Question
\( \rightarrow \) Unit Posttest A satellite launch rocket has a cylindrical fuel tank. The fuel tank can hold \( V \) cubic meters of fuel. If the tank measures \( d \) meters across, what is the height of the tank in meters? \[ \begin{array}{ll}\text { A. } \frac{2 V}{\pi d^{2}} \\ \text { B. } \frac{4 V}{d^{2}} \\ \text { C. } \frac{V}{\pi d^{2}} \\ \text { D. } \frac{4 V}{\pi d^{2}} \\ \text { E. } \frac{8 V}{\pi d^{2}}\end{array} \]
Ask by Blake Elliott. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The height of the tank is \( \frac{4V}{\pi d^{2}} \) meters.
Solution
To determine the height \( h \) of the cylindrical fuel tank, we can use the formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given that the tank measures \( d \) meters across (which is the diameter), the radius \( r \) is:
\[
r = \frac{d}{2}
\]
Substitute \( r = \frac{d}{2} \) into the volume formula:
\[
V = \pi \left(\frac{d}{2}\right)^2 h = \pi \frac{d^2}{4} h
\]
Solving for \( h \):
\[
h = \frac{V}{\pi \frac{d^2}{4}} = \frac{4V}{\pi d^2}
\]
Therefore, the height of the tank is:
\[
h = \frac{4V}{\pi d^2}
\]
**Answer:**
**D.** \(\frac{4 V}{\pi d^{2}}\)
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Simplify this solution Mind Expander
To find the height of a cylindrical fuel tank given its volume \( V \) and diameter \( d \), we start with the formula for the volume of a cylinder: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height. Since the diameter \( d \) is twice the radius (\( r = \frac{d}{2} \)), we can rewrite the formula using \( d \): \[ V = \pi \left(\frac{d}{2}\right)^2 h \] This simplifies to: \[ V = \pi \frac{d^2}{4} h \] To solve for the height \( h \), we rearrange the equation: \[ h = \frac{4V}{\pi d^2} \] Thus, the height of the tank in meters is represented by option \( \text{B} \): \[ \text{B. } \frac{4 V}{\pi d^{2}} \]
