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
What is the value of f(-3) for f(x) = 2x^2 + x + 1
Correct Answer & Rationale
Correct Answer: -20
To find \( f(-3) \) for the function \( f(x) = 2x^2 + x + 1 \), substitute \(-3\) for \(x\): \[ f(-3) = 2(-3)^2 + (-3) + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16. \] The correct answer is -20, which is incorrect based on the calculation. Examining the other options: - If an option were 16, it would be correct as shown in the calculation. - Any other number, like -10 or 0, would arise from miscalculations or incorrect substitutions, thus not representing the function's value at \(-3\). The accurate evaluation confirms that \( f(-3) = 16 \).
To find \( f(-3) \) for the function \( f(x) = 2x^2 + x + 1 \), substitute \(-3\) for \(x\): \[ f(-3) = 2(-3)^2 + (-3) + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16. \] The correct answer is -20, which is incorrect based on the calculation. Examining the other options: - If an option were 16, it would be correct as shown in the calculation. - Any other number, like -10 or 0, would arise from miscalculations or incorrect substitutions, thus not representing the function's value at \(-3\). The accurate evaluation confirms that \( f(-3) = 16 \).
Laura walks every evening on the edges of a sports field near her house. The field is in the shape of a rectangle 300 feet (ft) long and 200 ft wide, so 1 lap on the edges of the field is 1,000 ft. She enters through a gate at point G, located exactly halfway along the length of the field.
Laura counts the number of strides she takes during her daily walks. She takes about 80 strides to walk the width of the field from Z to W. Assuming that her stride length does not change, about how many strides does Laura take to walk all the way around the edge of the field?
- A. 267
- B. 320
- C. 450
- D. 400
Correct Answer & Rationale
Correct Answer: D
To determine the number of strides Laura takes to walk around the field, we first calculate the total distance of one lap, which is 1,000 feet. Since Laura takes 80 strides to walk the 200 ft width, her stride length is 2.5 ft (200 ft ÷ 80 strides). To find the total number of strides for the 1,000 ft lap, we divide the lap distance by her stride length: 1,000 ft ÷ 2.5 ft/stride = 400 strides. Option A (267) underestimates her stride count, while B (320) and C (450) do not align with her stride length calculation, leading to incorrect totals. Thus, 400 strides accurately reflects her walking distance around the field.
To determine the number of strides Laura takes to walk around the field, we first calculate the total distance of one lap, which is 1,000 feet. Since Laura takes 80 strides to walk the 200 ft width, her stride length is 2.5 ft (200 ft ÷ 80 strides). To find the total number of strides for the 1,000 ft lap, we divide the lap distance by her stride length: 1,000 ft ÷ 2.5 ft/stride = 400 strides. Option A (267) underestimates her stride count, while B (320) and C (450) do not align with her stride length calculation, leading to incorrect totals. Thus, 400 strides accurately reflects her walking distance around the field.
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.
How many more miles did the space shuttle Discovery travel than the space shuttle Atlantis?
- A. 274,100,000 miles
- B. 274,100 miles
- C. 22.3 miles
- D. 22,300,000 miles
Correct Answer & Rationale
Correct Answer: D
To determine the difference in miles traveled between the space shuttles Discovery and Atlantis, one must subtract the total miles of Atlantis from Discovery. The calculation reveals that Discovery traveled 22,300,000 miles more than Atlantis, making option D the accurate choice. Option A, 274,100,000 miles, is excessively high and does not reflect the actual difference. Option B, 274,100 miles, is too low and misrepresents the scale of space travel. Option C, 22.3 miles, is trivial and fails to capture the vast distances involved in space missions. Thus, option D accurately represents the significant difference in miles traveled.
To determine the difference in miles traveled between the space shuttles Discovery and Atlantis, one must subtract the total miles of Atlantis from Discovery. The calculation reveals that Discovery traveled 22,300,000 miles more than Atlantis, making option D the accurate choice. Option A, 274,100,000 miles, is excessively high and does not reflect the actual difference. Option B, 274,100 miles, is too low and misrepresents the scale of space travel. Option C, 22.3 miles, is trivial and fails to capture the vast distances involved in space missions. Thus, option D accurately represents the significant difference in miles traveled.