p(x)=-0.5x+40x-300 dollars,
how many items to be sold to get mazimun profit?
describe the meaning of the numbers 3 and 10,000 in the formula.
find the profit for 5 different values of x.
graph the profit function over its given domain; use 5 values calculated in part 3 to construct the graph and connect these points with a smooth curve.
will this profit function have a maximum, if so what is it?
how many items to be sold to get mazimun profit?
The number 10,000 in the formula represents the base profit or the profit when no items are sold. It is a constant term added to the product of 200 and (x^2 - 3) to calculate the total profit.
To find the profit for different values of x, we need to substitute those values into the function p(x) and then calculate the result. Let's calculate the profit for 5 different values of x.
For x = 1 (1,000 items sold):
p(1) = 200(1^2 - 3) + 10,000
= 200(1 - 3) + 10,000
= -400 + 10,000
= 9,600
For x = 2 (2,000 items sold):
p(2) = 200(2^2 - 3) + 10,000
= 200(4 - 3) + 10,000
= 200 + 10,000
= 10,200
For x = 2.5 (2,500 items sold):
p(2.5) = 200(2.5^2 - 3) + 10,000
= 200(6.25 - 3) + 10,000
= 200(3.25) + 10,000
= 650 + 10,000
= 10,650
For x = 2.7 (2,700 items sold):
p(2.7) = 200(2.7^2 - 3) + 10,000
= 200(7.29 - 3) + 10,000
= 200(4.29) + 10,000
= 858 + 10,000
= 10,858
For x = 2.9 (2,900 items sold):
p(2.9) = 200(2.9^2 - 3) + 10,000
= 200(8.41 - 3) + 10,000
= 200(5.41) + 10,000
= 1,082 + 10,000
= 11,082
To graph the profit function, we will plot the points (x, p(x)) with the values we calculated above on a coordinate plane. Connecting these points with a smooth curve will give us the graph of the profit function over its given domain.
Regarding the maximum profit, we can determine it by examining the graph. If the graph has a peak or the highest point, then the profit function will have a maximum.