Question

A rectangular painting has a diagonal measure of 5 inches and an area of 12 square inches. Use the formula for the area of a rectangle and the Pythagorean Theorem to find the length and width of the painting.

Answer 1

We have 2 different equations with which we need to work to find the answer to this. First off, the area formula is A = L * W. Our area is 12, so 12 = L * W. Then we have the Pythagorean's theorem we have to worry about, which says that $5^2=l^2+w^2$. We are going to solve the area equation for L to get $l=\frac{12}{w}$. What we are going to do now is sub that into the other formula in place of L to get $5^2=(\frac{12}{w})^2+w^2$ and $25=\frac{144}{w^2}+w^2$. In order to solve for w, we need to multiply all those terms by w^2 to get rid of the denominator. That leaves us with this then: $25w^2=144+w^4$. We need to factor that now to solve for w. We will bring everything over to the one side of the equation and set it equal to 0. $w^4-25w^2+144=0$. We are going to use a u substitution here to make the factoring easier. We are going to let $u=w^2$ so $u^2=w^4$. Now we will factor $u^2-25u+144=0$. When we factor that we get (u - 16)(u - 9). By the Zero Product Property, either u - 16 = 0 or u - 9 = 0. u = 16 and u = 9. But remember that u = w^2, so what we really have, after we make the replacement, is $w^2=16$ so w = 4, and $w^2=9$ and w = 3. Now it just so happens that if the length is 4 and the width is 3, the product of those is 12 (the area), and if you remember, in a right triangle with a hypotenuse of 5, the side lengths will be 3 and 4. So your dimensions of the rectangle are that the length is 4 and the width is 3. There you go!