Question

a septic tank has the shape shown to the right. how many gallons does it hold? (1 ft³ ≈ 7.48 gallons)
the tank holds gal. (round to the nearest gallon as needed)

Explanation:

Step1: Convert dimensions to feet

Since 1 foot = 12 inches, 6 ft 9 in = 6 + 9/12=6.75 ft and 3 ft 9 in = 3+9/12 = 3.75 ft.

Step2: Calculate the volume of the tank

The tank can be thought of as a rectangular - prism - like shape with semi - circular ends. The volume of the central rectangular part is \(V_{rect}=l\times w\times h\), and the two semi - circular ends together form a full circle. The radius of the circular part \(r=\frac{3.75}{2}\) ft and the length of the rectangular part \(l = 6.75\) ft, width \(w = 3.75\) ft. The volume of the tank \(V=V_{rect}+V_{circle - part}\). The volume of the rectangular part \(V_{rect}=6.75\times3.75\times3.75\) and the volume of the circular part \(V_{circle - part}=\pi r^{2}\times6.75=\pi(\frac{3.75}{2})^{2}\times6.75\).
\[

$$\begin{align*} V&=6.75\times3.75\times3.75+\pi(\frac{3.75}{2})^{2}\times6.75\\ &=6.75\times3.75\times3.75+\pi\times\frac{14.0625}{4}\times6.75\\ &=6.75\times3.75\times3.75 + 3.515625\pi\times6.75\\ &=6.75\times(3.75\times3.75+3.515625\pi)\\ &=6.75\times(14.0625 + 3.515625\times3.14)\\ &=6.75\times(14.0625+11.0446875)\\ &=6.75\times25.1071875\\ &\approx169.473516 \end{align*}$$

\]

Step3: Convert volume to gallons

Since \(1\ ft^{3}\approx7.48\) gallons, the volume in gallons \(V_{gal}=V\times7.48\).
\[V_{gal}=169.473516\times7.48\approx1267\]

Answer:

1267