A farmer finds that if she plants fifty trees per acre,
each tree will yield sixty-five bushels of fruit. She estimates that for each
additional tree planted per acre, the yield for each tree will decrease by three bushels.
How many trees should she plant per acre to maximize her harvest?
The total harvest per acre will be equal to the number of trees times
the number of bushels per tree. If we let X equal the number of additional trees,
we can say H of X, the harvest per acre, would be equal to the total number of trees,
which would be fifty plus X, times the number of bushels per tree. The bushels per tree
start at sixty-five, but it decreases by three bushels for every extra tree.
So, the total number of bushels per tree would be sixty-five minus three X. Therefore,
we have fifty plus X as the total number of trees
and sixty-five minus three X as the total number of bushels per tree.
Let's go ahead and multiply this out: H of X is equal to fifty times sixty-five, which is
three thousand two hundred fifty, and then minus fifty times three X,
which is minus one hundred fifty X, plus sixty-five X, minus three X squared.
Let's go ahead and combine like terms and write the terms in descending order. We'd
have H of X equals negative three X squared. Combining the X terms gives
us minus eighty-five X, then plus three thousand two hundred fifty.
So, our goal here is to maximize this function. For the next step, we'll determine the critical
numbers, which are where the first derivative is equal to zero or undefined.
So, H prime of X is equal to negative six X minus eighty-five. This would
equal zero. Notice how the derivative function is a linear function, which is
never undefined. To find the critical number, we'll set this equal to zero and solve for X.
So we'll add eighty-five to both sides. Negative six X equals eighty-five. Divide both sides by
negative six, and we have X is approximately negative fourteen point seventeen. Since
X is a number of additional trees, we're going to round to the nearest
whole number. So we'll use a critical number of X equals negative fourteen.
But we should still verify that the function is maximized and not minimized at this value.
We should recognize that the harvest function here is a quadratic function
with a negative leading coefficient. So we have a parabola that opens down or a
concave down function. Therefore, this X value would maximize the function.
We can also use the first or second derivative test.
Let's use the second derivative test to verify the concavity.
Here's the first derivative function. Notice how the second derivative function H double prime of
X would be equal to negative six. Because the second derivative is always negative,
the function is always concave down. This is good news because
that means at our critical number we have a maximum function value.
Therefore, the function is maximized at approximately X equals negative fourteen. But
we might be thinking to ourselves how can X be negative if X is the number of additional trees?
Well, the total number of trees is fifty plus X. So the ideal number
of trees would be fifty plus negative fourteen or thirty-six trees per acre.
So, fifty trees is too many to maximize the harvest. The harvest is maximized with
fifty plus X trees, which is fifty plus negative fourteen or thirty-six trees.
I hope you found this helpful!