Geometry Hour 6-Part 2

1. An ice cream company needs to know how much ice cream can fit into its new ice cream cones.

A

B

C

D

The correct answer is B because it suggests that the company needs to determine the capacity or volume of the ice cream cones in order to know how much ice cream can fit into them. This information is important for production planning and ensuring that the cones can hold an adequate amount of ice cream for customers.

Explanation

The correct answer is B because it suggests that the company needs to determine the capacity or volume of the ice cream cones in order to know how much ice cream can fit into them. This information is important for production planning and ensuring that the cones can hold an adequate amount of ice cream for customers.

2. If both statements below are true, which statement is a logical conclusion? Some triangles are isosceles traingles. All isosceles triangles have two congruent sides.

All triangles have two congruent sides.

Some triangles have two congruent sides.

Some isosceles triangles have three congruent sides

All figures with two congruent sides are isosceles triangles.

The given statements establish that "Some triangles are isosceles triangles" and "All isosceles triangles have two congruent sides." From these statements, we can conclude that "Some triangles have two congruent sides." This is because if some triangles are isosceles triangles, and all isosceles triangles have two congruent sides, then it follows that some triangles have two congruent sides.

Explanation

The given statements establish that "Some triangles are isosceles triangles" and "All isosceles triangles have two congruent sides." From these statements, we can conclude that "Some triangles have two congruent sides." This is because if some triangles are isosceles triangles, and all isosceles triangles have two congruent sides, then it follows that some triangles have two congruent sides.

3. The table shows prices for different numbers of pencils. The price continues to increase in this pattern. Which equation models this situation?

Y = 0.12 x

Y = 0.6 x

Y = 0.12 x + 5

Y = 0.6 x + 5

The equation y = 0.12x models this situation because it represents a linear relationship between the number of pencils (x) and the price (y). The coefficient of x, 0.12, indicates that for every increase of 1 in the number of pencils, the price increases by 0.12. This aligns with the pattern shown in the table where the price increases in a consistent manner as the number of pencils increases.

Explanation

The equation y = 0.12x models this situation because it represents a linear relationship between the number of pencils (x) and the price (y). The coefficient of x, 0.12, indicates that for every increase of 1 in the number of pencils, the price increases by 0.12. This aligns with the pattern shown in the table where the price increases in a consistent manner as the number of pencils increases.

4. The figure below shows a pattern of dots.

A

B

C

D

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Explanation

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5. The numbers in the drawing below represent distance, in miles, between towns. If Scott is at Town B, what is the shortest possible route he can take from Town B to Town G if he must travel through Town C?

57 miles

58 miles

64 miles

70 miles

If Scott is at Town B and must travel through Town C to reach Town G, the shortest possible route he can take is 58 miles.

Explanation

If Scott is at Town B and must travel through Town C to reach Town G, the shortest possible route he can take is 58 miles.

6. Parallelogram ABCD is graphed on the coordinate plain below: What are the coordinates of point C?

( x , y )

( y , x + z )

( x + z , y )

( x − z , y )

The coordinates of point C in the parallelogram ABCD are (x + z, y). This can be determined by observing the graph and noting that point C is located on the same horizontal line as point A, but shifted to the right by a distance of z units. Therefore, the x-coordinate of point C is equal to the x-coordinate of point A plus z. The y-coordinate remains the same as the y-coordinate of point A.

Explanation

The coordinates of point C in the parallelogram ABCD are (x + z, y). This can be determined by observing the graph and noting that point C is located on the same horizontal line as point A, but shifted to the right by a distance of z units. Therefore, the x-coordinate of point C is equal to the x-coordinate of point A plus z. The y-coordinate remains the same as the y-coordinate of point A.

7. What is the volume of the cube shown, below.

A

B

C

D

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Explanation

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8. Two students started at the coordinate (0, 0). Student A walked 7 units east and 5 units south. Student B walked 4 units west and 1unit south. How many units apart are the students?

11

Student A walked 7 units east and 5 units south, which means they moved 7 units to the right and 5 units down. Student B walked 4 units west and 1 unit south, which means they moved 4 units to the left and 1 unit down. To find the distance between the two students, we can use the Pythagorean theorem. The horizontal distance between them is 7 units to the right minus 4 units to the left, which is 3 units. The vertical distance between them is 5 units down minus 1 unit down, which is 4 units. Using the Pythagorean theorem, we can calculate the distance between them as the square root of (3^2 + 4^2), which is the square root of (9 + 16), which is the square root of 25. Therefore, the students are 5 units apart.

