If a manufacturer charges p dollars each for rugby shirts, then he expects to sell 2000 – 100p shirts per week, What polynomial represents the total revenue expected for a week? How many shirts will be sold if the manufacturer charges $200 each for the shirts? Find the total revenue when the shirts are sold for $20 each.

To find the polynomial that represents the total revenue expected for a week, we need to multiply the price per shirt by the number of shirts sold in that week.

Let's break down the problem step by step:

Step 1: Determine the number of shirts sold per week
The number of shirts sold per week is given by the formula: 2000 – 100p

Step 2: Calculate the total revenue
The total revenue is given by the formula: price per shirt * number of shirts sold

Let's substitute the values from step 1 into step 2 to find the polynomial:

Total revenue = (2000 – 100p) * p

Now, let's simplify this equation:

Total revenue = 2000p - 100p^2

So, the polynomial that represents the total revenue expected for a week is 2000p - 100p^2.

Now, let's move on to the second part of the question:

If the manufacturer charges $200 each for the shirts, we can substitute p = 200 into the equation we just found to calculate the number of shirts sold:

Number of shirts sold = 2000 - 100(200)
Number of shirts sold = 2000 - 20000
Number of shirts sold = -18000

Since the number of shirts sold cannot be negative, we can conclude that 0 shirts will be sold if the manufacturer charges $200 each for the shirts.

Lastly, let's find the total revenue when the shirts are sold for $20 each. We can substitute p = 20 into the equation:

Total revenue = 2000(20) - 100(20^2)
Total revenue = 40000 - 100(400)
Total revenue = 40000 - 40000
Total revenue = 0

Therefore, the total revenue when the shirts are sold for $20 each is $0.