Question

A piece of wire 13 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. (a) How much wire should be used for the square in order to maximize the total area? (b) How much wire should be used for the square in order to minimize the total area?

Answer 1

The function you seek to minimize is

()=3‾√4(3)2+(13−4)2

f

(

x

)

=

3

4

(

x

3

)

2

+

(

13

−

x

4

)

2

Then

′()=3‾√18−13−8=(3‾√18+18)−138

f

′

(

x

)

=

3

x

18

−

13

−

x

8

=

(

3

18

+

1

8

)

x

−

13

8

Note that ″()>0

f

″

(

x

)

>

0

so that the critical point at ′()=0

f

′

(

x

)

=

0

will be a minimum. The critical point is at

=1179+43‾√≈7.345m

x

=

117

9

+

4

3

≈

7.345

m

So that the amount used for the square will be 13−

13

−

x

, or

13−=524+33‾√≈5.655m