a pyramid has a height of 5 inches and a surface area of 90 inches squared. A similar pyramid has a height of 10 inches what is it's surface area?

the surface area of similar solids is proportional to the square of their corresponding sides

Area / 90 = 10^2/5^2 = 100/25 = 4
Area = 90(4) = 360 inches^2

To find the surface area of the similar pyramid with a height of 10 inches, we can use the concept of similarity between two pyramids.

When two pyramids are similar, their corresponding sides are proportional. This means that if the height of the first pyramid doubles, the corresponding sides of the second pyramid will also double. Since the height of the first pyramid is 5 inches and the height of the second pyramid is 10 inches (double of the first), we can infer that the corresponding sides of the second pyramid are also double.

Since the surface area of a pyramid is calculated by adding the area of its base to the sum of the areas of its lateral faces, and the lateral faces are triangles, we can say that the surface area is directly proportional to the square of the corresponding side lengths.

Let's denote the surface area of the first pyramid as A1, and the surface area of the second pyramid as A2. Also, let's denote the corresponding side lengths of the first pyramid as s1, and the corresponding side lengths of the second pyramid as s2.

We have the following proportional relationship:
(A2 / A1) = (s2^2 / s1^2)

Since we know the surface area of the first pyramid is 90 inches squared, we can substitute the values into the equation:
(A2 / 90) = (s2^2 / s1^2)

We've established that the corresponding side lengths of the second pyramid are twice those of the first pyramid, so we can write:
(A2 / 90) = (4^2 / 2^2)

Now we can solve for A2:
A2 = (90) * (4^2 / 2^2)
A2 = (90) * (16 / 4)
A2 = 90 * 4
A2 = 360

Therefore, the surface area of the similar pyramid with a height of 10 inches is 360 inches squared.