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A right prism has a polygonal base with n sides. Each side has a length of 5 in. The altitude of the prism is 20 in.

If the base had n+1 sides, each still having a side of length 5 in, by how much, in square inches, would the lateral area of the prism increase by? Express your answer as a number only.

by 100 in

To find the lateral area of a prism, we need to calculate the perimeter of the base and then multiply it by the height of the prism.

The perimeter of a polygon with n sides, each side having a length of 5 in, can be calculated as n * 5 in.

So, the perimeter of the base of the original prism is 5n in.

The lateral area of the original prism is then the product of the base perimeter and the height, which is (5n in) * (20 in) = 100n in^2.

Now, let's consider the prism with a base of n+1 sides. The perimeter of this base can be calculated as (n+1) * 5 in.

The lateral area of the prism with the new base is then (5(n+1) in) * (20 in).

To find the increase in the lateral area, we need to subtract the lateral area of the original prism from the lateral area of the prism with the new base:

[(5(n+1) in) * (20 in)] - [100n in^2]

Simplifying the expression gives us:

100n in^2 + 100 in - 100n in^2

The terms with n^2 cancel out, leaving us with:

100 in

So, the lateral area of the prism would increase by 100 square inches.

To find the increase in the lateral area of the prism when the base has n+1 sides instead of n sides, we need to first calculate the lateral areas of both prisms.

The lateral area of a right prism is given by the formula:

Lateral Area = Perimeter of the base * Height

In our case, the perimeter of the base of the original prism with n sides is simply n * 5 inches since each side has a length of 5 inches. The height of the prism is given as 20 inches.

Therefore, the lateral area of the original prism is:

Lateral Area (Original) = (n * 5) * 20 = 100n square inches

Now, let's calculate the lateral area of the prism with n+1 sides.

The perimeter of the base of the prism with n+1 sides is (n+1) * 5 inches since each side still has a length of 5 inches. The height remains 20 inches.

Therefore, the lateral area of the prism with n+1 sides is:

Lateral Area (New) = ((n+1) * 5) * 20 = 100(n+1) square inches

To find the increase in the lateral area, we subtract the original lateral area from the new lateral area:

Increase in Lateral Area = Lateral Area (New) - Lateral Area (Original)

= 100(n+1) - 100n

Simplifying this expression, we get:

Increase in Lateral Area = 100n + 100 - 100n

The n terms cancel out:

Increase in Lateral Area = 100

Therefore, the increase in the lateral area of the prism, when the base has n+1 sides instead of n sides, is 100 square inches.