The radius of the sphere below is 6 centimeters (cm). What is the volume, in cubic centimeters, of the sphere?
- A. 904.32
- B. 150.72
- C. 25.12
- D. 75.36
Correct Answer & Rationale
Correct Answer: A
To find the volume of a sphere, the formula \( V = \frac{4}{3} \pi r^3 \) is used, where \( r \) is the radius. For a radius of 6 cm, the calculation is: \[ V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) \approx 904.32 \, \text{cm}^3 \] Option A (904.32) correctly represents this volume. Option B (150.72) and Option C (25.12) are significantly lower than the actual volume, indicating miscalculations or incorrect application of the formula. Option D (75.36) is also incorrect, as it does not appropriately reflect the cubic growth of the volume with respect to the radius, resulting in an underestimation.
To find the volume of a sphere, the formula \( V = \frac{4}{3} \pi r^3 \) is used, where \( r \) is the radius. For a radius of 6 cm, the calculation is: \[ V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) \approx 904.32 \, \text{cm}^3 \] Option A (904.32) correctly represents this volume. Option B (150.72) and Option C (25.12) are significantly lower than the actual volume, indicating miscalculations or incorrect application of the formula. Option D (75.36) is also incorrect, as it does not appropriately reflect the cubic growth of the volume with respect to the radius, resulting in an underestimation.
Other Related Questions
What is the slope of a line that is perpendicular to the line y = -9x + 7?
- A. 1\9
- B. -0.111111111
- C. 9
- D. -9
Correct Answer & Rationale
Correct Answer: A
To find the slope of a line perpendicular to the line given by the equation \(y = -9x + 7\), first identify the slope of the original line, which is \(-9\). The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \(-9\) is \(\frac{1}{9}\). Option A, \(\frac{1}{9}\), is the correct slope. Option B, \(-0.111111111\), is incorrect as it represents \(-\frac{1}{9}\), not the positive reciprocal. Option C, \(9\), is incorrect because it is the opposite sign of the required reciprocal. Option D, \(-9\), is simply the original slope and does not represent a perpendicular relationship.
To find the slope of a line perpendicular to the line given by the equation \(y = -9x + 7\), first identify the slope of the original line, which is \(-9\). The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \(-9\) is \(\frac{1}{9}\). Option A, \(\frac{1}{9}\), is the correct slope. Option B, \(-0.111111111\), is incorrect as it represents \(-\frac{1}{9}\), not the positive reciprocal. Option C, \(9\), is incorrect because it is the opposite sign of the required reciprocal. Option D, \(-9\), is simply the original slope and does not represent a perpendicular relationship.
Acceleration, a, in meters per second squared (m/s^2), is found by the formula a = (V2 - V1)/t where V1, is the beginning velocity, V2 is the end velocity, and t is time. What is the acceleration, in m/s^2, of an object with a beginning velocity of 14 m/s and end velocity of 8 m/s over a time of 4 seconds?
- A. 1.5
- B. -1.5
- C. 4.5
- D. -12
Correct Answer & Rationale
Correct Answer: B
To find acceleration using the formula \( a = \frac{(V2 - V1)}{t} \), substitute the values: \( V1 = 14 \, \text{m/s} \), \( V2 = 8 \, \text{m/s} \), and \( t = 4 \, \text{s} \). This results in \( a = \frac{(8 - 14)}{4} = \frac{-6}{4} = -1.5 \, \text{m/s}^2 \). Option A (1.5) is incorrect as it does not account for the decrease in velocity. Option C (4.5) miscalculates the difference and time. Option D (-12) incorrectly computes the acceleration by misapplying the formula or misinterpreting the values. Thus, the only accurate calculation reflects a deceleration, resulting in -1.5 m/s².
To find acceleration using the formula \( a = \frac{(V2 - V1)}{t} \), substitute the values: \( V1 = 14 \, \text{m/s} \), \( V2 = 8 \, \text{m/s} \), and \( t = 4 \, \text{s} \). This results in \( a = \frac{(8 - 14)}{4} = \frac{-6}{4} = -1.5 \, \text{m/s}^2 \). Option A (1.5) is incorrect as it does not account for the decrease in velocity. Option C (4.5) miscalculates the difference and time. Option D (-12) incorrectly computes the acceleration by misapplying the formula or misinterpreting the values. Thus, the only accurate calculation reflects a deceleration, resulting in -1.5 m/s².
Ricardo has two bank accounts. Each month, he will withdraw a certain amount of money from the first account and deposit a different amount of money into the second account. The inequality 8,000 – 200x ? 5,000 + 300x can be solved to find the number of months, x, for which the account has more money than the second account. What is the solution to this inequality?
- A. x ? 6
- B. x ? 30
- C. x ? 30
- D. x ? 6
Correct Answer & Rationale
Correct Answer: D
To solve the inequality \( 8,000 - 200x > 5,000 + 300x \), we first isolate \( x \). Rearranging gives \( 8,000 - 5,000 > 300x + 200x \), simplifying to \( 3,000 > 500x \). Dividing by 500 results in \( x < 6 \). Thus, the solution indicates that for \( x \) to ensure the first account has more money, it must be less than 6 months. Option A incorrectly states \( x \geq 6 \), which contradicts the solution. Options B and C mistakenly suggest \( x \geq 30 \), which is not relevant to the problem.
To solve the inequality \( 8,000 - 200x > 5,000 + 300x \), we first isolate \( x \). Rearranging gives \( 8,000 - 5,000 > 300x + 200x \), simplifying to \( 3,000 > 500x \). Dividing by 500 results in \( x < 6 \). Thus, the solution indicates that for \( x \) to ensure the first account has more money, it must be less than 6 months. Option A incorrectly states \( x \geq 6 \), which contradicts the solution. Options B and C mistakenly suggest \( x \geq 30 \), which is not relevant to the problem.
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.