To solve this problem, we need the following formulas:
• The volume of a cylinder:
[tex]V_c=\pi\cdot l\cdot r^2=\frac{1}{4}\cdot\pi\cdot l\cdot d^2,[/tex]where l is the length, r the radius and d the diameter.
• The volume of a sphere:
[tex]V_s=\frac{4}{3}\cdot\pi\cdot r^3=\frac{1}{6}\cdot\pi\cdot d^3,[/tex]where r is the radius and d is the diameter.
Now, from the figure, we see that the septic tank is given by:
1) A cylinder in the middle, of dimensions:
• length, l = 6' 3'' = 6.25ft,
,• diameter, d = 4' 9'' = 4.75ft,
Replacing these values in the formula above, the volume of the cylinder is:
[tex]V_c=\frac{1}{4}\cdot\pi\cdot(6.25ft)\cdot(4.75ft)^2\cong110.753ft^3\text{.}[/tex]2) Two semi-spheres in the extremes, of:
• diameter, d = 4' 9'' = 4.75ft.
Combining the two parts, we have a complete sphere.
Replacing these values in the formula above, the volume of the sphere is:
[tex]V_s=\frac{1}{6}\cdot\pi\cdot(4.75ft)^3\cong56.115ft^3\text{.}[/tex]The total volume of the tank is the sum of the volumes of its constituents:
[tex]V=V_c+V_s\cong110.753ft^3+56.115ft^3\cong166.868ft^3\text{.}[/tex]Now, we know that 1 ft3 ≅ 7.48 gal, so the volume in gallons is:
[tex]V\cong166.868ft^3\cdot\frac{7.48gal}{1ft^3}\cong1248.176gal\cong1248gal.[/tex]Answer
The volume of the septic tank to the nearest gallon is 1248 gallons.
