Question

A septic tank has the shape shown to the right. How many gallons does it hold? (1 ft3≈ 7.48 gal.) Round to the nearest tenth

Answer 1

To solve this problem, we need the following formulas:

• The volume of a cylinder:

`V_c=\pi\cdot l\cdot r^2=(1)/(4)\cdot\pi\cdot l\cdot d^2,`

where l is the length, r the radius and d the diameter.

• The volume of a sphere:

`V_s=(4)/(3)\cdot\pi\cdot r^3=(1)/(6)\cdot\pi\cdot d^3,`

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:

`V_c=(1)/(4)\cdot\pi\cdot(6.25ft)\cdot(4.75ft)^2\cong110.753ft^3\text{.}`

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:

`V_s=(1)/(6)\cdot\pi\cdot(4.75ft)^3\cong56.115ft^3\text{.}`

The total volume of the tank is the sum of the volumes of its constituents:

`V=V_c+V_s\cong110.753ft^3+56.115ft^3\cong166.868ft^3\text{.}`

Now, we know that 1 ft3 ≅ 7.48 gal, so the volume in gallons is:

`V\cong166.868ft^3\cdot(7.48gal)/(1ft^3)\cong1248.176gal\cong1248gal.`

Answer

The volume of the septic tank to the nearest gallon is 1248 gallons.