A rectangular prism is 12 inches long and 4 inches wide. What is the prism's total surface area, in square inches, if the greatest possible straight-line distance between any two vertices is 13 inches?

No answer

To find the total surface area of a rectangular prism, we need to know the length, width, and height. We have the length (12 inches) and width (4 inches), but we don't have the height. However, we know that the greatest possible straight-line distance between any two vertices (which is the diagonal of the prism) is 13 inches.

We can use the Pythagorean theorem to find the height. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, the hypotenuse is the diagonal of the prism, and the other two sides are the length and width of the prism.

So, we have:

13^2 = 12^2 + 4^2 + h^2

Solving for h (height), we get:

h = sqrt(13^2 - 12^2 - 4^2) h = sqrt(169 - 144 - 16) h = sqrt(9) h = 3 inches

Now that we have the height, we can find the total surface area of the prism. The formula for the surface area of a rectangular prism is 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.

Substituting the given values, we get:

Surface area = 2(12)(4) + 2(12)(3) + 2(4)(3) Surface area = 96 + 72 + 24 Surface area = 192 square inches

So, the total surface area of the prism is 192 square inches.