A new smog-control device will reduce the output of sulfur oxides from automobil

Question

A new smog-control device will reduce the output of sulfur oxides from automobile exhausts. It is estimated that the rate of savings to the community from use of the device will be approximated by S(x) = -x^2 + 4x + 8 where S(x) is the rate of savings (in millions of dollars) after x years of use of the device. The new device cuts down on the production of sulfur oxides, but it causes an increase in the production of nitrous oxides. The rate of additional costs (in millions) to the community after x years is approximated by C(x) = 3/25 x^2. This device isn’t producing money – so what then is meant by the community will save or the community will have additional costs? For how many years will it pay to use the new devices? What will be the net savings over this period of time?

Explanation / Answer

1) community will have additional costs

2)s(x)=c(x)

-x2 + 4x + 8= (3/25) x2

(3/25 +1) x2 -4x -8 =0

(28/25) x2 -4x -8 =0

x2 -(100/28)x -(200/28) =0

by quadratic formula

x=[(100/28)+((-100/28)2-4(100/28)(-200/28)) ]/2,x=[(100/28)-((-100/28)2-4(100/28)(-200/28)) ]/2

x=50/7 , x=-25/7

x cannot be negative

so x =50/7 =7.14 years

3)net savings =[0 to 50/7]-x2 + 4x + 8-(3/25) x2 dx

=[0 to 50/7] 4x + 8-(28/25) x2 dx

=[0 to 50/7] 2x2 + 8x-(28/75)x3

=2(50/7)2 + 8(50/7)-(28/75)(50/7)3

=3400/147

=23.13 million$

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.