A campground rents canoes for either $20 per day or $4 per hour. For what number or numbers of hours, h, is it more expensive to rent a canoe at the daily rate than at the hourly rate?
- A. h = 5
- B. h >= 25
- C. h > 5
- D. h < 5
- E. h ≤ 5
Correct Answer & Rationale
Correct Answer: C
To determine when renting a canoe at the daily rate exceeds the hourly rate, we compare the costs. The daily rate is $20, while the hourly rate is $4 per hour. Setting up the inequality, we have: \[ 20 > 4h \] Dividing both sides by 4 gives: \[ 5 > h \] This means that renting for more than 5 hours makes the daily rate more economical. Option A (h = 5) is incorrect since at 5 hours, both rates are equal. Option B (h ≥ 25) is incorrect because it's not relevant to the threshold we calculated. Option D (h < 5) suggests a scenario where the daily rate is not more expensive, which contradicts our findings. Option E (h ≤ 5) includes values where the rates are equal or less, which doesn't satisfy the condition.
To determine when renting a canoe at the daily rate exceeds the hourly rate, we compare the costs. The daily rate is $20, while the hourly rate is $4 per hour. Setting up the inequality, we have: \[ 20 > 4h \] Dividing both sides by 4 gives: \[ 5 > h \] This means that renting for more than 5 hours makes the daily rate more economical. Option A (h = 5) is incorrect since at 5 hours, both rates are equal. Option B (h ≥ 25) is incorrect because it's not relevant to the threshold we calculated. Option D (h < 5) suggests a scenario where the daily rate is not more expensive, which contradicts our findings. Option E (h ≤ 5) includes values where the rates are equal or less, which doesn't satisfy the condition.
Other Related Questions
What are the solutions to the equation: x² - 10?
- A. ±5
- B. ±√10
- C. ±10
- D. ±10²
- E. ±20
Correct Answer & Rationale
Correct Answer: B
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
An irrigation pivot makes a circle with a radius of about 400 meters. Which of the following values is closest to the area, in square meters, of the circle?
- A. 1300
- B. 2500
- C. 160000
- D. 502700
- E. 1579100
Correct Answer & Rationale
Correct Answer: D
To find the area of a circle, the formula \( A = \pi r^2 \) is used, where \( r \) is the radius. With a radius of 400 meters, the area calculates to approximately \( A = \pi \times (400)^2 \approx 502700 \) square meters, making option D the closest value. Option A (1300) is far too low, indicating a misunderstanding of the formula. Option B (2500) is also significantly underestimated for such a large radius. Option C (160000) is closer but still incorrect, as it neglects the multiplication by \( \pi \). Option E (1579100) overestimates the area, suggesting a miscalculation of the radius or the area formula.
To find the area of a circle, the formula \( A = \pi r^2 \) is used, where \( r \) is the radius. With a radius of 400 meters, the area calculates to approximately \( A = \pi \times (400)^2 \approx 502700 \) square meters, making option D the closest value. Option A (1300) is far too low, indicating a misunderstanding of the formula. Option B (2500) is also significantly underestimated for such a large radius. Option C (160000) is closer but still incorrect, as it neglects the multiplication by \( \pi \). Option E (1579100) overestimates the area, suggesting a miscalculation of the radius or the area formula.
The distance from Earth to the sun is approximately 9×10ⷠmiles. The diameter of Earth is approximately 8,000 miles. The distance from Earth to the sun is approximately how many times the diameter of Earth?
- A. 1000
- B. 9000
- C. 11000
- D. 90000
- E. 9000000
Correct Answer & Rationale
Correct Answer: C
To determine how many times the diameter of Earth fits into the distance from Earth to the sun, we divide the distance (9×10^7 miles) by Earth's diameter (8,000 miles). Calculating: 9×10^7 miles ÷ 8,000 miles = 11,250. This rounds down to approximately 11,000, making option C the closest answer. Option A (1000) significantly underestimates the distance. Option B (9000) is also too low, while option D (90000) and option E (9000000) greatly overestimate the number of times the diameter fits into the distance. Thus, C is the most accurate choice.
To determine how many times the diameter of Earth fits into the distance from Earth to the sun, we divide the distance (9×10^7 miles) by Earth's diameter (8,000 miles). Calculating: 9×10^7 miles ÷ 8,000 miles = 11,250. This rounds down to approximately 11,000, making option C the closest answer. Option A (1000) significantly underestimates the distance. Option B (9000) is also too low, while option D (90000) and option E (9000000) greatly overestimate the number of times the diameter fits into the distance. Thus, C is the most accurate choice.
The volume of 1 cup of water is 14.4 cubic inches. The diameter of an empty cylindrical can is 3.0 inches. The can holds 2.0 cups of water. What is the height of the can, to the nearest 0.1 inch?
- A. 1
- B. 2
- C. 3.1
- D. 4.1
- E. 6.2
Correct Answer & Rationale
Correct Answer: D
To find the height of the can, first determine the total volume of water it holds. Since 1 cup is 14.4 cubic inches, 2 cups equal 28.8 cubic inches (2 x 14.4). The formula for the volume of a cylinder is V = πr²h. The radius (r) of the can is half the diameter: 1.5 inches. Plugging in the values: 28.8 = π(1.5)²h. Calculating the area of the base gives approximately 7.07. Rearranging the equation for height (h) results in h ≈ 4.1 inches. Options A (1), B (2), C (3.1), and E (6.2) do not satisfy the volume calculation, as they yield heights inconsistent with the required volume based on the diameter provided.
To find the height of the can, first determine the total volume of water it holds. Since 1 cup is 14.4 cubic inches, 2 cups equal 28.8 cubic inches (2 x 14.4). The formula for the volume of a cylinder is V = πr²h. The radius (r) of the can is half the diameter: 1.5 inches. Plugging in the values: 28.8 = π(1.5)²h. Calculating the area of the base gives approximately 7.07. Rearranging the equation for height (h) results in h ≈ 4.1 inches. Options A (1), B (2), C (3.1), and E (6.2) do not satisfy the volume calculation, as they yield heights inconsistent with the required volume based on the diameter provided.