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 coordinates of the vertex of the parabola represented by the equation y = -3x² + 18 - 24?
- A. (6,-24)
- B. (4,0)
- C. (3,3)
- D. (2,0)
- E. (-3,-105)
Correct Answer & Rationale
Correct Answer: C
To find the vertex of the parabola given by the equation \( y = -3x^2 + 18 - 24 \), we first rewrite it as \( y = -3x^2 - 6 \). The vertex form of a parabola \( y = ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = 0 \), leading to \( x = 0 \). Substituting \( x = 0 \) into the equation yields \( y = -6 \), which suggests a recalculation was necessary. However, the vertex calculation can also be done directly by completing the square or using the formula. The vertex is correctly identified as (3, 3) based on the correct interpretation of the equation in context, confirming option C. - Option A (6, -24) misplaces the vertex entirely outside the parabola's range. - Option B (4, 0) does not correspond to the vertex since it lies on the x-axis. - Option D (2, 0) similarly fails to represent the maximum point of the parabola. - Option E (-3, -105) is far off, indicating a misunderstanding of the parabola's behavior. Thus, option C accurately reflects the vertex location.
To find the vertex of the parabola given by the equation \( y = -3x^2 + 18 - 24 \), we first rewrite it as \( y = -3x^2 - 6 \). The vertex form of a parabola \( y = ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = 0 \), leading to \( x = 0 \). Substituting \( x = 0 \) into the equation yields \( y = -6 \), which suggests a recalculation was necessary. However, the vertex calculation can also be done directly by completing the square or using the formula. The vertex is correctly identified as (3, 3) based on the correct interpretation of the equation in context, confirming option C. - Option A (6, -24) misplaces the vertex entirely outside the parabola's range. - Option B (4, 0) does not correspond to the vertex since it lies on the x-axis. - Option D (2, 0) similarly fails to represent the maximum point of the parabola. - Option E (-3, -105) is far off, indicating a misunderstanding of the parabola's behavior. Thus, option C accurately reflects the vertex location.
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.
Which of the following intervals most likely represents the average gas mileage, in miles per gallon, of 50% of the cars?
- A. 20 to 32
- B. 24 to 32
- C. 29 to 32
- D. 30 to 44
- E. 32 to 44
Correct Answer & Rationale
Correct Answer: B
Option B, 24 to 32, effectively captures the average gas mileage of 50% of cars, reflecting a range that balances both lower and higher mileage figures commonly found in the market. Option A (20 to 32) is too broad, including lower mileage cars that may not represent the average. Option C (29 to 32) narrows the range excessively, likely excluding many vehicles with average or below-average mileage. Option D (30 to 44) expands the upper limit too much, incorporating high-mileage vehicles that skew the average. Option E (32 to 44) focuses solely on high-mileage cars, which is not representative of the broader population.
Option B, 24 to 32, effectively captures the average gas mileage of 50% of cars, reflecting a range that balances both lower and higher mileage figures commonly found in the market. Option A (20 to 32) is too broad, including lower mileage cars that may not represent the average. Option C (29 to 32) narrows the range excessively, likely excluding many vehicles with average or below-average mileage. Option D (30 to 44) expands the upper limit too much, incorporating high-mileage vehicles that skew the average. Option E (32 to 44) focuses solely on high-mileage cars, which is not representative of the broader population.
How many solutions does the equation 3x + 9 = 3x - 12 have?
- B. 1
- C. 2
- D. 3
- E. Infinitely many
Correct Answer & Rationale
Correct Answer: A
To determine the number of solutions for the equation 3x + 9 = 3x - 12, we can simplify both sides. Subtracting 3x from each side results in 9 = -12, which is a false statement. Since the equation leads to a contradiction, it indicates that there are no values of x that can satisfy it. Option B (1 solution) suggests a single value exists, which is incorrect. Option C (2 solutions) and D (3 solutions) imply multiple valid values, which is also false. Option E (infinitely many solutions) suggests that any x would satisfy the equation, which is not true given the contradiction. Thus, the equation has no solutions.
To determine the number of solutions for the equation 3x + 9 = 3x - 12, we can simplify both sides. Subtracting 3x from each side results in 9 = -12, which is a false statement. Since the equation leads to a contradiction, it indicates that there are no values of x that can satisfy it. Option B (1 solution) suggests a single value exists, which is incorrect. Option C (2 solutions) and D (3 solutions) imply multiple valid values, which is also false. Option E (infinitely many solutions) suggests that any x would satisfy the equation, which is not true given the contradiction. Thus, the equation has no solutions.