Find the dimensions of a rectangle with a perimeter of 180 feet that has the maximum area.
The side lengths are feet.
(Use a comma to separate answers as needed.)

To maximize the area of a rectangle with a given perimeter, we can use the relationship between the dimensions. For a rectangle, the perimeter is given by the formula , where is the length and is the width. Given that feet, we have:

Now, the area of the rectangle is given by . Substituting into the area formula gives:

This is a quadratic function that opens downward (the coefficient of is negative), so its maximum value occurs at the vertex. The vertex can be found using the formula , where . Here, and :

Thus, feet and substituting back to find :

The side lengths are feet.