Gymnast Clothing manufactures expensive hockey jerseys for sale to college bookstores in runs of up to 250. Its cost (in dollars) for a run of x hockey jerseys is
C(x) = 2500 + 10x + 0.2x2
Domain(0 ≤ x ≤ 250)
Gymnast Clothing sells the jerseys at $105 each. Find the revenue function. R(x)=105x
Find the profit function.
p(x)= -0.2x^2+95x-2500
How many should Gymnast Clothing manufacture to make a profit? HINT [See Example 2.] (Round your answer up to the nearest whole number.)
___Jerseys
I can't get the correct # of jerseys. it has to be within the domain, but I keep getting numbers higher than it. I don't know what is wrong.
Thank you
It is 28 jerseys to obtain a profit
To find the number of jerseys Gymnast Clothing needs to manufacture in order to make a profit, we need to determine the value of x where the profit function, p(x), is greater than zero.
The profit function is given as:
p(x) = -0.2x^2 + 95x - 2500
To find the number of jerseys, we need to solve the inequality p(x) > 0:
-0.2x^2 + 95x - 2500 > 0
To solve this inequality, you can use a quadratic equation or graphing. I will use graphing to find the solutions.
Let's graph the function p(x) = -0.2x^2 + 95x - 2500:
First, find the x-intercepts of the quadratic equation by setting p(x) equal to zero:
-0.2x^2 + 95x - 2500 = 0
Solve this equation to find the x-values where the graph crosses the x-axis. The solutions are x = 5 and x = 250.
Now, graph the function p(x) = -0.2x^2 + 95x - 2500, where x is within the given domain, 0 ≤ x ≤ 250.
The graph of the function will be a downward-opening parabola that intersects the x-axis at x = 5 and x = 250. We need to find the range of x-values where p(x) is greater than zero, which is the area above the x-axis.
Looking at the graph, we see that the function p(x) is positive (greater than zero) within the interval x = 5 to x = 250.
Therefore, the number of jerseys Gymnast Clothing should manufacture to make a profit is in the range of 5 < x < 250.
However, the exact number within this range cannot be determined without additional information. If you have any specific constraints or requirements, please provide them, and I will be happy to assist you further.
To find the number of jerseys Gymnast Clothing should manufacture to make a profit, we need to determine the value of x where the profit function, p(x), is greater than zero.
The profit function is given by p(x) = -0.2x^2 + 95x - 2500.
To solve for x, we set the profit function equal to zero and solve for x:
-0.2x^2 + 95x - 2500 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.2, b = 95, and c = -2500. Plugging these values into the quadratic formula, we get:
x = (-95 ± √(95^2 - 4(-0.2)(-2500))) / (2(-0.2))
Simplifying further:
x = (-95 ± √(9025 - 2000)) / (-0.4)
x = (-95 ± √(7025)) / (-0.4)
Now, we can calculate the two possible values of x:
x1 = (-95 + √(7025)) / (-0.4)
x2 = (-95 - √(7025)) / (-0.4)
Calculating these values:
x1 ≈ 50.562
x2 ≈ 144.437
Since the domain for x is 0 ≤ x ≤ 250, we round up these values to the nearest whole number:
The minimum number of jerseys Gymnast Clothing should manufacture to make a profit is 51 jerseys.