Suppose a local vendor charges $2 per hot dog and that the number of hot dogs sold per hour x is given by x(t)=−4t^2+20t+92, where t is the number of hours since 10 AM, 0≤t≤4.

Question

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

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

c. What is the revenue per hour at noon?(SOLVED)

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?

I NEED D AND E

(when answering pls label the variables (1,2,3.. or a,b,c...) when writing the answers, THANK YOU)

Answer 1

d.

x(t) =- 4t² + 20t + 92

drive the equation you will have

dx/dt = - 8t + 20

to find the max value let dx/ dt = 0

0= - 8t + 20

t= 20/8= 5/2 = 2.5

if we start at 10 AM , the max seal will be at 12:30

and the revenue is

R(t) = $3 * x(t) = 3 [- 4t² + 20t + 92]

R(5/2) = 3 * [ -4* ( 5/2) ²+ 20 * 5/2 +92 ]

= 3{ -25 + 50 + 92]= $351

Hint : if you graph the function x(t) we will see that the max occur at t= 2.5