We are given a propane tank that is shaped like a cylinder with hemispherical ends. The diameter of the tank is 10 feet, and the total length of the tank including the hemispherical ends is 40 feet. We need to find the surface area of the tank to determine how many square feet need to be painted.

1. Problem Description

We are given a propane tank that is shaped like a cylinder with hemispherical ends. The diameter of the tank is 10 feet, and the total length of the tank including the hemispherical ends is 40 feet. We need to find the surface area of the tank to determine how many square feet need to be painted.

2. Solution Steps

First, find the radius of the tank:

$r = \frac{diameter}{2} = \frac{10}{2} = 5$ feet

The two hemispheres form a sphere. The surface area of a sphere is given by:

$A_{sphere} = 4 \pi r^2$

The length of the cylindrical part of the tank can be found by subtracting the diameter (which is the combined length of the two hemispheres) from the total length:

$L_{cylinder} = 40 - 10 = 30$ feet

The surface area of the cylindrical part of the tank (excluding the ends) is given by:

$A_{cylinder} = 2 \pi r L_{cylinder}$

The total surface area of the tank is the sum of the surface area of the sphere and the surface area of the cylinder:

$A_{total} = A_{sphere} + A_{cylinder} = 4 \pi r^2 + 2 \pi r L_{cylinder}$

Substitute the values of $r$ and $L_{cylinder}$ into the equation:

$A_{total} = 4 \pi (5^2) + 2 \pi (5)(30)$

$A_{total} = 4 \pi (25) + 2 \pi (150)$

$A_{total} = 100 \pi + 300 \pi$

$A_{total} = 400 \pi$

Approximating $\pi$ with 3.14:

$A_{total} = 400 \times 3.14 = 1256$ square feet

3. Final Answer

The surface area of the tank that needs to be painted is $400\pi$ square feet, which is approximately 1256 square feet.