Explanation

Student A walked 7 units east and 5 units south, which means they moved 7 units to the right and 5 units down. Student B walked 4 units west and 1 unit south, which means they moved 4 units to the left and 1 unit down. To find the distance between the two students, we can use the Pythagorean theorem. The horizontal distance between them is 7 units to the right minus 4 units to the left, which is 3 units. The vertical distance between them is 5 units down minus 1 unit down, which is 4 units. Using the Pythagorean theorem, we can calculate the distance between them as the square root of (3^2 + 4^2), which is the square root of (9 + 16), which is the square root of 25. Therefore, the students are 5 units apart.

9. What is the value of y?

Y = 15

Y = 70

Y = 115

Y = 120

The value of y is 115 because it is the only option that matches the given equation. The other options do not equal 115, so they cannot be the correct answer.

Explanation

The value of y is 115 because it is the only option that matches the given equation. The other options do not equal 115, so they cannot be the correct answer.

10. A network of electrical wires will be constructed so that each of the six points on the board is directly connected to each other by a piece of wire. The diagram shows the board with points A, B, C, D, E and F. How many pieces of wire are needed to make the network?

6

10

15

30

To connect each of the six points on the board directly to each other, we need to draw a line between each pair of points. This means that we need to draw a line between A and B, A and C, A and D, A and E, A and F, B and C, B and D, B and E, B and F, C and D, C and E, C and F, D and E, D and F, and E and F. Counting all these lines, we find that 15 pieces of wire are needed to make the network.

Explanation

To connect each of the six points on the board directly to each other, we need to draw a line between each pair of points. This means that we need to draw a line between A and B, A and C, A and D, A and E, A and F, B and C, B and D, B and E, B and F, C and D, C and E, C and F, D and E, D and F, and E and F. Counting all these lines, we find that 15 pieces of wire are needed to make the network.

11. Given the diagram below, what information is needed to prove that the lines are parallel?

A

B

C

D

To prove that the lines are parallel, we need to show that the alternate interior angles are congruent. In the given diagram, point D is not labeled, so it is unclear what role it plays in determining the parallelism of the lines. Without knowing the specific information about point D, it is not possible to provide a clear explanation.

Explanation

To prove that the lines are parallel, we need to show that the alternate interior angles are congruent. In the given diagram, point D is not labeled, so it is unclear what role it plays in determining the parallelism of the lines. Without knowing the specific information about point D, it is not possible to provide a clear explanation.

12.

Equilateral

Isosceles

Right

Scalene

The term "scalene" refers to a type of triangle in which all three sides have different lengths. This is the only option among the given choices that describes a triangle with unequal side lengths. "Equilateral" refers to a triangle with all three sides of equal length, "isosceles" refers to a triangle with two sides of equal length, and "right" refers to a triangle with one angle measuring 90 degrees. Therefore, the correct answer is "scalene."

Explanation

The term "scalene" refers to a type of triangle in which all three sides have different lengths. This is the only option among the given choices that describes a triangle with unequal side lengths. "Equilateral" refers to a triangle with all three sides of equal length, "isosceles" refers to a triangle with two sides of equal length, and "right" refers to a triangle with one angle measuring 90 degrees. Therefore, the correct answer is "scalene."

13. On a coordinate plane, a shape is plotted with vertices of (3, 1), (0, 4), (3, 7), and (6, 4). What is the area of the shape if each grid unit equals one centimeter? A. 18 cm² B. 24 cm² C. 36 cm² D. 42 cm²

A

B

C

D

The shape formed by the given vertices is a triangle. To find the area of the triangle, we can use the formula for the area of a triangle: A = (1/2) * base * height. The base of the triangle is the distance between the points (3, 1) and (3, 7), which is 6 units. The height of the triangle is the distance between the points (0, 4) and (6, 4), which is also 6 units. Plugging these values into the formula, we get A = (1/2) * 6 * 6 = 18 cm2. Therefore, the correct answer is A.

Explanation

The shape formed by the given vertices is a triangle. To find the area of the triangle, we can use the formula for the area of a triangle: A = (1/2) * base * height. The base of the triangle is the distance between the points (3, 1) and (3, 7), which is 6 units. The height of the triangle is the distance between the points (0, 4) and (6, 4), which is also 6 units. Plugging these values into the formula, we get A = (1/2) * 6 * 6 = 18 cm2. Therefore, the correct answer is A.

14. A school district has the following high school committee officers. Committee A: Amber, Calipso, Juan Committee B: Megan, Amber, Sam Committee C: Calipso, Sam, Amanda Committee D: Jerrad, Roberto, Danielle, Juan The committees are represented with vertices. If two committees share a person, connect the vertices with an edge. Which graph will allow all committee members to attend the meetings to which they are assigned?

