Sam's Science Fair Quiz

1.

Only I

Only II

Only III

I, II, and III

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Explanation

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2. Jack has a garden with 20 rows of pepper plants. Ten of the rows have 55 plants each, and there are 1200 plants total. How many plants are in each of the other ten rows? (Assume that there are equal amounts of plants in each.)

The total number of plants in the garden is given as 1200. Ten rows have 55 plants each, which means there are a total of 10 * 55 = 550 plants in these rows. To find the number of plants in each of the other ten rows, we subtract the number of plants in the first ten rows from the total number of plants: 1200 - 550 = 650. Therefore, each of the other ten rows must have 650/10 = 65 plants.

Explanation

The total number of plants in the garden is given as 1200. Ten rows have 55 plants each, which means there are a total of 10 * 55 = 550 plants in these rows. To find the number of plants in each of the other ten rows, we subtract the number of plants in the first ten rows from the total number of plants: 1200 - 550 = 650. Therefore, each of the other ten rows must have 650/10 = 65 plants.

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3. Evaluate. 4^3/2

The expression 43/2 can be evaluated by dividing 43 by 2, which equals 21.5. However, since the answer options are whole numbers, we need to round down to the nearest whole number, which is 21. Therefore, the correct answer is 8.

Explanation

The expression 43/2 can be evaluated by dividing 43 by 2, which equals 21.5. However, since the answer options are whole numbers, we need to round down to the nearest whole number, which is 21. Therefore, the correct answer is 8.

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5. The grocery store has apples that sell for $1.75 per pound and pears that sell for $2.45 per pound. Write a linear equation that represents how much of each type of fruit can be bought with $25.

The equation 1.75x + 2.45y = 25 represents the total cost of buying x pounds of apples and y pounds of pears. The coefficient 1.75 represents the cost per pound of apples, and the coefficient 2.45 represents the cost per pound of pears. The sum of the costs of the two types of fruit should equal $25, which is represented by the constant term on the right side of the equation. Therefore, this equation accurately represents the relationship between the amount of each type of fruit that can be bought with $25.

Explanation

The equation 1.75x + 2.45y = 25 represents the total cost of buying x pounds of apples and y pounds of pears. The coefficient 1.75 represents the cost per pound of apples, and the coefficient 2.45 represents the cost per pound of pears. The sum of the costs of the two types of fruit should equal $25, which is represented by the constant term on the right side of the equation. Therefore, this equation accurately represents the relationship between the amount of each type of fruit that can be bought with $25.

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6. Find the equation given the zeros:
x = 4 + 3i
x = 4 - 3i

The given equation has the zeros x = 4 + 3i and x = 4 - 3i. Since complex zeros occur in conjugate pairs, we can write the factors of the equation as (x - (4 + 3i))(x - (4 - 3i)). Simplifying this expression gives us (x - 4 - 3i)(x - 4 + 3i). Expanding the expression further, we get x^2 - 8x + 25. Therefore, the equation is x^2 - 8x + 25 = 0.

Explanation

The given equation has the zeros x = 4 + 3i and x = 4 - 3i. Since complex zeros occur in conjugate pairs, we can write the factors of the equation as (x - (4 + 3i))(x - (4 - 3i)). Simplifying this expression gives us (x - 4 - 3i)(x - 4 + 3i). Expanding the expression further, we get x^2 - 8x + 25. Therefore, the equation is x^2 - 8x + 25 = 0.

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7.

Write any remainder as a fraction.

The given expression is x^2 - x + 2 - (2/3x + 1). To simplify the expression, we need to distribute the negative sign to each term inside the parentheses. Thus, the expression becomes x^2 - x + 2 - 2/3x - 1. We can combine like terms by combining the x terms and the constant terms. Therefore, the simplified expression is x^2 - (4/3)x + 1.

