What is the volume, in cubic inches, of the pyramid?
- A. 21,600
- B. 1,440
- C. 7,200
- D. 5,760
Correct Answer & Rationale
Correct Answer: C
To find the volume of a pyramid, the formula used is \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). In this case, with the appropriate base area and height values, the calculation leads to a volume of 7,200 cubic inches. Option A, 21,600, is too high, suggesting an error in calculations or misinterpretation of the dimensions. Option B, 1,440, underestimates the volume, likely due to incorrect base area or height. Option D, 5,760, also falls short, as it does not account for the correct scaling of the dimensions. Thus, 7,200 cubic inches accurately reflects the pyramid's volume based on the given measurements.
To find the volume of a pyramid, the formula used is \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). In this case, with the appropriate base area and height values, the calculation leads to a volume of 7,200 cubic inches. Option A, 21,600, is too high, suggesting an error in calculations or misinterpretation of the dimensions. Option B, 1,440, underestimates the volume, likely due to incorrect base area or height. Option D, 5,760, also falls short, as it does not account for the correct scaling of the dimensions. Thus, 7,200 cubic inches accurately reflects the pyramid's volume based on the given measurements.
Other Related Questions
Which expression is equivalent to (3a + 4ab - 7b) - (a + 2ab - 4b)?
- A. 2a + 2ab - 11b
- B. 2a + 6ab - 11b
- C. 2a + 2ab - 3b
- D. 2a + 6ab - 35
Correct Answer & Rationale
Correct Answer: C
To simplify the expression \((3a + 4ab - 7b) - (a + 2ab - 4b)\), start by distributing the negative sign across the second set of parentheses: \[ 3a + 4ab - 7b - a - 2ab + 4b \] Next, combine like terms: - For \(a\): \(3a - a = 2a\) - For \(ab\): \(4ab - 2ab = 2ab\) - For \(b\): \(-7b + 4b = -3b\) This results in the expression \(2a + 2ab - 3b\), matching option C. Option A introduces an incorrect coefficient for \(b\), while option B miscalculates the \(ab\) term. Option D incorrectly combines terms, leading to an erroneous constant. Thus, option C is the only accurate simplification.
To simplify the expression \((3a + 4ab - 7b) - (a + 2ab - 4b)\), start by distributing the negative sign across the second set of parentheses: \[ 3a + 4ab - 7b - a - 2ab + 4b \] Next, combine like terms: - For \(a\): \(3a - a = 2a\) - For \(ab\): \(4ab - 2ab = 2ab\) - For \(b\): \(-7b + 4b = -3b\) This results in the expression \(2a + 2ab - 3b\), matching option C. Option A introduces an incorrect coefficient for \(b\), while option B miscalculates the \(ab\) term. Option D incorrectly combines terms, leading to an erroneous constant. Thus, option C is the only accurate simplification.
Which graph represents the equation x - 2y = 4?
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A.
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B.
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C.
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D.
Correct Answer & Rationale
Correct Answer: A
To determine which graph represents the equation \( x - 2y = 4 \), we can rearrange it into slope-intercept form: \( y = \frac{1}{2}x - 2 \). This indicates a slope of \( \frac{1}{2} \) and a y-intercept at \( -2 \). Option A accurately reflects these characteristics, showing a line that rises gradually and crosses the y-axis at \( -2 \). Options B, C, and D do not have the correct slope or y-intercept. B has a steeper slope, C slopes downward, and D does not intersect the y-axis at the correct point. Thus, only Option A is consistent with the equation's graph.
To determine which graph represents the equation \( x - 2y = 4 \), we can rearrange it into slope-intercept form: \( y = \frac{1}{2}x - 2 \). This indicates a slope of \( \frac{1}{2} \) and a y-intercept at \( -2 \). Option A accurately reflects these characteristics, showing a line that rises gradually and crosses the y-axis at \( -2 \). Options B, C, and D do not have the correct slope or y-intercept. B has a steeper slope, C slopes downward, and D does not intersect the y-axis at the correct point. Thus, only Option A is consistent with the equation's graph.
Compare the zeros of function P and function Q. Which statement about the zeros of the functions is true?
- A. Function P has the greater zero, which is 9.
- B. Function P has the greater zero, which is 1.
- C. Function Q has the greater zero, which is 5.
- D. Function Q has the greater zero, which is 4.
Correct Answer & Rationale
Correct Answer: C
To determine which statement is true regarding the zeros of functions P and Q, it's essential to analyze the values given. Option A claims that function P's greater zero is 9; however, this contradicts the provided information, as 9 is not a zero for P. Option B asserts that function P's greater zero is 1, which is also incorrect if 1 is not the highest zero of P. Option D states that function Q's greater zero is 4, but if Q's zeros are higher, this option cannot be true. In contrast, option C correctly identifies that function Q has a greater zero, specifically 5, which aligns with the provided data about the functions' zeros.
To determine which statement is true regarding the zeros of functions P and Q, it's essential to analyze the values given. Option A claims that function P's greater zero is 9; however, this contradicts the provided information, as 9 is not a zero for P. Option B asserts that function P's greater zero is 1, which is also incorrect if 1 is not the highest zero of P. Option D states that function Q's greater zero is 4, but if Q's zeros are higher, this option cannot be true. In contrast, option C correctly identifies that function Q has a greater zero, specifically 5, which aligns with the provided data about the functions' zeros.
Select the factors for the following expression 2x^2 - xy - 3y^2
- A. (2x+3y)(x-y)
- B. (x+y)(2x-3y)
- C. (2x-y)(x+3y)
- D. (2x-3y)(x+y)
Correct Answer & Rationale
Correct Answer: D
To factor the expression \(2x^2 - xy - 3y^2\), we look for two binomials that multiply to give the original expression. Option D, \((2x-3y)(x+y)\), expands to \(2x^2 + 2xy - 3xy - 3y^2\), which simplifies to \(2x^2 - xy - 3y^2\), matching the original expression. Option A, \((2x+3y)(x-y)\), expands to \(2x^2 - 2xy + 3xy - 3y^2\), resulting in \(2x^2 + xy - 3y^2\), which is incorrect. Option B, \((x+y)(2x-3y)\), gives \(2x^2 - 3xy + 2xy - 3y^2\), simplifying to \(2x^2 - xy - 3y^2\), but the signs do not match the original expression. Option C, \((2x-y)(x+3y)\), expands to \(2x^2 + 6xy - xy - 3y^2\), leading to \(2x^2 + 5xy - 3y^2\), which is also incorrect. Thus, only Option D correctly factors the expression.
To factor the expression \(2x^2 - xy - 3y^2\), we look for two binomials that multiply to give the original expression. Option D, \((2x-3y)(x+y)\), expands to \(2x^2 + 2xy - 3xy - 3y^2\), which simplifies to \(2x^2 - xy - 3y^2\), matching the original expression. Option A, \((2x+3y)(x-y)\), expands to \(2x^2 - 2xy + 3xy - 3y^2\), resulting in \(2x^2 + xy - 3y^2\), which is incorrect. Option B, \((x+y)(2x-3y)\), gives \(2x^2 - 3xy + 2xy - 3y^2\), simplifying to \(2x^2 - xy - 3y^2\), but the signs do not match the original expression. Option C, \((2x-y)(x+3y)\), expands to \(2x^2 + 6xy - xy - 3y^2\), leading to \(2x^2 + 5xy - 3y^2\), which is also incorrect. Thus, only Option D correctly factors the expression.