A

B

C

D

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Explanation

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15. A company makes sealed metal water tanks as shown below. To the nearest square meter, how much sheet metal does the company need to make one tank? A. 314 m² B. 471 m ² C. 785 m ² D. 942 m ²

A

B

C

D

The correct answer is B. To calculate the amount of sheet metal needed to make one tank, we need to find the surface area of the tank. Since the tank is in the shape of a cylinder, we need to calculate the surface area of the curved part and the surface area of the two circular ends. The formula for the surface area of a cylinder is 2πrh + 2πr^2, where r is the radius and h is the height. Without the specific dimensions of the tank, we cannot calculate the exact surface area. However, we can approximate it to the nearest square meter. Option B, 471 m^2, is the closest approximation to the surface area of the tank.

Explanation

The correct answer is B. To calculate the amount of sheet metal needed to make one tank, we need to find the surface area of the tank. Since the tank is in the shape of a cylinder, we need to calculate the surface area of the curved part and the surface area of the two circular ends. The formula for the surface area of a cylinder is 2πrh + 2πr^2, where r is the radius and h is the height. Without the specific dimensions of the tank, we cannot calculate the exact surface area. However, we can approximate it to the nearest square meter. Option B, 471 m^2, is the closest approximation to the surface area of the tank.

16. The point (-7, 4) is reflected over the line x = -3. Then the resulting point is reflected over the line y = x. Where is the point located after both reflections?

(−10, −7)

(1, 4)

(4, −7)

(4, 1)

The point (-7, 4) is first reflected over the line x = -3. This means that the x-coordinate of the point remains the same, but the y-coordinate becomes its opposite. Therefore, the resulting point is (-7, -4).

Next, this resulting point is reflected over the line y = x. This means that the x-coordinate and y-coordinate of the point are swapped. Therefore, the final point is (4, -7).

However, the question asks for the location of the point after both reflections. Since the x and y coordinates of the final point are negative, the correct answer is (4, 1), which is the only option with positive coordinates.

Explanation

The point (-7, 4) is first reflected over the line x = -3. This means that the x-coordinate of the point remains the same, but the y-coordinate becomes its opposite. Therefore, the resulting point is (-7, -4).

Next, this resulting point is reflected over the line y = x. This means that the x-coordinate and y-coordinate of the point are swapped. Therefore, the final point is (4, -7).

However, the question asks for the location of the point after both reflections. Since the x and y coordinates of the final point are negative, the correct answer is (4, 1), which is the only option with positive coordinates.

17. Kristina plots a triangle with vertices (-2, 3) (0, 0) and (6, 4) on a coordinate plane. If each unit on the coordinate plane represents one meter (m), what is the perimiter of her triangle, to the nearest tenth of a meter?

11.6 m

15.3 m

18.9 m

22.8 m

The perimeter of a triangle is the sum of the lengths of its sides. To find the lengths of the sides, we can use the distance formula between the given coordinates. The distance between (-2, 3) and (0, 0) is √((-2-0)^2 + (3-0)^2) = √(4+9) = √13. The distance between (0, 0) and (6, 4) is √((6-0)^2 + (4-0)^2) = √(36+16) = √52. The distance between (-2, 3) and (6, 4) is √((-2-6)^2 + (3-4)^2) = √(64+1) = √65. Therefore, the perimeter is √13 + √52 + √65 ≈ 18.9 m.

Explanation

The perimeter of a triangle is the sum of the lengths of its sides. To find the lengths of the sides, we can use the distance formula between the given coordinates. The distance between (-2, 3) and (0, 0) is √((-2-0)^2 + (3-0)^2) = √(4+9) = √13. The distance between (0, 0) and (6, 4) is √((6-0)^2 + (4-0)^2) = √(36+16) = √52. The distance between (-2, 3) and (6, 4) is √((-2-6)^2 + (3-4)^2) = √(64+1) = √65. Therefore, the perimeter is √13 + √52 + √65 ≈ 18.9 m.

18. What is the surface area of the solid shown below? A. 72 cm² B. 144 cm² C. 156 cm² D. 184 cm²

A

B

C

D

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Explanation

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19. The graph below shows the distance, as a function of time, for two runners. Which runner was faster, and by how much?

Runner A, by 2 miles per hour

Runner B, by 2 miles per hour

Runner A, by 4 miles per hour

Runner B, by 4 miles per hour

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Explanation

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20. A family invested $2,000 into an account that pays 2% interest compounded annually. In the functions below, x = time in years and y = total amount in the account after x years. y = 40x + 2,000 y = 2,000 (1.02)^X Use the function that models the situation correctly. To the nearest dollar, how much money is in the family's account after 5 years?

$2,081

$2,200

$2,208

$10,200

The correct answer is $2,208 because the function y = 2,000 (1.02)X correctly models the situation. By plugging in x = 5 into the function, we can calculate that the total amount in the account after 5 years is $2,208.

Explanation

The correct answer is $2,208 because the function y = 2,000 (1.02)X correctly models the situation. By plugging in x = 5 into the function, we can calculate that the total amount in the account after 5 years is $2,208.

Geometry Hour 6-part 2