p = 300e−2q; p = 90

Question

Calculate the consumers' surplus at the indicated unit price p

 for the demand equation. HINT [See Example 1.] (Round your answer to the nearest cent.)

p = 300e^−2q;  p = 90

Answer 1

set the two equations equal to find the point of intersection

300e^-2q =90

e^-2q = 90/300 = 3/10 = 0.3

-2q = ln.3

q = ln.3/-2

q = -(1/2)ln.3 = about 0.602 = upper limit to evaluate the definite integral

q=0 is the lower limit

integral of 300e^-2q -90

= -600e^-2q -90q

evaluate from q=0 to -.5ln.3

= -600(e^-2(-.5ln.3) - 90(-.5ln.3) - (--600(e^0)

= -600e^ln.3 +45ln.3 +600

= -600(.3) + 45ln.3 +600

= 45ln.3 - 180 +600

= about -54.18 -180 +600

= -234.18 +600

= $365.02 = consumer surplus