Calculus question

Question

a florida citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 400 oranges. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. Express the grower's total yield as a function of the number of additional trees planted, draw the graph and estimate the total number of trees the grower should plant to maximize yield.

Answer 1

In many word problems, the hardest part of solving is writing the equation; in order to do it you must go over the question until you fully understand it. Drawing a diagram and labeling quantities can often be very helpful, depending on the nature of the problem. In this case, we are told to "express the grower's total yield as a function of the number of additional trees planted," that is, the number of trees past 60. Let y be the total yield, and x be the number of trees past the first 60. Thus, the total number of trees is (60 + x), and the average yield per tree is y/(60 + x). Using these expressions and the information given, we can write the following equations:

 

Average yield: y/(60 + x) = 400 - 4x

Total yield: y = (400 - 4x)(60 + x)

                        = 24000 + 160x - 4x²

In standard form: y = -4x² + 160x + 24000

 

 

If we have a graphing calculator or other utility available, at this point we can simply graph the equation, noticing that at the maximum, x appears to be around 20, and y is about 25,600. If we go this route, we can skip the next paragraph and proceed to the answer. Otherwise, we need to consider how to graph this equation by hand and find its maximum.

 

Notice that the graph will be a parabola with open end downwards (there is a negative coefficient on x²). Therefore, the maximum point should occur at the vertex. Fortunately, we learn the formula for finding the vertex of a parabola in algebra 1: x = -b/2a. In this case, x = -160/2(-4) = -160/(-8) = 20. Plugging this value back into our equation, we find that y(20) = 25,600, which will be our maximum yield. To graph the rest of the curve, simply plug in x-values on either side of the vertex (18, 19, 21, 22,...) find the corresponding y-values, plot the points and connect them.

 

Now that we have found the maximum yield, are we finished? No! It is tempting to stop here, but we have to make sure we've answered the question that was asked. In this case, the original question was to estimate the total number of trees to plant for maximum yield. Remember that we defined x to be the number of trees beyond the first 60. Therefore, we must add 60 to x to find the answer. Now, 20 + 60 = 80, so our final answer is "In order to maximize yield, the grower should plant 80 trees."

Answer 2

YIELD = (NUMBER OF TREES)*(NUMBER OF ORANGES PER TREE)

 

Let's assume NUMBER OF TREES = 60 + x, where x is the number of additional trees above 60

 

The NUMBER OF ORANGES PER TREE will = (400-4x).  Hence:

 

YIELD = (60+x)*(400-4x) = 24000-240x+400x-4x² = -4x² + 160x + 24,000

 

To find the maximum YIELD, take the derivative of YIELD wrt x, set it to zero, and solve for x:

 

d(YIELD)/dx = -8x + 160

 

0 = -8x +160

 

8x = 160

 

x = 20

 

The grower should grow 60 + 20 = 80 trees to maximize yield.