Question

A farmer has $1500 available to build an E-shaped fence along a straight river so as to create two identical pastures. The materials for the side parallel to the river cost $6 per foot, and the materials for the three sections perpendicular to the river cost $5 per foot. Find the dimensions for which the total area is as large as possible.

Answer 1

Answer: The largest dimensions that are possible would be 50 foot by 125 foot.

Step-by-step explanation:

Since we have given that

Amount a farmer has available = $1500

Let x be the side perpendicular to the river.

Let y be the side parallel to the river.

Number of section perpendicular to the river = 3

cost of material for the side parallel to the river = $6 per foot

Cost of material for the side perpendicular to the river = $5 per foot

So, total cost becomes

$Cost=6y+5(3x)=1500\\\\Cost=6y+15x=1500\\\\y=\dfrac{1500-15x}{6}}$

Area would be

$A=x\times y\\\\A=x(\dfrac{1500-15x}{6})\\\\A=\dfrac{1}{6}(1500x-15x^2)\\\\A'=\dfrac{1}{6}(1500-30x)$

Now, put A' = 0 to get the critical points.

So, it becomes,

$\dfrac{1}{6}(1500-30x)=0\\\\1500-30x=0\\\\1500=30x\\\\x=\dfrac{1500}{30}\\\\x=50$

$A''=\dfrac{-30}{6}=-5$

so, at x= 50 it will give maximum dimensions.

$y=\dfrac{1500-15x}{6}=\dfrac{1500-15\times 50}{6}=\dfrac{750}{6}=125$

So, the largest dimensions that are possible would be 50 foot by 125 foot.