A company's weekly profit, in riyals, is modeled by the function P(u)=-0.032u^2+46u-3000. where u is the number of units sold each week.
a) the maximum weekly profit. Answer: 13531.25 riyals
b) the loss for week's holiday period, where no units are sold .Answer:3000 riyals
c) the number of units sold each at break-even point for the company. Answer: 69 or 1369 units
Thank you so, so much for everything. Really thanks a lot!!!
your a) and b) are correct
in c) small detail,
my answer was 68.479.. which would round off to 68 , not 69
the other answer of 1369 is correct
To find the maximum weekly profit, we need to determine the vertex of the parabolic function.
The vertex of a parabola is given by the formula (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c and f(x) is the function.
In this case, the function is P(u) = -0.032u^2 + 46u - 3000, so a = -0.032, b = 46, and c = -3000.
Using the formula, we can calculate the x-coordinate of the vertex as -b/2a. Plugging in the values, we get:
x = -46 / (2 * (-0.032)) = 714.2857 (approx)
To find the y-coordinate of the vertex, we substitute this x-value into the function:
P(714.2857) = -0.032 * (714.2857)^2 + 46 * 714.2857 - 3000 ≈ 13531.25 riyals
Therefore, the maximum weekly profit is approximately 13531.25 riyals.
To find the loss for the week's holiday period where no units are sold, we need to substitute u = 0 into the function:
P(0) = -0.032 * 0^2 + 46 * 0 - 3000 = -3000 riyals
So, the loss for the week's holiday period is 3000 riyals.
To find the break-even point, we need to determine where the profit function equals zero. In other words, we need to solve the quadratic equation -0.032u^2 + 46u - 3000 = 0.
There are two ways to approach this:
1. Factoring: Unfortunately, this quadratic equation cannot be factored easily, so we move on to the next approach.
2. Quadratic formula: The quadratic formula can be used to solve any quadratic equation in the form ax^2 + bx + c = 0. In this case, a = -0.032, b = 46, and c = -3000.
The quadratic formula states that the solutions for x can be found using the formula:
u = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values, we get:
u = (46 ± √(46^2 - 4 * (-0.032) * -3000)) / (2 * -0.032)
Calculating the values inside the square root, we get:
u = (46 ± √(2116 - 3840)) / (-0.064)
Simplifying further, we have:
u = (46 ± √(-1724)) / (-0.064)
Since the square root of a negative number is not defined in the real number system, it means that the quadratic equation has no real solutions. Therefore, there is no break-even point for this company in terms of units sold.
The two values, 69 and 1369 units, mentioned as possible answers, are not accurate break-even points.