1. A company has determined that its monthly profit ( P ) depends on the amount
ID: 2865024 • Letter: 1
Question
1. A company has determined that its monthly profit (P) depends on the amount of money spent on advertising (x). The relationship is given by the equation P(x) = (4x)/(3x^2+27), where P and x are measured in thousands of dollars.
a). What amount should the company spend on advertising to assure maximum profit?
b). What is the maximum profit?
2. A farmer wants to make three identical rectangular enclosures next to his barn, as shown below. If he has only 1500 ft. of fencing and if the side along the barn needs no fencing, what is the largest total area that can be enclosed in this way?
B A R N
3. By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 12 inches long and 12 inches wide, find the dimensions of the box that will yield the maximum volume.
Explanation / Answer
1)a)profit P(x) = (4x)/(3x^2+27)
differentiate with respect to x
P'(x) = [(4*1)(3x2+27) -(4x)(6x+0)]/(3x2+27)2
P'(x) = [12x2+108 -24x2]/(3x2+27)2
P'(x) = [108 -12x2]/(3x2+27)2
for critical points P'(x)=0
[108 -12x2]/(3x2+27)2=0
108 -12x2=0
x2=108/12
x2=9
x=3
for x <3 ,P'(x) .0 , for x >3,P'(x) <0
so maximum profit whem 3 thousand dollars are spent on advertising
b)
maximum profit P(3) =(4*3)/(3*32 +27)
maximum profit =(12)/(27+27)
maximum profit =12/54
maximum profit =2/9
maximum profit =0.222 thousand dollars
2)let sides are represeted as x,y
4x+y =1500
y =1500-4x
total area A=x*y
total area A=x(1500-4x)
total area A=1500x-4x2
for largest total area dA/dx =0 ,d2A/dx2<0
dA/dx =1500-8x =0
x=1500/8
x=187.5
d2A/dx2=-8<0
largest total area A=187.5(1500-4*187.5)
largest total area A=140625 ft2
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