Question

Dorothy Wagner is currently selling 80 "I ♥ Calculus" T-shirts per day, but sales are dropping at a rate of 2 per day. She is currently charging $7 per T-shirt, but to compensate for dwindling sales, she is increasing the unit price by $1 per day. How fast, and in what direction, is her daily revenue currently changing?

Answer 1

Answer:

Increasing by $66 per day

Step-by-step explanation:

If sales are dropping at a rate of  2 per day, the sales function is:

$S = -2t +80$

If price is increasing by $1 per day, the daily price function is:

$P=1t+7$

Revenue is given by daily sales multiplied by daily price:

$R = S(t)*P(t) = (-2t+80)*(t+7)\\R(t) = -2t^2+66t+560$

The derivate of the revenue function gives us the daily rate of change in revenue:

$R(t) = -2t^2+66t+560\\R'(t) = -4t+66$

Currently (t=0) her daily revenue is changing by:

$R'(0) = -4*0+66\\R'(0) = \ 66$

Her revenue is increasing by $66 per day.