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2 years ago

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A rectangular area of 36 f t2 is to be fenced off. Three sides will use fencing costing $1 per foot and the remaining side will

use fencing costing $3 per foot. Find the dimensions of the rectangle of least cost. Make sure to use a careful calculus argument, including the argument that the dimensions you find do in fact result in the least cost (i.e. minimizes the cost function).

1 answer:

OLga [1]2 years ago

7 0

Answer:

x = 8,49 ft

y = 4,24 ft

Step-by-step explanation:

Let x be the longer side of rectangle and y the shorter

Area of rectangle = 36 ft² 36 = x* y ⇒ y =36/x

Perimeter of rectangle:

P = 2x + 2y for convinience we will write it as P = ( 2x + y ) + y

C(x,y) = 1 * ( 2x + y ) + 3* y

The cost equation as function of x is:

C(x) = 2x + 36/x + 108/x

C(x) = 2x + 144/x

Taking derivatives on both sides of the equation

C´(x) = 2 - 144/x²

C´(x) = 0 2 - 144/x² = 0 ⇒ 2x² -144 = 0 ⇒ x² = 72

x = 8,49 ft y = 36/8.49 y = 4,24 ft

How can we be sure that value will give us a minimun

We get second derivative

C´(x) = 2 - 144/x² ⇒C´´(x) = 2x (144)/ x⁴

so C´´(x) > 0

condition for a minimum

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Answer:

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2. The auditorium can accommodate up to 180 spectators, hence

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Plot all solutions sets to each inequality and the common region is the solution set to the system of inequalities. This region is not empty, so the charity will reach its goal. For example, if they sell 50 students tickets and 125 adult tickets, they will raise $\$(15\cdot 50+25\cdot 125)=\$3,875$

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