You are having a conference call with the CEO of a paper company. You have interpreted the number of trees cut down versus profi
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Let 
. Then the definite integral can be split up at 
so that
Now take the derivative. By the fundamental theorem of calculus, you have
![\displaystyle\frac{\mathrm dF}{\mathrm dx}=\frac{\mathrm d}{\mathrm dx}\left[\int_{t=c}^{t=5x}\frac{\mathrm dt}t-\int_{t=c}^{t=2x}\frac{\mathrm dt}t\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7B%5Cmathrm%20dF%7D%7B%5Cmathrm%20dx%7D%3D%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B%5Cint_%7Bt%3Dc%7D%5E%7Bt%3D5x%7D%5Cfrac%7B%5Cmathrm%20dt%7Dt-%5Cint_%7Bt%3Dc%7D%5E%7Bt%3D2x%7D%5Cfrac%7B%5Cmathrm%20dt%7Dt%5Cright%5D)
![=\displaystyle\frac1{5x}\dfrac{\mathrm d}{\mathrm dx}[5x]-\frac1{2x}\dfrac{\mathrm d}{\mathrm dx}[2x]](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac1%7B5x%7D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5B5x%5D-%5Cfrac1%7B2x%7D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5B2x%5D)
Then integrating with respect to
, we recover 
and find that
where
is an arbitrary constant.
Now take the derivative. By the fundamental theorem of calculus, you have
Then integrating with respect to
where
Answer:
Length = 150 yards
Width = 100 yards
Step-by-step explanation:
We want 600 yards of fencing that will result in the largest 2 fenced corrals, sharing a common border.
It will take the shape of a rectangle, with a dividing fence down the center.
Let W and L, Width and Length of the larger enclosure.
See attachment.
W= Area of the larger enclosure.
The perimeter is 2W + 2L.
The dividing fence is 1W
We know that we only have 600 yards of fence, so:
2W + 2L + 1W = 600 yards
Area = W x L
---
3W + 2L = 600 (yards)
2L = 600 -3W
L = (600-3W)/2
L = 300 -(3/2)W
---
Use this expression in the Area calculation:
Area = W x L
Area = W x (300 -(3/2)W)
Area = 300W -(3/2)W^2)
To find the maximum area, take the first derivative and set to zero to find the value of W that results in the greatest area.
Area' = 300 -2(3/2)W)
0 = 300 - 3W
3W = 300
W = 100 yards
Since 3W + 2L = 600
L = (600 - 3W)/2
L = (600 - 3(100))/2
L = 150 yards
Area = 150*100 = 15,000 yards^2
