A rectangular ceiling tile has a perimeter of 64 inches. Its area is 247 square inches. What are the dimensions of the tile? inches by inches

To find the dimensions of the rectangular tile, we need to solve for length (L) and width (W). We know that the perimeter (P) of a rectangle is given by the formula P = 2L + 2W and the area (A) is given by the formula A = LW.

Given: P = 64 inches A = 247 square inches

We can set up the following system of equations:

2L + 2W = 64 LW = 247

To solve this system, we can start by simplifying the perimeter equation:

L + W = 32

Next, we can express one variable in terms of the other using the simplified perimeter equation. Let's solve for L:

L = 32 - W

Now we can substitute L in the area equation:

(32 - W)W = 247 32W - W^2 = 247 W^2 - 32W + 247 = 0

This is a quadratic equation in the form of ax^2 + bx + c = 0. We can solve for W using the quadratic formula, W = [-b ± sqrt(b^2 - 4ac)] / 2a:

W = [32 ± sqrt((32)^2 - 41247)] / 2*1 W = [32 ± sqrt(1024 - 988)] / 2 W = [32 ± sqrt(36)] / 2 W = [32 ± 6] / 2 W = 18 or W = 13

Substitute W = 18 and W = 13 into the equation L = 32 - W to find the corresponding lengths:

When W = 18, L = 32 - 18 = 14 When W = 13, L = 32 - 13 = 19

So, the dimensions of the tile are 14 inches by 18 inches or 19 inches by 13 inches.