The profit in dollars in producing x- items of some commodity is given by the equation P = - 37 x^2 + 1073 x - 7548 .
How many items should be produced to break even? (If there are two break-even points, then enter the smaller value of x. Your solution may not be an integer. Use your calculator for irrational square roots.)
How many items should be produced to maximize the profit?
What is the maximum profit? (You may need your calculator to compute the value.)
Profit = 0 is break even.
So solve for x in
P(x)=-37x²+1073x-7548 = 0
to give x=12 or x=17
The profit is given by the maximum of the P(x) function, which is located at x=(12+17)/2=15.5, or the maximum profit is
P(15.5). (substitute 15.5 for x and evaluate the profit).
To find the number of items needed to break even, we need to set the profit equation equal to zero:
-37x^2 + 1073x - 7548 = 0
We can solve this quadratic equation by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -37, b = 1073, and c = -7548. Plugging these values into the quadratic formula, we can solve for x.
x = (-1073 ± √(1073^2 - 4(-37)(-7548))) / (2(-37))
Simplifying this equation gives us:
x = (-1073 ± √(1149529 - 1115688)) / (-74)
x = (-1073 ± √43841) / (-74)
Using a calculator, we can find the approximate values of the square root and simplify further:
x ≈ (-1073 ± 209.355) / (-74)
Now, we have two possible solutions:
x ≈ (-1073 + 209.355) / (-74) ≈ -8.671
x ≈ (-1073 - 209.355) / (-74) ≈ 22.657
Since we cannot produce a negative quantity, the smallest value of x that satisfies the equation is approximately 22.657. Therefore, to break even, approximately 23 items should be produced.
To find the number of items that maximize the profit, we can find the vertex of the quadratic equation. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = -37 and b = 1073 in this case.
x = -1073 / (2(-37))
Simplifying this equation gives us:
x = -1073 / -74 ≈ 14.500
So, approximately 14.5 items should be produced to maximize profit.
To find the maximum profit, we substitute this value of x back into the profit equation:
P = -37(14.500)^2 + 1073(14.500) - 7548
Calculating this will give us the maximum profit.