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An apple farm yields an average of 38 bushels of apples per tree when 22 trees are planted on an...

Question:

An apple farm yields an average of 38 bushels of apples per tree when 22 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?

Quadratic Equations

Two types of connections are possible between coefficients and zeroes of the quadratic equation. The addition of the zeroes is given by {eq}- \dfrac{b}{a} {/eq} where b is the coefficient of the second term and the multiplication of the zeroes is written as {eq}\dfrac{c}{a} {/eq}.

Answer and Explanation:

Let the number of additional trees planted be x.

Total number of trees planted is {eq}\left( {22 + x} \right) {/eq},

Average bushels of apple per tree is {eq}\left( {38 - x} \right) {/eq}.

Yield Y is equal to the total number of trees planted times the average bushels of apple per tree,

Therefore,

{eq}\begin{align*} Y &= \left( {22 + x} \right) \times \left( {38 - x} \right)\\ &= 1064 + 10x - {x^2}\\ &= - {x^2} + 10x + 1064 \end{align*} {/eq}

In order to maximize the yield, take the derivative with respect to x,

{eq}\dfrac{{dY}}{{dx}} = - 2x + 10 {/eq}

Equating it zero to find the maxima,

{eq}\begin{align*} \dfrac{{dY}}{{dx}} &= 0\\ - 2x + 10 &= 0\\ x &= 5 \end{align*} {/eq}

Hence, number of trees planted are

{eq}22 + x = 22 + 5 = 27 {/eq}

In order to get the highest yield, 27 trees should be planted on an acre.


Learn more about this topic:

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How to Use the Quadratic Formula to Solve a Quadratic Equation

from Math 101: College Algebra

Chapter 4 / Lesson 10
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