The triangle shown in the diagram has an area of 24 square centimeters. What is h, the height in centimeters, of the triangle?
- A. 9
- B. 4
- C. 8
- D. 2
Correct Answer & Rationale
Correct Answer: C
To find the height \( h \) of the triangle, we use the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Given the area is 24 cm², we can rearrange the formula to solve for \( h \): \( h = \frac{2 \times \text{Area}}{\text{base}} \). Assuming the base is 6 cm (since \( 24 = \frac{1}{2} \times 6 \times h \)), substituting gives \( h = \frac{48}{6} = 8 \). - Option A (9) is too high, as it would yield an area greater than 24 cm². - Option B (4) results in an area of only 12 cm², which is insufficient. - Option D (2) yields an area of 6 cm², far below the required area. Thus, only option C (8) satisfies the area requirement.
To find the height \( h \) of the triangle, we use the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Given the area is 24 cm², we can rearrange the formula to solve for \( h \): \( h = \frac{2 \times \text{Area}}{\text{base}} \). Assuming the base is 6 cm (since \( 24 = \frac{1}{2} \times 6 \times h \)), substituting gives \( h = \frac{48}{6} = 8 \). - Option A (9) is too high, as it would yield an area greater than 24 cm². - Option B (4) results in an area of only 12 cm², which is insufficient. - Option D (2) yields an area of 6 cm², far below the required area. Thus, only option C (8) satisfies the area requirement.
Other Related Questions
Simplify 6^2 - 3^2
- A. 6
- B. 9
- C. 27
- D. 3
Correct Answer & Rationale
Correct Answer: C
To simplify \(6^2 - 3^2\), we apply the difference of squares formula, which states \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = 6\) and \(b = 3\). Thus, we have: \[ 6^2 - 3^2 = (6 - 3)(6 + 3) = 3 \times 9 = 27 \] Option A (6) is incorrect as it miscalculates the expression. Option B (9) mistakenly considers only one of the squared terms. Option D (3) misinterprets the operations involved, leading to an incorrect result. The correct evaluation yields 27, confirming option C as the accurate answer.
To simplify \(6^2 - 3^2\), we apply the difference of squares formula, which states \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = 6\) and \(b = 3\). Thus, we have: \[ 6^2 - 3^2 = (6 - 3)(6 + 3) = 3 \times 9 = 27 \] Option A (6) is incorrect as it miscalculates the expression. Option B (9) mistakenly considers only one of the squared terms. Option D (3) misinterprets the operations involved, leading to an incorrect result. The correct evaluation yields 27, confirming option C as the accurate answer.
The Great Pyramid at Giza in Egypt is a square pyramid that measures approximately 756 feet on each side. The height of the pyramid is approximately 450 feet. What is the approximate volume, in cubic feet, of the pyramid?
- A. 51,030,000
- B. 85,730,400
- C. 226,800
- D. 453,600
Correct Answer & Rationale
Correct Answer: B
To find the volume of a pyramid, the formula used is \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). The base area of the Great Pyramid, being a square, is calculated as \( 756 \times 756 = 571,536 \) square feet. Multiplying this by the height of 450 feet gives \( 571,536 \times 450 = 257,184,000 \). Dividing by 3 yields a volume of approximately 85,728,000 cubic feet, which rounds to 85,730,400. Option A (51,030,000) underestimates the height and base area. Option C (226,800) miscalculates the base area significantly. Option D (453,600) incorrectly applies the volume formula, failing to account for the correct base area and height.
To find the volume of a pyramid, the formula used is \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). The base area of the Great Pyramid, being a square, is calculated as \( 756 \times 756 = 571,536 \) square feet. Multiplying this by the height of 450 feet gives \( 571,536 \times 450 = 257,184,000 \). Dividing by 3 yields a volume of approximately 85,728,000 cubic feet, which rounds to 85,730,400. Option A (51,030,000) underestimates the height and base area. Option C (226,800) miscalculates the base area significantly. Option D (453,600) incorrectly applies the volume formula, failing to account for the correct base area and height.
Solve the inequality for x: -4/3 x + 4 ? 16
- A. x??9
- B. x??9
- C. x??9
- D. x?9
Correct Answer & Rationale
Correct Answer: A
To solve the inequality \(-\frac{4}{3}x + 4 < 16\), first isolate \(x\) by subtracting 4 from both sides, resulting in \(-\frac{4}{3}x < 12\). Next, multiply both sides by \(-\frac{3}{4}\), remembering to reverse the inequality sign, yielding \(x > 9\). Options B and C incorrectly suggest \(x < 9\), which contradicts our solution. Option D, stating \(x \leq 9\), also misrepresents the inequality since it does not include values greater than 9. Thus, only option A accurately reflects the solution \(x > 9\).
To solve the inequality \(-\frac{4}{3}x + 4 < 16\), first isolate \(x\) by subtracting 4 from both sides, resulting in \(-\frac{4}{3}x < 12\). Next, multiply both sides by \(-\frac{3}{4}\), remembering to reverse the inequality sign, yielding \(x > 9\). Options B and C incorrectly suggest \(x < 9\), which contradicts our solution. Option D, stating \(x \leq 9\), also misrepresents the inequality since it does not include values greater than 9. Thus, only option A accurately reflects the solution \(x > 9\).
The daily cost, C(x), tor a company to produce x microscopes is given by the equation C(x) = 300 + 10.5x. What is the cost of producing 50 microscopes?
- A. $41,250
- B. $360.50
- C. $15,525
- D. $825
Correct Answer & Rationale
Correct Answer: D
To find the cost of producing 50 microscopes, substitute x = 50 into the cost equation C(x) = 300 + 10.5x. This yields C(50) = 300 + 10.5(50), resulting in C(50) = 300 + 525 = 825. Thus, the cost for 50 microscopes is $825. Option A ($41,250) is incorrect as it likely results from a miscalculation or misunderstanding of the equation. Option B ($360.50) underestimates the production cost by omitting the correct multiplication factor. Option C ($15,525) suggests an error in the calculation, possibly misinterpreting the coefficients in the equation.
To find the cost of producing 50 microscopes, substitute x = 50 into the cost equation C(x) = 300 + 10.5x. This yields C(50) = 300 + 10.5(50), resulting in C(50) = 300 + 525 = 825. Thus, the cost for 50 microscopes is $825. Option A ($41,250) is incorrect as it likely results from a miscalculation or misunderstanding of the equation. Option B ($360.50) underestimates the production cost by omitting the correct multiplication factor. Option C ($15,525) suggests an error in the calculation, possibly misinterpreting the coefficients in the equation.