Explanation

The given expression is x^2 - x + 2 - (2/3x + 1). To simplify the expression, we need to distribute the negative sign to each term inside the parentheses. Thus, the expression becomes x^2 - x + 2 - 2/3x - 1. We can combine like terms by combining the x terms and the constant terms. Therefore, the simplified expression is x^2 - (4/3)x + 1.

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10. Solve the system of equations.
3x - 5y = 4
-2x + 4y = 3

The answer (15.5, 8.5) is the solution to the given system of equations. This means that when we substitute x = 15.5 and y = 8.5 into both equations, we get true statements. The parentheses and commas are used to separate the x and y values, and they are repeated three times to indicate that (15.5, 8.5) is the correct answer.

Explanation

The answer (15.5, 8.5) is the solution to the given system of equations. This means that when we substitute x = 15.5 and y = 8.5 into both equations, we get true statements. The parentheses and commas are used to separate the x and y values, and they are repeated three times to indicate that (15.5, 8.5) is the correct answer.

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11. Evaluate f(3x + 2) when f(x) = x² + 5.

The given question asks us to evaluate the function f(3x + 2) when f(x) = x^2 + 5. To do this, we substitute 3x + 2 into the function in place of x. So, f(3x + 2) = (3x + 2)^2 + 5. Expanding this expression gives us 9x^2 + 12x + 4 + 5, which simplifies to 9x^2 + 12x + 9. Therefore, the correct answer is 9x^2 + 12x + 9.

Explanation

The given question asks us to evaluate the function f(3x + 2) when f(x) = x^2 + 5. To do this, we substitute 3x + 2 into the function in place of x. So, f(3x + 2) = (3x + 2)^2 + 5. Expanding this expression gives us 9x^2 + 12x + 4 + 5, which simplifies to 9x^2 + 12x + 9. Therefore, the correct answer is 9x^2 + 12x + 9.

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13. Factor completely.
x³ + 5x² - 4x - 20

The given expression, x^3 + 5x^2 - 4x - 20, can be factored completely as (x + 2)(x - 2)(x + 5) or (x - 2)(x + 2)(x + 5). Both of these factorizations are equivalent and represent the complete factorization of the given expression.

Explanation

The given expression, x^3 + 5x^2 - 4x - 20, can be factored completely as (x + 2)(x - 2)(x + 5) or (x - 2)(x + 2)(x + 5). Both of these factorizations are equivalent and represent the complete factorization of the given expression.

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14. Evaluate: log₂(1/32)

The expression log2(1/32) can be rewritten as log2(2^-5), since 1/32 is equal to 2^-5. Using the property of logarithms, logb(b^x) = x, we can simplify the expression to -5. Therefore, the correct answer is -5.

Explanation

The expression log2(1/32) can be rewritten as log2(2^-5), since 1/32 is equal to 2^-5. Using the property of logarithms, logb(b^x) = x, we can simplify the expression to -5. Therefore, the correct answer is -5.

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15. How would this equation look when graphed?
(x - 3)² + (y + 4)² = 16

A line with a slope of (-3/4) and y-intercept at 16

A line with a slope of 16 intersecting the point (-3, 4)

A parabola with a vertex of (-3, 4)

A circle with a center at (3, -4) and radius of 4

A circle with a center at (-3, 4) and a radius of 4

The equation (x - 3)2 + (y + 4)2 = 16 represents a circle in the coordinate plane. The center of the circle is given by the point (3, -4) and the radius is 4 units. This means that all the points on the graph of this equation are equidistant from the center (3, -4) and the distance is always 4 units. Therefore, the correct answer is "A circle with a center at (3, -4) and a radius of 4."

Explanation

The equation (x - 3)2 + (y + 4)2 = 16 represents a circle in the coordinate plane. The center of the circle is given by the point (3, -4) and the radius is 4 units. This means that all the points on the graph of this equation are equidistant from the center (3, -4) and the distance is always 4 units. Therefore, the correct answer is "A circle with a center at (3, -4) and a radius of 4."