Find the dimensions of the rectangle with area 289 square inches that has minimum perimeter, and then find the minimum perimeter.

To minimize the perimeter you should always have a square.

sqrt (289) = 17

The dimensions should be 17 X 17

To see, try starting at length 1, and gradually increase the length.

The height decreases at a faster rate than the length increases, up until you reach a square.

Or if you want to use algebra, Say the width is 17-x

Then the length is 289 / (17-x)

Now, this is bigger than 17+x, as shown here:

289 / (17-x) > 17+x

289 > 289 - x^2

which is true.

so the perimeter would be bigger than 2 * (17 - x + 17 + x) = 2 * (2 * 17) = 4 * 17

Again, the dimensions should be a square. 17 X 17.