A backyard has an area of 72 square feet. which is the smallest and biggest possible perimeter of the backyard? .

Question

Please I need to know the process to this problem. Have many excercises on what is the smallest perimeter and biggest and don't know how to calculate to get answer. Please help! Thanks!

Answer 1

Is this for calculus?  If not, you can draw various rectangles with area 72 (1 × 72, 2 × 36, ... , 8 × 9, maybe even √72 × √72), and compare their perimeters (2·1 + 2·72, ...).  Your answer for the smallest should match your intuition.

 

If this is for calculus, you can verify your intuition by calculating the dimensions.  Call the length and width x and y.  Then the area xy = 72, and the perimeter 2x + 2y is to be optimized.  To make the perimeter a function of one variable, substitute y = 72/x from the area constraint in for y.  Then the perimeter is

 

    P(x) = 2x + 2(72/x)

 

Now take the derivative of P(x) and set it to 0.  (This is where the graph of P(x) levels off.)

 

    P'(x) = 0

 

Solve for x, plug that into P(x), and you'll get the smallest perimeter.  As for the largest perimeter, there is none.