# The number of vines that should be planted to maximize the grape production.

### Precalculus: Mathematics for Calcu...

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

### Precalculus: Mathematics for Calcu...

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

#### Solutions

Chapter 3.1, Problem 70E
To determine

## To evaluate: The number of vines that should be planted to maximize the grape production.

Expert Solution

The number of vines that should be planted to maximize the grape production is 150 and the maximum grape production per acre is 7225pounds .

### Explanation of Solution

Given:

The number of grape produced per acre is given by the function A(n)=(700+n)(100.01n) , where n is the number of additional vines planted.

Calculation:

The function of the number of grape produced per acre is given by

A(n)=(700+n)(100.01n)=700(100.01n)+n(100.01n)=70007n+10n0.01n2=0.01n2+3n+7000 (1)

The standard form of function,

f(n)=an2+bn+c (2)

The maximum or minimum value of the function occurs at,

n=b2a (3)

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

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

From the equation (1) and (2),

a=0.01b=3

Substitute 0.01 for a and 3 for b in the equation (3),

n=32×0.01=30.02=150

The number of vines per acre is 150 .

The function has maximum value,

a=9<0 .

Substitute 150 for n in the equation (1),

A(n)=0.01×(150)2+3×150+7000=0.01×22500+450+7000=225+450+7000=7225

The maximum grape production is 7225pounds .

Thus, the number of vines that should be planted to maximize the grape production is 150 and the maximum grape production per acre is 7225pounds .

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