Question

You have been raising the price of shirts by 50 cents per week and sales have been falling at a continuously compounding rate of 5% per week. Assuming you are now selling 50 shirts per week and charging $10 per shirt, how much revenue will be generated in the coming year?

Answer 1

The price after x weeks is (assuming the falling value is calculated after the price increase)
p(x) = 10 + 0.50x
the sales, q(x) are
q(x) = 50 * 0.95^x
The revenue is price * sales, so in week #x, the revenue is
r(x) = (10 + 0.50x)*(50 * 0.95^x) = 25(x+20)*0.95^x
That means the revenue for the year is
∫[0,52] 25(x+20)*0.95^x dx = 16153.20

Answer 2

Okay and so I would simplify ∫[0,52] 25(x+20)*0.95^x dx by taking the constant out so it is 25 times (∫[0,20] (x^2-400)/(x-20)*0.95^x dx + ∫[20,52] (x^2-400)/(x-20)*0.95^x dx) right? But how do I keep simplifying that so that it is an exact number without ∫ in it?

Answer 3

Not sure if you still need help but given what you are looking for, I believe you would look into functions such as Pe^rt. So you would probably want ∫[0,52] 50 (aka the quantity you started selling) e^(0.05(aka the rate)t)(10+0.05t)dt. Then you can solve. You should probably get an answer around $16583.42 if done correctly.

Answer 4

you have to integrate x e^x using integration by parts
and of course, ∫a^x = 1/lna a^x
Not sure why you want to change (x+20) to (x^2-400)/(x-20) ,
especially since we have x(10+0.5x) = 1/2 (x^2+20x)
That said, you get
∫(x^2+20x)*0.95^x dx
= ln0.95 ∫(x^2+20x)e^x dx
= ln0.95 e^x (x^2+18x-18)