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?
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
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?
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.
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)
To calculate the revenue generated in the coming year, we need to consider the changing factors: the price of shirts and the number of shirts sold.
Let's break down the steps to find the revenue generated:
Step 1: Determine the number of shirts sold after one year.
Since the sales are falling at a continuously compounding rate of 5% per week, we can use the formula for exponential growth/decay:
N = N0 * (1 - r)^t
where N0 is the initial value, r is the rate of growth/decay, and t is the time in weeks.
Given that you are currently selling 50 shirts per week, the rate of decay is 5% or 0.05, and the time is 52 weeks (in a year), we can calculate the number of shirts sold after one year:
N = 50 * (1 - 0.05)^52
N ≈ 50 * 0.662
N ≈ 33.1
Therefore, it is estimated that you will sell approximately 33 shirts in the coming year.
Step 2: Calculate the price of shirts after one year.
Since the price is raised by 50 cents per week, we can multiply the number of weeks in a year (52) by the increment per week ($0.50) to find the total price increase after one year:
Price increase = 0.50 * 52
Price increase = 26
Starting from $10 per shirt, the price after one year will be $10 + $26 = $36 per shirt.
Step 3: Calculate the revenue generated in the coming year.
Revenue = Number of shirts sold * Price per shirt
Revenue = 33 * $36
Revenue ≈ $1,188
Therefore, it is estimated that you will generate approximately $1,188 in revenue in the coming year.