The surface area of a rectangular prism is 729 square meters. What is the surface area of a similar prism that has edge lengths that are larger by a scale factor of 3

area changes by the square of the linear ratio

so, the area becomes 729*3^2

To find the surface area of a similar rectangular prism, we need to understand that the surface area is directly proportional to the square of the scale factor.

Given that the surface area of the original prism is 729 square meters, let's denote it as A1. The scale factor, denoted as k, relates the dimensions (length, width, and height) of the original prism to the dimensions of the similar prism.

We can use the formula for the surface area of a rectangular prism: A = 2lw + 2lh + 2wh, where l represents the length, w represents the width, and h represents the height.

In this case, the scale factor is 3. So, the edge lengths of the similar prism are larger by a factor of 3.

Let's call the dimensions of the original prism l1, w1, and h1, and the dimensions of the similar prism l2, w2, and h2.

According to the given information, the surface area of the original prism is 729 square meters, so we can set up the equation A1 = 2(l1w1 + l1h1 + w1h1) = 729.

Now, let's find the dimensions of the similar prism. Since the edge lengths are larger by a scale factor of 3, the dimensions of the similar prism are l2 = 3l1, w2 = 3w1, and h2 = 3h1.

Using these new dimensions, we can find the surface area of the similar prism, which we'll denote as A2.

A2 = 2(l2w2 + l2h2 + w2h2)
= 2[(3l1)(3w1) + (3l1)(3h1) + (3w1)(3h1)]
= 2(9l1w1 + 9l1h1 + 9w1h1)
= 18(l1w1 + l1h1 + w1h1)

Since l1w1 + l1h1 + w1h1 represents the surface area of the original prism (A1), we can substitute the value of A1 into the expression.

A2 = 18(A1)
= 18(729)
= 13,122

Therefore, the surface area of the similar prism with edge lengths that are larger by a scale factor of 3 is 13,122 square meters.