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Chapter 9: Problem 40

A strawberry farmer will receive \(\$ 30\) per bushel of strawberries during the first week of harvesting. Each week after that, the value will drop \(\$ 0.80\) per bushel. The farmer estimates that there are approximately 120 bushels of strawberries in the fields, and that the crop is increasing at a rate of four bushels per week. When should the farmer harvest the strawberries to maximize their value? How many bushels of strawberries will yield the maximum value? What is the maximum value of the strawberries?

Short Answer

Expert verified
The farmer should harvest the strawberries after 20 weeks for maximum value. At this time, the farmer will have 200 bushels of strawberries that will yield a maximum value of $7200.

Step by step solution

01

Developing the equation

Let's symbolize the number of weeks of growing before harvesting starts as \(x\) weeks. Hence, the total number of bushels available for harvest after \(x\) weeks will be (120 + 4x) bushels. Now, the rate of decrease in the value per bushel is \(\$0.80\) per week. Its initial price during the first week of the harvest is \(\$30.00\), so after \(x\) weeks its price will be \(\$30 - 0.8x\) Therefore, the total value \(V\) of the strawberries harvested after \(x\) weeks can be represented by the equation: \(V = (120 + 4x)(30 - 0.8x)\)
02

Maximisation of the Value function

To maximize this value, we need to handle it as an optimization problem. To achieve this, we need to find the first derivative of the function \(V\), set it to zero, and solve for \(x\). \(V\) is a function of \(x\) and is given as \(V = (120 + 4x)(30 - 0.8x) = 3600 + 32x - 0.8x^2\). The derivative \(V'\) of this function is \(V' = 32 - 1.6x\) Setting this equal to zero and solving for \(x\) gives \(x = 32/1.6 = 20\) weeks.
03

Validation

To ensure that this value maximizes the function and is not a minimum or inflection point, check the second derivative. It is \(V'' = -1.6\) which is always negative indicating that \(x = 20\) guarantees a maximum for the problem. Now, substituting \(x = 20\) into \(V\), Catching the strawberries at \(x = 20\) gives a maximum value \(V = (120+4*20)(30-0.8*20) = \$7200\). Also, the total number of bushels of crop harvested for maximum reward is \(120 + 4*20 = 200\) bushels.

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Most popular questions from this chapter

The gross domestic product (GDP) of the United States for 2001 through 2005 is modeled by \(G=0.0026 x^{2}-7.246 x+14,597.85\) where \(G\) is the GDP (in billions of dollars) and \(x\) is the capital outlay (in billions of dollars). Use differentials to approximate the change in the GDP when the capital outlays change from \(\$ 2100\) billion to \(\$ 2300\) billion.

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