Question

A farmer plans to build a 250 square feet rectangular chicken pen adjacent toher chicken coop. The coop wall will form one side of the fence but she needs tofence the otherthreesides. The fencing costs$10 per linear foot. Her math-whizdaughter determines that if the side of pen parallel to the coop wall isxft long,then the cost in dollars of fencing will be given by the functionC(x) = 10x+5000x.(a) [4 points] Use a diagram and the information provided above toshowthatthe cost function isC(x) = 10x+5000x

Answer 1

The perimeter of a rectangle is the sum of twice the width and twice the length

The function that gives the cost of fencing is given from the graph by the following function;

$C(x) = 10 \cdot x + \dfrac{5,000}{x}$

Please find attached the required diagram

The given parameter are;

The area of the square rectangle = 250 m²

The length of the side of the pen wall parallel to the coop = x

The cost of fencing per foot, c = $10

The function for the cost in dollars of fencing, C(x) = 10x + 5000

Solution:

Let y represent the length of the other side, we have;

x × y = 250

Perimeter, P = x + 2·y

$y = \dfrac{250}{x}$

The length of the fencing required, P = x + 2·y

$P = x + 2 \times \dfrac{250}{x} = x + \dfrac{500}{x}$

$P = x + \dfrac{500}{x}$

The total cost of the fencing, C(x) = c × P

Therefore;

$C(x) = 10 \times \left(x + \dfrac{500}{x} \right) = 10 \cdot x + \dfrac{5,000}{x}$

$C(x) = 10 \cdot x + \dfrac{5,000}{x}$

Learn more about functions here:

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