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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Revenue The owner of an orange grove must decide when to pick one variety of oranges. She can sell them for $24 a bushel if she sells them now, with each tree yielding an average of 5 bushels. The yield increases by half a bushel per week for the next 5 weeks, but the price per bushel decreases by $1.50 per bushel each week. When should the oranges be picked for maximum return?

To determine

To calculate: The time for which oranges should be picked for maximum return.

Explanation

Given Information:

If sold now for $24 a bushel, each tree yields an average of 5 bushels. The yield increases at the rate of half a bushel per week for the next 5 weeks and the price decreases at the rate of $1.50 per bushel each week.

Formula used:

To find the maximum value, calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to 0. the relevant value is the stationary value that satisfies the provided conditions.

If the second derivative of the provided function is less than zero, then substituting the value of the independent variable will give the maximum value of the equation.

The product rule is used when the function is a product of two differentiable terms. If p(x)=f(x)g(x), then the derivative is

p'(x)=f'(x)g(x)+f(x)g'(x)

Calculation:

Consider the provided statement,

If sold now for $24 a bushel, each tree yields an average of 5 bushels.

The yield increases at the rate of half a bushel per week for the next 5 weeks and the price decreases at the rate of $1.50 per bushel each week.

Let x is the time to pick oranges. The return is

R=(241.5x)(5+0.5x)

To find the stationary value for R=(241.5x)(5+0.5x), differentiate the equation with respect to the independent variable x to get

R'=(1.5)(5+0.5x)+0.5(241.5x)=7

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