Determine the break even points of the profit function P(x) = -2x^2 + 7x + 8, where x is the number of dirt bikes produced, in thousands
a) 4408
b) 4468
c) 4448
d) 4400
To determine the break even points of the profit function, we need to find the values of x where P(x) = 0.
Given: P(x) = -2x^2 + 7x + 8
Setting P(x) = 0, we have:
0 = -2x^2 + 7x + 8
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -2, b = 7, and c = 8.
Substituting the values into the quadratic formula, we get:
x = (-7 ± √(7^2 - 4(-2)(8))) / (2(-2))
Simplifying further, we have:
x = (-7 ± √(49 + 64)) / (-4)
x = (-7 ± √113) / (-4)
Now, let's calculate the values of x:
x1 = (-7 + √113) / (-4)
x2 = (-7 - √113) / (-4)
Let's evaluate these expressions:
x1 ≈ 4.408
x2 ≈ -0.318
Since the number of dirt bikes produced cannot be negative, we can disregard the negative value of x2.
Therefore, the break-even point is approximately x = 4.408.
To convert this to the number of dirt bikes produced in thousands, we multiply x by 1000:
Break-even point = 4.408 * 1000 ≈ 4408
So the correct answer is option a) 4408.
Well, determining the break-even points involves finding the x-values where the profit function equals zero. In other words, we need to solve the equation -2x^2 + 7x + 8 = 0.
Now, let's use some humor to approach this problem. Why did the dirt bike factory start losing money? Because they couldn't find the break-even point! It's like they were riding in circles, always falling short of making a profit.
Anyway, let's get serious and solve this quadratic equation. We have:
-2x^2 + 7x + 8 = 0
To make things simpler, let's divide the whole equation by -1:
2x^2 - 7x - 8 = 0
Now, we can either use the quadratic formula or factor the equation. Let's go with factoring:
(2x + 1)(x - 8) = 0
Now, we set each factor equal to zero:
2x + 1 = 0 OR x - 8 = 0
Solving these equations individually, we get:
x = -1/2 OR x = 8
However, we need to remember that 'x' represents the number of dirt bikes produced in thousands. So, our answer should be in terms of thousands. Therefore, we have:
a) -500
b) 8000
c) None of the above
So, the correct answer is none of the above. We don't have any of the options listed in terms of thousands. Looks like even the options couldn't find the break-even point!
To determine the break-even points of the profit function, we need to find the values of x where the profit P(x) equals zero.
The profit function is given as P(x) = -2x^2 + 7x + 8.
Setting P(x) to zero, we have:
0 = -2x^2 + 7x + 8
To solve this equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Where a, b, and c are the coefficients of the quadratic equation.
In this case, a = -2, b = 7, and c = 8.
Substituting these values into the quadratic formula, we get:
x = (-7 ± √(7^2 - 4(-2)(8))) / (2(-2))
Simplifying further:
x = (-7 ± √(49 + 64)) / (-4)
x = (-7 ± √113) / -4
Now, we can calculate the approximate values of x using a calculator:
x ≈ (-7 + √113) / -4 ≈ 4.508
x ≈ (-7 - √113) / -4 ≈ 0.492
Since x represents the number of dirt bikes produced in thousands, the break-even points are approximately 4508 and 492.
None of the answer choices provided match these values exactly, so it seems there might be an error in the given answer choices.
break-even is where profit=0
so just solve in the usual way, using the quadratic formula