BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 2, Problem 22P

(a)

To determine

To find: The function that defines the total area of the four pens.

Expert Solution

Answer to Problem 22P

The function that defines the total area of four pens is P(x)=375x52x2 .

Explanation of Solution

Given:

A perimeter rectangular is 750ft . The rectangular area divided into four pens with fencing parallel to one side of the rectangle.

Formula used:

Perimeter of the rectangle is,

Perimeterofrectangle=2(length+breadth) (1)

Area of rectangle is,

Areaofrectangle=length×breadth (2)

Calculation:

Let the length of the one rectangular pen is x and the breadth of the rectangular pen is y.

Substitute x for length and y for breadth in equation (1),

Perimeterofonerectanglepen=2(length+breadth)=2(x+y)=2x+2y

Perimeter of four pens,

Perimeterof4rectanglepen=2x+2y+3x+6y750=5x+8y (3)

Now, get the value of y in terms of x from the equation (3),

750=5x+8y7505x=8yy=7505x8

Substitute x for length and y for breadth in equation (2) to find the area of the rectangular pens,

Areaof4rectangular pens=4lengthbreadth=4xy (4)

Substitute 7505x8 for y in equation (4),

Areaof4rectangular pens=4lengthbreadth=4x(7505x8)=375x52x2

Thus, the function that defines the total area of four pens is P(x)=375x52x2 .

(b)

To determine

To find: The largest possible area of the four pens.

Expert Solution

Answer to Problem 22P

The largest possible area of the four pens is 14062.5squaremeters .

Explanation of Solution

Given:

A perimeter rectangular is 750ft . The rectangular area divided into four pens with fencing parallel to one side of the rectangle.

Calculation:

Area is maximum when the derivative of the function P is zero.

From the part (a) the function that defines the total area of four pens is P(x)=375x52x2 .

Differentiate the function P(x)=375x52x2 with respect to x both sides,

dP(x)dx=d(375x52x2)dx=3755x

Now, take the differentiation equal to zero to find the value of x,

dP(x)dx=03755x=05x=375x=75

Substitute 75 for x in equation (3),

y=7505758=7503758=3758

Now, substitute 75 for x and 3758 for y in equation (4),

Areaof4rectangular pens=4×length×breadth=4×75×3758=14062.5squaremeters

Thus, the largest possible area of the four pens is 14062.5squaremeters .

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