Question

A small dog kennel with 8 individual rectangular holding pens of equal size is to be
constructed using 144 ft of chain link fencing material. One side of the kennel is to
be placed against a building and requires no fencing, as shown in the figure below.

a. Find the dimensions (for each holding
pen) that produce a maximum area for
each pen.
b. What is that maximum area for each
holding pen?

Answer 1

Answer:

(a)

• length = 9 ft
• width = 8 ft

(b) area = 72 ft²

Step-by-step explanation:

Let:

• length of each kennel = x ft
• width of each kennel = y ft

The total length of chain = 8x + 9y

$8x+9y=144$

$8x=144-9y$

$x=18-\frac{9}{8} y$ ... [1]

Area of each kennel (A) = length × width

$A=x\times y$

(substitute x with [1])

$A=(18-\frac{9}{8} y)\times y$

$A=18y-\frac{9}{8} y^2$

When A is maximum → the 1st derivative of A = 0

$\displaystyle \frac{dA}{dy} =0$

$\displaystyle \frac{d(18y-\frac{9}{8}y^2) }{dy} =0$

$18(1)y^{1-1}-\frac{9}{8} (2)y^{2-1}=0$

$18-\frac{9}{4} y=0$

$y=18\div\frac{9}{4}$

$\bf y=8\ ft$

[1]

$x=18-\frac{9}{8} y$

$x=18-\frac{9}{8} (8)$

$\bf x=9\ ft$

(a)

• length = 9 ft
• width = 8 ft

(b)

Area = length × width

        = 9 × 8

        = 72 ft²