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A chemical company is researching the effect of a new pesticide on crop yields. Preliminary results show that the extra yield per hectare is given by the expression \({900p \over 2 + p}\), where p is the mass of pesticide, in kilograms. The extra yield is also measured in kilograms.

 

Substitute several values for p and determine what seems to be the greatest extra yield possible.

 

Here's what I perceived from the question:

\({900p \over 2 + p}\) = \({extra \space yield \over hectare}\) so that means 900 is the number of maximum yields...?

 

When I tried substituting several values, I would still be under 900... so it could possibly go up to ∞

 

However, in the answers in the back of the text said that 900 kg is the greatest... I don't understand how they got 900?

Guest Apr 1, 2018
 #1
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Do we have to specifically substitute values to find the answer, or can we use a different method?

Mathhemathh  Apr 1, 2018
 #2
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It's what they said, but I guess we can use a different method... the thing is, I don't know any other method :')

Guest Apr 1, 2018
 #3
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So...

 

We are trying to find the maximum value of the function \(Y(p)={900p\over2+p}\) within the domain \(x\ge0\). This is the graph of a hyperbola, and it is positive. So, in order to find the maximum value of the function, we have to get its horizontal asymptote. An asymptote is a line that a graph approaches but never touches.  The rules are as follows:

 

1. If the numerator's degree is less than the denominator's degree, then the horizontal asymptote is y = 0.

2. If the numerator's degree is equal to the denominator's degree, the the horizontal asymptote is the quotient of the coefficients of the highest degree terms.

3. If the numerator's degree is greater than the denominator's degree, then it does not have a horizontal asymptote.

 

The degree of 900p is 1, and the degree of 2+p is 1. Therefore, the horizontal asymptote is \(y={900\over1}\), or just \(y=900\). This means that the maximum value of the function is 900.

 

What this actually means is that the function approaches 900 as you go farther along the x-axis, or, as you increase the amount of pesticide. So, theoretically, if you had infinite pesticide, you would have 900kg of extra yield. But... infinity is tricky. Just trust me, the answer is 900.

Mathhemathh  Apr 1, 2018

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