Yield/acre using fertilizer

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Paul Kersey

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Hi, I'm taking a business analysis class and we just started doing derivatives. I haven't had any calculus classes before but I'm doing all right. Not sure about this word problem however.

The yield, Y, of an apple orchard (measured in bushels of apples per acre) is a function of the amount "x" of fertilizer in pounds used per acre. Suppose:

Y= f(x) = 320+140x-10x²

(a) What is the yield if 5 pounds of fertilizer are used per acre?

To solve this I simply plugged 5 into the x values. Y=320+700-250 simplified to Y=770. That is to say the yield is 770 bushels of apples per acre if five pounds of fertilizer are used per acre.

(b) Find f'(5). Give units with your answer and interpret it in terms of apples and fertilizer.

To solve this I took the derivative of 320+140x-10x², plugging 5 in for x.
f'(5)=140-10x(2)
f'(5)=140-20x
f'(5)=140-20(5)
f'(5)= 40

Ok, so I have f'(5)=40. I am not sure what this number means. I do not know how it relates in terms of apples and fertilizer. I hope I did everything right so far. Thanks for any help.
 
The derivative is the slope at a point. At x=5, you get 40.

This means when 5 pounds of fertilizer are used, the bushels of apples is increasing by 40.

Notice if you set the derivative to 0 and solve for x you get x=7. This means they get the most bushels when they use 7 pounds of fertilizer. The yield increases up to 7, then the yield begins to drop after that.

\(\displaystyle \displaystyle \frac{dy}{dx}=\frac{\Delta \text{bushels}}{\Delta \text{lbs of fertilizer}}\)

At the instant 7 pounds are used, the yield is not changing. It has peaked. It then begins to go down. Too much fertilizer can ruin the crop.
 
(b) Find f'(5). Give units with your answer and interpret it in terms of apples and fertilizer.

To solve this I took the derivative of 320+140x-10x², plugging 5 in for x.
f'(5)=140-10x(2)
f'(5)=140-20x
f'(5)=140-20(5)
f'(5)= 40

Ok, so I have f'(5)=40.
Click to expand...

To elucidate on Galactus' comments, forget the calculus for a minute. Draw a graph of the function. The graph of f(x) is a parabola that opens downward. To the left of x=7, the slope of the parabola is always positive, meaning the function (and the yield) are increasing. At x = 7, the slope is zero, and the function has reached its maximum value (highest yield). To the right of x=7, the slope of the parabola is negative, meaning the function (and yield) are decreasing.

Edited to correct brain-dead error. :)
 
Last edited:
wjm11 said:
No. f'(5)= -40. The negative sign means the slope is negative and that f(x) is decreasing. In this case, that means the yield is decreasing.

To elucidate on Galactus' comments, forget the calculus for a minute. Draw a graph of the function. The graph of f(x) is a parabola that opens downward. To the left of x=7, the slope of the parabola is always positive, meaning the function (and the yield) are increasing. At x = 7, the slope is zero, and the function has reached its maximum value (highest yield). To the right of x=7, the slope of the parabola is negative, meaning the function (and yield) are decreasing.
Click to expand...

No, wjm11, f'(5) = 40. To the left of x=7 we have an increasing function so the value of the derivative to the left of x=7 would be positive.
 
srmichael said:
No, wjm11, f'(5) = 40. To the left of x=7 we have an increasing function so the value of the derivative to the left of x=7 would be positive.
Click to expand...

Dang. Thanks Srmichael. Just woke up; should have had my coffee first this morning. Of course it's positive; it's to the left of x = 7!
 
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