Question

Perimeter of a rectangle is 24. Find the maximum area such a rectangle can have

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Answer 1

Let a and b be the length and width of the rectangle.

We are given 2(a+b) = 24

Or a+b = 12. From this we can write b = 12-a

We know area is the product of length and width = ab

Product of these two numbers = a . (12 - a)

Here the product is dependent on a. Let it be a function P(a).

P(a) = a. (12- a)

Now to find the maximum value of this, we will differentiate P(a) and equate it to zero.(We know that at maximum, the derivative is zero)

P’(a) = -2a + 12

P’(a) = 0

-2a + 12 = 0

Or a = 6

Let’s check by second derivative test whether we have a maximum or minimum at x =6.

P”(a) = -2

P”(6) = -2

Which is negative and according to the test we have a maximum at a = 6.

So, the numbers are 6 and 12- 6 = 6

Product of numbers = 6 $\times$ 6 = 36 = maximum area