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.
Other Related Questions
A temperature of F degrees Fahrenheit will be converted to C degrees Celsius. Given F = 9/5C + 32, which of the following expressions represents that temperature in degrees Celsius?
- A. 5/9(F-32)
- B. 5/9F-32
- C. 9/5(F-32)
- D. 9/5(F+32)
- E. 9/5F+32
Correct Answer & Rationale
Correct Answer: A
To convert Fahrenheit (F) to Celsius (C), the formula is rearranged from F = 9/5C + 32 to isolate C. Starting with F = 9/5C + 32, subtracting 32 from both sides gives F - 32 = 9/5C. Multiplying both sides by 5/9 yields C = 5/9(F - 32), which matches option A. Option B (5/9F - 32) incorrectly places 32 outside the parentheses, misrepresenting the conversion. Option C (9/5(F - 32)) incorrectly applies the conversion factor, while D (9/5(F + 32)) and E (9/5F + 32) misapply the formula entirely by not correctly isolating C.
To convert Fahrenheit (F) to Celsius (C), the formula is rearranged from F = 9/5C + 32 to isolate C. Starting with F = 9/5C + 32, subtracting 32 from both sides gives F - 32 = 9/5C. Multiplying both sides by 5/9 yields C = 5/9(F - 32), which matches option A. Option B (5/9F - 32) incorrectly places 32 outside the parentheses, misrepresenting the conversion. Option C (9/5(F - 32)) incorrectly applies the conversion factor, while D (9/5(F + 32)) and E (9/5F + 32) misapply the formula entirely by not correctly isolating C.
Which of the following equations does not represent y as a function of x in the standard (x, y) coordinate plane?
- A. y = x
- B. y = x + 2
- C. y = x² + 2
- D. x = y + 2
- E. x = y² + 2
Correct Answer & Rationale
Correct Answer: E
Option E, \( x = y^2 + 2 \), does not represent \( y \) as a function of \( x \) because it can yield multiple \( y \) values for a single \( x \) value. For example, when \( x = 6 \), \( y \) can be either 2 or -2, violating the definition of a function. In contrast, options A, B, and C express \( y \) explicitly in terms of \( x \), allowing only one output for each input. Option D, while rearranging the equation, can also be transformed into a function of \( y \) in terms of \( x \) (i.e., \( y = x - 2 \)). Thus, options A, B, C, and D all represent \( y \) as a function of \( x \).
Option E, \( x = y^2 + 2 \), does not represent \( y \) as a function of \( x \) because it can yield multiple \( y \) values for a single \( x \) value. For example, when \( x = 6 \), \( y \) can be either 2 or -2, violating the definition of a function. In contrast, options A, B, and C express \( y \) explicitly in terms of \( x \), allowing only one output for each input. Option D, while rearranging the equation, can also be transformed into a function of \( y \) in terms of \( x \) (i.e., \( y = x - 2 \)). Thus, options A, B, C, and D all represent \( y \) as a function of \( x \).
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.
Let f(x) = 3x². What is f(-2x)?
- A. -36x²
- B. -12x²
- C. -6x²
- D. 12x²
- E. 36x²
Correct Answer & Rationale
Correct Answer: D
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.