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erastova [34]
3 years ago
7

A fence must be built to enclose a rectangular area of 45 comma 000 ftsquared. Fencing material costs $ 3 per foot for the two s

ides facing north and south and ​$6 per foot for the other two sides. Find the cost of the least expensive fence.
Mathematics
1 answer:
Tatiana [17]3 years ago
6 0

Answer:

150 feet by 300 feet.

Step-by-step explanation:

The fence is to enclose a rectangular area of 45,000 ft squared.

If the dimensions of the rectangle are x and y

Area of a rectangle = xy

  • xy=45000
  • x=\frac{45000}{y}

Perimeter of the Rectangle =2x+2y

Fencing material costs $ 3 per foot for the two sides facing north and south and ​$6 per foot for the other two sides.

  • Cost of Fencing, C=$(6*2x+3*2y)=$(12x+6y)

Substitute x=\frac{45000}{y} into the Cost to get C(y)

C=12x+6y

C(y)=12(\frac{45000}{y})+6y\\C(y)=\frac{540000+6y^2}{y}

The value at which the cost is least expensive is at the minimum point of C(y), when the derivative is zero.

C^{'}(y)=\dfrac{6y^2-540000}{y^2}

\dfrac{6y^2-540000}{y^2}=0\\6y^2-540000=0\\6y^2=540000\\y^2=\frac{540000}{6} =90000\\y=\sqrt{90000}=300

Recall,

x=\frac{45000}{y}=\frac{45000}{300}=150

Since x=150, y=300

The dimensions that will be least expensive to build is 150 feet by 300 feet.

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