p = 7 + 2q1⁄3; p = 12

Question

Calculate the producers' surplus (in dollars) for the supply equation at the indicated unit price p.

 (Round your answer to the nearest cent.)

p = 7 + 2q^1⁄3;  p = 12

Answer 1

Producer's Surplus is the area between the Supply curve and the horizontal price line= $19.53 per unit, rounded off to nearest cent

calculate the area below the Supply curve and the x (q) axis, then subtract that from

the rectangular area between the price line and the x axis from q=0 to q= the x coordinate of the intersection point of the price line and the Supply curve.

take the integral of the supply curve equation, evaluated from zero to the x (q) value of the intersection of the Supply curve and the price line

the intersection point is found by setting 7+2(cube root of q) = 12

cube root of q = (12-7)/2 = 5/2

cube both sides

q = (5/2)^3

= 125/8

= 15 5/8

= 15.625

integral of 7 + 2q^(1/3)

=7q + 2q^(4/3)/(4/3)

= 7q + (3/2)q^(4/3)

evaluate between q= 0 and 15.625 or 125/8

= 7(125/8) + (3/2)(125/8)^(4/3)

= 7(125/8) + (3/2)(5/2)^4

=7(4)(125)/8(4) + (3/2)(625)/16

= 3500/32 + 1875/32

= 5375/32

rectangular area betwen the price line and x axis between q=0 and q=15 5/8

= 12 x 15 5/8

= 12(125/8)

=12(4)(125)/4(8)

= 6000/32

subtract the Supply curve area from the rectangular price line area

6000/32 - 5375/32

= 625/32

= 19 17/32

= 19.53125 = Producer's Surplus

=$19.53 per unit