if a vendor charges p dollars each for rugby shirts, then he expects to sell 2000-100p shirts at a tournament.
a. Find a polynomial R(p) that represents the total revenue when the shirts are p dollars each
b. Find r(5), R(10), and R(20)
C. Use the bar graph to determine the price that will give the maximum total revenue
R(p) = p*(2000 - 100p)
= 2000p - 100 p^2
You do the calculations. The highest R value will be at p = 10, which is what they call R(10). It equals $10,000
Note that there will be zero sales when p = $20
To find the polynomial that represents the total revenue, we need to multiply the price per shirt (p) by the number of shirts sold at that price.
a. Let's break down the problem:
- The vendor charges p dollars each for rugby shirts.
- The vendor expects to sell 2000 - 100p shirts at a tournament.
The formula for the total revenue (R) is given by:
R(p) = p * (2000 - 100p)
b. To find R(5), R(10), and R(20), substitute the given values into the formula R(p) = p * (2000 - 100p):
R(5) = 5 * (2000 - 100 * 5)
= 5 * (2000 - 500)
= 5 * 1500
= 7500
R(10) = 10 * (2000 - 100 * 10)
= 10 * (2000 - 1000)
= 10 * 1000
= 10,000
R(20) = 20 * (2000 - 100 * 20)
= 20 * (2000 - 2000)
= 20 * 0
= 0
c. To determine the price that will give the maximum total revenue, we can create a bar graph of the different revenue values at different prices. The price that corresponds with the highest bar on the graph will give the maximum total revenue.
Unfortunately, as a text-based AI, I am unable to create or interpret visual elements such as a bar graph. However, you can create a table or a graph with different price values and corresponding revenue values to identify the highest revenue.
Alternatively, to find the price that will give the maximum total revenue algebraically, we can find the vertex of the quadratic function R(p) = p * (2000 - 100p). The vertex represents the maximum point of the graph. We can use the vertex formula:
Vertex(p) = -b / (2a)
In this case, a = -100 and b = 2000:
Vertex(p) = -2000 / (2 * -100)
= -2000 / (-200)
= 10
Therefore, the price that will give the maximum total revenue is p = 10 dollars.