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
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.
The graph shows data for a 5-hour glucose tolerance test for four patients.
Symptoms of a patient with diabetes during a 5-hour glucose tolerance test include a high blood-glucose level that increases quickly and then decreases only minimally over the 5-hour period. Which patient displays symptoms of diabetes?
- A. patient 2
- B. patient 1
- C. patient 4
- D. patient 3
Correct Answer & Rationale
Correct Answer: C
Patient 4 exhibits a rapid increase in blood glucose levels followed by a minimal decrease over the 5-hour test, indicating poor glucose regulation typical of diabetes. This pattern reflects the body's inability to effectively utilize insulin. In contrast, Patient 1 shows a quick rise followed by a significant decline, suggesting normal glucose metabolism. Patient 2 may demonstrate a slight increase but returns to baseline, indicating no diabetes. Patient 3's levels remain stable, which is also indicative of normal glucose tolerance. Thus, only Patient 4 aligns with the expected symptoms of diabetes during the test.
Patient 4 exhibits a rapid increase in blood glucose levels followed by a minimal decrease over the 5-hour test, indicating poor glucose regulation typical of diabetes. This pattern reflects the body's inability to effectively utilize insulin. In contrast, Patient 1 shows a quick rise followed by a significant decline, suggesting normal glucose metabolism. Patient 2 may demonstrate a slight increase but returns to baseline, indicating no diabetes. Patient 3's levels remain stable, which is also indicative of normal glucose tolerance. Thus, only Patient 4 aligns with the expected symptoms of diabetes during the test.
Robert has $50 to spend on his utility bills each month. The basic monthly charge for water and sewer is $23.77. Electricity costs $0.1116 for each kilowatt hour used. The inequality 0.1116x + 23.77 ? 50 represents Robert's monthly utility budget. To the nearest kilowatt hour, what is the maximum number of kilowatt hours of electricity that Robert can Use without going over his monthly budget amount?
- A. 661
- B. 235
- C. 448
- D. 424
Correct Answer & Rationale
Correct Answer: B
To determine the maximum kilowatt hours (kWh) Robert can use without exceeding his budget, we start with the inequality \(0.1116x + 23.77 \leq 50\). Solving for \(x\), we first subtract 23.77 from both sides, yielding \(0.1116x \leq 26.23\). Dividing by 0.1116 gives \(x \leq 235\). Thus, Robert can use a maximum of 235 kWh. Option A (661) exceeds the budget significantly. Option C (448) and Option D (424) also surpass the budget when calculated with the fixed water charge. Only option B (235) fits within the constraints of Robert's budget.
To determine the maximum kilowatt hours (kWh) Robert can use without exceeding his budget, we start with the inequality \(0.1116x + 23.77 \leq 50\). Solving for \(x\), we first subtract 23.77 from both sides, yielding \(0.1116x \leq 26.23\). Dividing by 0.1116 gives \(x \leq 235\). Thus, Robert can use a maximum of 235 kWh. Option A (661) exceeds the budget significantly. Option C (448) and Option D (424) also surpass the budget when calculated with the fixed water charge. Only option B (235) fits within the constraints of Robert's budget.
A figure is formed by shaded squares on a grid. Which figure has a perimeter of 12units and an area of 8 square units?
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A.
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B.
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C.
- D. None of the above
Correct Answer & Rationale
Correct Answer: C
To determine the figure that meets the criteria of having a perimeter of 12 units and an area of 8 square units, we analyze each option. Option C achieves both requirements: it has a perimeter of 12 units, calculated by adding the lengths of all sides, and an area of 8 square units, determined by multiplying its length and width (2 x 4). In contrast, Option A has a perimeter exceeding 12 units, while its area is less than 8 square units. Option B has a perimeter of 10 units and an area of 6 square units, failing both criteria. Option D is not applicable since Option C meets the conditions. Thus, Option C stands out as the only figure that satisfies both the perimeter and area requirements.
To determine the figure that meets the criteria of having a perimeter of 12 units and an area of 8 square units, we analyze each option. Option C achieves both requirements: it has a perimeter of 12 units, calculated by adding the lengths of all sides, and an area of 8 square units, determined by multiplying its length and width (2 x 4). In contrast, Option A has a perimeter exceeding 12 units, while its area is less than 8 square units. Option B has a perimeter of 10 units and an area of 6 square units, failing both criteria. Option D is not applicable since Option C meets the conditions. Thus, Option C stands out as the only figure that satisfies both the perimeter and area requirements.