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
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.
The daily cost, C(x), for 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 determine the cost of producing 50 microscopes, substitute x = 50 into the equation C(x) = 300 + 10.5x. This gives C(50) = 300 + 10.5(50) = 300 + 525 = 825. Thus, the total cost is $825. Option A ($41,250) is incorrect as it miscalculates the cost by multiplying incorrectly. Option B ($360.50) results from a misunderstanding of the equation, possibly neglecting the fixed cost. Option C ($15,525) likely arises from an error in multiplying the variable cost without adding the fixed cost. Each incorrect option fails to follow the proper calculation method outlined in the cost equation.
To determine the cost of producing 50 microscopes, substitute x = 50 into the equation C(x) = 300 + 10.5x. This gives C(50) = 300 + 10.5(50) = 300 + 525 = 825. Thus, the total cost is $825. Option A ($41,250) is incorrect as it miscalculates the cost by multiplying incorrectly. Option B ($360.50) results from a misunderstanding of the equation, possibly neglecting the fixed cost. Option C ($15,525) likely arises from an error in multiplying the variable cost without adding the fixed cost. Each incorrect option fails to follow the proper calculation method outlined in the cost equation.
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 estimates that she can walk the length of the field from corner W to corner X in 55 seconds. To the nearest tenth of a mile per hour, what is her walking speed? (1 mile = 5,280 feet)
- A. 3.7
- B. 5.5
- C. 3.4
- D. 5.3
Correct Answer & Rationale
Correct Answer: B
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
An advertisement poster in the window of a shoe store is in the shape of a rectangle. The length of the poster is 9 less than 4 times the width. Which expression represents the length of the poster when w is the width
- A. 4w - 9
- B. 9 - 4w
- C. 4w + 9
- D. 9w - 4
Correct Answer & Rationale
Correct Answer: A
The expression for the length of the poster is determined by the relationship given in the problem. The length is described as "9 less than 4 times the width," which translates mathematically to \(4w - 9\). Option A (4w - 9) accurately reflects this relationship. Option B (9 - 4w) incorrectly suggests that the length is greater than 9 and decreases as width increases, which contradicts the problem's description. Option C (4w + 9) implies that the length increases by 9, rather than decreasing, which is not aligned with the original statement. Option D (9w - 4) introduces an incorrect multiplication factor and does not adhere to the given relationship, making it invalid.
The expression for the length of the poster is determined by the relationship given in the problem. The length is described as "9 less than 4 times the width," which translates mathematically to \(4w - 9\). Option A (4w - 9) accurately reflects this relationship. Option B (9 - 4w) incorrectly suggests that the length is greater than 9 and decreases as width increases, which contradicts the problem's description. Option C (4w + 9) implies that the length increases by 9, rather than decreasing, which is not aligned with the original statement. Option D (9w - 4) introduces an incorrect multiplication factor and does not adhere to the given relationship, making it invalid.