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 3.1, Problem 76E

(a)

To determine

To evaluate: The function that models the cross section area of gutter in terms of x .

Expert Solution

Answer to Problem 76E

The function that models area of gutter is A(x)=30x2x2 .

Explanation of Solution

Given:

A rain gutter is formed by bending up the side of 30 in. wide rectangular metal sheet.

Calculation:

The width of the gutter is x .

The length of the corral is 302x .

The area of the gutter is,

A=Length×width=x(302x) (1)

From the equation (1)

A(x)=x×(302x)=30x2x2 (2)

The area of gutter in the term of x is,

A(x)=30x2x2 .

Thus, the function that models area of gutter is A(x)=30x2x2 .

(b)

To determine

To evaluate: The value of x to maximize the area of gutter.

Expert Solution

Answer to Problem 76E

The value of x to maximize the area of gutter is 7.5in .

Explanation of Solution

Given:

A rain gutter is formed by bending up the side of 30 in wide rectangular metal sheet.

Calculation:

The width of the gutter is x .

The length of the corral is 302x .

The area of the gutter is,

A=Length×width=x(302x) (1)

From the equation (1),

A(x)=x×(302x)=30x2x2 (2)

The area of gutter in the term of x is,

A(x)=2x2+30x (3)

The standard form of function,

f(x)=ax2+bx+c (4)

The maximum or minimum value of the function occurs at,

x=b2a (5)

If a>0 , then the minimum value is f(b2a) .

If a<0 , then the maximum value is f(b2a) .

From the equation (3) and (4),

a=2b=30

Substitute 2 for a and 30 for b in the equation (5),

x=302×(2)=7.5

The value of x is 7.5 in.

The function has maximum value,

a=2<0 .

Thus, the value of x to maximize the area of gutter is 7.5in .

(c)

To determine

To evaluate: The maximum cross-sectional area of the gutter.

Expert Solution

Answer to Problem 76E

The maximum cross sectional area of the gutter is 112.5in2 .

Explanation of Solution

Given:

A rain gutter is formed by bending up the side of 30 in. wide rectangular metal sheet.

Calculation:

The width of the gutter is x .

The length of the corral is 302x .

The area of the gutter is,

A=Length×width=x(302x) (1)

From the equation (1)

A(x)=x×(302x)=30x2x2 (2)

The area of gutter in the term of x is,

A(x)=2x2+30x (3)

The standard form of function,

f(x)=ax2+bx+c (4)

The maximum or minimum value of the function occurs at,

x=b2a (5)

If a>0 , then the minimum value is f(b2a) .

If a<0 , then the maximum value is f(b2a) .

From the equation (3) and (4),

a=2b=30

Substitute 2 for a and 30 for b in the equation (5),

x=302×(2)=7.5

The value of x is 7.5 in.

The function has maximum value,

a=2<0 .

Substitute 7.5 for x in the equation (3),

A(x)=2×(7.5)2+30×7.5=2×56.25+225=112.5

The maximum cross sectional area of the gutter is 112.5 square inch.

Thus, the maximum cross sectional area of the gutter is 112.5in2 .

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