Vendor charges $2 per hot dog and hot dogs sold per hour x is given by x(t)=-4t^2+20t+92 , t is the number of hours since 10 AM, 0 is less than/equal to t which is less than/equal to 4

Question

a. Find an expression for the revenue per hour R as a function of x.

b. Find and simplify (R∘x)(t). What does this represent?

c. What is the revenue per hour at noon?

d. If the price were raised to $3 per hot dog with no change in the x(t) equation, which hour would produce the most revenue? Why?

e. If the price were dropped to $1 per hot dog, but that price drop caused the number of sales to increase according to the function x(t)=−9t2+22t+138, would the vendor make more money at the original $2 price, or at the $1 price?

Answer 1

Revenue = Price times quantity sold: R(x) = Px = 2(-4t^2+20t + 92) = -8t^2 + 40t + 184

at noon t=2, R = -8(2^2 + 40(2) + 184 = -32 + 80 + 182 = $130

if price were raised to $3 then R = -12t^2 +60t + 92

take the derivative and set = 0, then solve for t

R'=-24t + 60 = 0, t = 60/24= 5/2 = 2.5 plus 10 = 12:30 pm

-9t^2 + 22t + 138 at noon would be 9-(2)^2 + 22(2) + 138 = -36+44+ 138 = $146 > $130 at noon with $1