Answer
The volume of the trough is 141.5 gallons
SOLUTION
Problem Statement
The question gives us a trough with semi-circular bases with a diameter of 17 inches and a length of 6 feet. We are asked to calculate the volume of the trough in gallons.
Method
- Before we begin solving, we should make all the units uniform.
- 12 inches make 1 foot. Thus, we can say that:
[tex]17\text{inches}=\frac{17}{12}ft[/tex]- To find the volume of the trough, we can simply find the volume of a cylinder that was cut in half. We simply find the volume of the full cylinder and then divide the result by 2.
- When the volume has been calculated, we should convert from cubic feet to gallons. We use the following conversion:
[tex]1cubic\text{ foot}\to7.48052\text{ gallons}[/tex]- Thus, we can write solve the question using the following steps:
Step 1: Find the full volume of the cylinder.
Step 2: Find the volume of the trough.
Step 3: Convert the result from Step 2 from cubic feet to gallons
Implementation
Step 1: Find the full volume of the cylinder.
[tex]\begin{gathered} \text{Volume of a cylinder }=\pi\times(\frac{d}{2})^2\times h \\ \text{where,} \\ d=diameter \\ h=\text{height or length of the cylinder} \\ \\ V=\pi\times(\frac{17}{12})^2\times6 \\ \\ V=37.83ft^3 \end{gathered}[/tex]Step 2: Find the volume of the trough.
[tex]\begin{gathered} \text{Volume of trough}=\frac{\text{Volume of cylinder}}{\text{2}} \\ \\ \therefore\text{Volume of trough}=\frac{37.83}{2}=18.915ft^3 \end{gathered}[/tex]Step 3: Convert the result from Step 2 from cubic feet to gallons
[tex]\begin{gathered} \text{ Using the following conversion:} \\ 1ft^3\to7.48052\text{gallons} \\ \therefore18.915ft^3\to\frac{18.915ft^3}{1ft^3}\times7.48052gallons \\ \\ =141.494\text{gallons}\approx141.5\text{gallons (To the nearest Tenth)} \end{gathered}[/tex]Final Answer
The volume of the trough is 141.5 gallons
