Question
7. A right rectangular prism with a square base is shown. Dimensions are given in units.
The volume of the prism is 192 cubic units. What is the height, h, of the prism in units?
(HINT: You are going to need to find the AREA of Base to solve.
A. 48 units
B. 12 units
C. 24 units
D. 64 units
The volume of the prism is 192 cubic units. What is the height, h, of the prism in units?
(HINT: You are going to need to find the AREA of Base to solve.
A. 48 units
B. 12 units
C. 24 units
D. 64 units
Answers
GPT-4o mini
To find the height \( h \) of a right rectangular prism with a square base given the volume, we can use the formula for the volume of the prism:
\[
V = \text{Base Area} \times h
\]
Since the base is a square, if we let \( s \) be the side length of the base, then the area of the base is:
\[
\text{Base Area} = s^2
\]
Thus, we can rewrite the volume formula as:
\[
V = s^2 \times h
\]
We know that the volume \( V \) is 192 cubic units:
\[
192 = s^2 \times h
\]
To find \( h \), we need to express \( h \) in terms of \( s^2 \):
\[
h = \frac{192}{s^2}
\]
Next, we need to choose values for \( s^2 \) that make sense given the problem hinting at the options provided (A. 48 units, B. 12 units, C. 24 units, D. 64 units).
Let's check possible values for \( s^2 \):
1. **If \( h = 48 \)**:
\[
s^2 = \frac{192}{48} = 4 \quad (\text{so } s = 2)
\]
2. **If \( h = 12 \)**:
\[
s^2 = \frac{192}{12} = 16 \quad (\text{so } s = 4)
\]
3. **If \( h = 24 \)**:
\[
s^2 = \frac{192}{24} = 8 \quad (\text{so } s = \sqrt{8} \approx 2.83)
\]
4. **If \( h = 64 \)**:
\[
s^2 = \frac{192}{64} = 3 \quad (\text{so } s = \sqrt{3} \approx 1.73)
\]
Each option provided a valid \( s^2 \), but since \( s^2 \) must be a perfect square for the square base, we can limit our options. Among the choices, \( s^2 = 16 \) from option B (height = 12) is the only one that results in an integer side length for the square base.
Thus, the height of the prism is:
\[
\boxed{12} \text{ units}
\]
\[
V = \text{Base Area} \times h
\]
Since the base is a square, if we let \( s \) be the side length of the base, then the area of the base is:
\[
\text{Base Area} = s^2
\]
Thus, we can rewrite the volume formula as:
\[
V = s^2 \times h
\]
We know that the volume \( V \) is 192 cubic units:
\[
192 = s^2 \times h
\]
To find \( h \), we need to express \( h \) in terms of \( s^2 \):
\[
h = \frac{192}{s^2}
\]
Next, we need to choose values for \( s^2 \) that make sense given the problem hinting at the options provided (A. 48 units, B. 12 units, C. 24 units, D. 64 units).
Let's check possible values for \( s^2 \):
1. **If \( h = 48 \)**:
\[
s^2 = \frac{192}{48} = 4 \quad (\text{so } s = 2)
\]
2. **If \( h = 12 \)**:
\[
s^2 = \frac{192}{12} = 16 \quad (\text{so } s = 4)
\]
3. **If \( h = 24 \)**:
\[
s^2 = \frac{192}{24} = 8 \quad (\text{so } s = \sqrt{8} \approx 2.83)
\]
4. **If \( h = 64 \)**:
\[
s^2 = \frac{192}{64} = 3 \quad (\text{so } s = \sqrt{3} \approx 1.73)
\]
Each option provided a valid \( s^2 \), but since \( s^2 \) must be a perfect square for the square base, we can limit our options. Among the choices, \( s^2 = 16 \) from option B (height = 12) is the only one that results in an integer side length for the square base.
Thus, the height of the prism is:
\[
\boxed{12} \text{ units}
\]