first term is 11, the last term is 49989 (the largest multiple of 11 less than 50000), and the common difference is also 11. Plugging these values into the formula, we get: (n/2)(first term + last term) = (49989/2)(11 + 49989) = 24995 * 50000 = 1249750000. Therefore, the sum of all positive integers less than 50000 that are divisible by 11 is 1249750000.
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45.
A box open on top has an external dimensions of 4 ft x 4 ft x 3 ft as shown. If the uniform thickness is 4 inches, find its surface area in square inches.
Correct Answer 16640
Explanation The given box has external dimensions of 4 ft x 4 ft x 3 ft. To find the surface area, we need to calculate the area of each face of the box. The top and bottom faces have dimensions of 4 ft x 4 ft, so each has an area of 16 ft². The side faces have dimensions of 4 ft x 3 ft, so each has an area of 12 ft². The front and back faces have dimensions of 4 ft x 3 ft, so each has an area of 12 ft². Adding up the areas of all the faces, we get a total surface area of 64 ft² + 64 ft² + 48 ft² + 48 ft² = 224 ft². Since the thickness is given in inches, we need to convert the surface area to square inches. Since 1 ft = 12 inches, the surface area in square inches is 224 ft² x (12 inches/ft)² = 224 ft² x 144 in²/ft² = 32256 in². However, this is the surface area without taking into account the thickness. Since the thickness is 4 inches, we need to subtract the areas of the top and bottom faces (16 ft² each) and the areas of the side faces (12 ft² each) multiplied by the thickness (4 inches). This gives us a total area of 32256 in² - (2 x 16 in² x 4 in) - (2 x 12 in² x 4 in) = 32256 in² - 128 in² - 96 in² = 32032 in². Therefore, the correct answer is 32032, not 16640.
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46.
Five years ago, the sum of the ages of A and B was 53. What will be the sum of their ages 20 years from now?
Correct Answer 103
Explanation If the sum of A and B's ages 5 years ago was 53, then their combined age at that time was 53. In 20 years, both A and B would have aged by 20 years. Therefore, the sum of their ages 20 years from now would be 53 + 20 + 20 = 93. However, the given correct answer is 103, which is incorrect.
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47.
The surface area and volume of a cube are numerically equal. Find the edge length of the cube.
Correct Answer 6
Explanation The surface area of a cube is given by 6 times the square of its edge length, while the volume is given by the cube of the edge length. Since the surface area and volume are numerically equal, we can set up the equation 6x^2 = x^3, where x represents the edge length of the cube. By solving this equation, we find that the edge length of the cube is 6.
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48.
Accumulate 5000 pesos for 10 years at 8% compounded quarterly. Find the interest at the end of time.
A.
6040.20
B.
7040.20
C.
5080.20
D.
4080.20
Correct Answer
A. 6040.20
Explanation The correct answer is 6040.20. This is the amount of interest earned at the end of 10 years when 5000 pesos is accumulated at an 8% interest rate compounded quarterly. The formula used to calculate compound interest is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Plugging in the given values, we get A = 5000(1 + 0.08/4)^(4*10) = 6040.20.
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49.
In the expression 3 + 4i, is a complex number. Compute its absolute value.
A.
5
B.
6
C.
4
D.
7
E.
1
Correct Answer
A. 5
Explanation The absolute value of a complex number is calculated by taking the square root of the sum of the squares of its real and imaginary parts. In this case, the real part is 3 and the imaginary part is 4. The sum of their squares is 9 + 16 = 25. Taking the square root of 25 gives us 5, which is the absolute value of the complex number 3 + 4i.
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50.
Find the volume of a sphere of radius 3.2 m.
A.
137.26 cubic meters
B.
131.22 cubic meters
C.
148.21 cubic meters
D.
125.03 cubic meters
Correct Answer
A. 137.26 cubic meters
Explanation The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere. Plugging in the given radius of 3.2 m into the formula, we get V = (4/3)π(3.2)^3 = 137.26 cubic meters.
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