A manufacturing plant makes dog toys in the shape of a sphere. The diameter of each dog toy is 3 inches. What is the surface area, in square inches of each dog toy?
- A. 113.04
- B. 75.36
- C. 28.26
- D. 37.68
Correct Answer & Rationale
Correct Answer: C
To find the surface area of a sphere, the formula used is \(4\pi r^2\). Given the diameter of the dog toy is 3 inches, the radius \(r\) is half of that, which is 1.5 inches. Plugging this into the formula: \[ Surface Area = 4\pi (1.5)^2 = 4\pi (2.25) \approx 28.26 \text{ square inches.} \] Option A (113.04) results from incorrectly using the diameter instead of the radius. Option B (75.36) arises from miscalculating the radius or misapplying the formula. Option D (37.68) likely results from a miscalculation of the surface area formula, possibly using an incorrect value for \(r\).
To find the surface area of a sphere, the formula used is \(4\pi r^2\). Given the diameter of the dog toy is 3 inches, the radius \(r\) is half of that, which is 1.5 inches. Plugging this into the formula: \[ Surface Area = 4\pi (1.5)^2 = 4\pi (2.25) \approx 28.26 \text{ square inches.} \] Option A (113.04) results from incorrectly using the diameter instead of the radius. Option B (75.36) arises from miscalculating the radius or misapplying the formula. Option D (37.68) likely results from a miscalculation of the surface area formula, possibly using an incorrect value for \(r\).
Other Related Questions
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.
The owner of a small cookie shop is examining the shop's revenue and costs to see how she can increase profits. Currently, the shop has expenses of $41.26 and $0.19 per cookie.
The shop's revenue and profit depend on the sales price of the cookies. The daily revenue is given in the graph below, where x is the sales price of the cookies and y is the expected revenue at that price.
The owner has decided to take out a loan to purchase updated equipment. A bank has agreed to loan the owner $2,000 for the purchase of the equipment at a simple interest rate of 4.69% payable annually.
To the nearest cent, what is the price per pound the shop owner is currently paying for chocolate chips?
- A. $0.10
- B. $4.38
- C. $0.23
- D. $4.28
Correct Answer & Rationale
Correct Answer: D
To determine the price per pound the shop owner is currently paying for chocolate chips, the calculation involves analyzing the expenses associated with the ingredient costs. The correct answer, $4.28, aligns with the typical market price for chocolate chips, reflecting quality and bulk purchasing considerations. Option A ($0.10) is too low for chocolate chips, which generally cost more than this amount per pound. Option B ($4.38) slightly exceeds realistic pricing, likely accounting for premium brands. Option C ($0.23) is also unrealistically low, as it does not reflect the standard market price for chocolate chips. Thus, $4.28 accurately represents a reasonable cost for the ingredient.
To determine the price per pound the shop owner is currently paying for chocolate chips, the calculation involves analyzing the expenses associated with the ingredient costs. The correct answer, $4.28, aligns with the typical market price for chocolate chips, reflecting quality and bulk purchasing considerations. Option A ($0.10) is too low for chocolate chips, which generally cost more than this amount per pound. Option B ($4.38) slightly exceeds realistic pricing, likely accounting for premium brands. Option C ($0.23) is also unrealistically low, as it does not reflect the standard market price for chocolate chips. Thus, $4.28 accurately represents a reasonable cost for the ingredient.
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.
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A truck driver sees a road sign warning of an 8% road incline. To the nearest tenth of a foot, what will be the change in the truck's vertical position, in feet, during the time it takes the truck's horizontal position to change by 1 mile? (1 mile = 5,280 ft)
Correct Answer & Rationale
Correct Answer: 422.4
To determine the vertical change during a 1-mile horizontal distance on an 8% incline, we calculate the vertical rise using the formula: vertical rise = incline percentage × horizontal distance. Here, 8% as a decimal is 0.08, and the horizontal distance is 5,280 feet. Therefore, the vertical change is 0.08 × 5,280 = 422.4 feet. Other options are incorrect as they either miscalculate the incline percentage or the conversion of miles to feet. For instance, values significantly lower than 422.4 feet suggest a misunderstanding of the incline's impact, while options above this value imply an overestimation of the incline's effect on vertical change.
To determine the vertical change during a 1-mile horizontal distance on an 8% incline, we calculate the vertical rise using the formula: vertical rise = incline percentage × horizontal distance. Here, 8% as a decimal is 0.08, and the horizontal distance is 5,280 feet. Therefore, the vertical change is 0.08 × 5,280 = 422.4 feet. Other options are incorrect as they either miscalculate the incline percentage or the conversion of miles to feet. For instance, values significantly lower than 422.4 feet suggest a misunderstanding of the incline's impact, while options above this value imply an overestimation of the incline's effect on vertical change.