Everything Maths Grade 12

7.6 CHAPTER 7. DIFFERENTIAL CALCULUS

However, we can use (7.25) to write w in terms of l:

w = 160 m− 2l (7.26)

Substitute (7.26) into (7.24) to get:

A = (160 m− 2l)l = 160 m− 2l^2 (7.27)

Step 2 : Differentiate

Since we are interestedin maximising the area,we differentiate (7.27) toget:

A�(l) = 160 m− 4l

Step 3 : Find the stationary point

To find the stationary point, we set A�(l) = 0 and solve for the valueof l that

maximises the area.

A�(l) = 160 m− 4l

0 = 160 m− 4l

∴ 4 l = 160 m

l =

160 m

4

l = 40 m

Substitute into (7.26) tosolve for the width.

w = 160 m− 2l

= 160 m− 2(40 m)

= 160 m− 80 m

= 80 m

Step 4 : Write the final answer

A width of 80 m and a length of 40 m will yield the maximalarea fenced off.

Exercise 7 - 5

1. The sum of two positive numbers is 20. One of the numbers is multiplied by the square of the
   other. Find the numbersthat make this product amaximum.


2. A wooden block is made as shown in the diagram. The ends are right-angled triangles having
   sides 3 x, 4 x and 5 x. The length of the blockis y. The total surface area of the block is 3 600 cm^2.