Determine the break-even points of the profit function P(x)=-2x^2+7x+8, where xis the number of dirt bikes produced, in thousands.
Well, determining the break-even points involves finding the values of x where the profit function, P(x), equals zero. In other words, we want to find the number of dirt bikes produced that results in zero profit.
Now, let's look at the profit function P(x) = -2x^2 + 7x + 8. To find the break-even points, we set P(x) equal to zero:
-2x^2 + 7x + 8 = 0
Now, solving this equation might not be as straightforward as an accountant's tax return, but fear not! We can apply the quadratic formula to find the solutions. Remember, the quadratic formula is like the superhero cape of equations. It goes like this:
x = (-b ± √(b^2 - 4ac)) / (2a)
So, plugging in our values into the quadratic formula, we get:
x = (-(7) ± √((7)^2 - 4(-2)(8))) / (2(-2))
After some number crunching and a bit of clown math, we simplify this to:
x = (-7 ± √(49 + 64)) / (-4)
x = (-7 ± √113) / (-4)
Voila! We have our break-even points, -7 plus or minus the square root of 113, all divided by -4. So, the break-even points are approximately:
x ≈ (-7 + √113) / (-4) and x ≈ (-7 - √113) / (-4)
Of course, since x represents the number of dirt bikes produced in thousands, we should divide these values by 1000 to get the final answer in the original units.
To determine the break-even points of the profit function P(x) = -2x^2 + 7x + 8, where x represents the number of dirt bikes produced in thousands, we need to find the values of x where P(x) equals zero.
To find the break-even points, we set P(x) equal to zero and solve for x as follows:
-2x^2 + 7x + 8 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = -2, b = 7, and c = 8.
Plugging these values into the quadratic formula:
x = (-(7) ± √((7)^2 - 4(-2)(8))) / (2(-2))
Simplifying further:
x = (-7 ± √(49 + 64)) / (-4)
x = (-7 ± √113) / (-4)
The two solutions for x are:
x1 = (-7 + √113) / (-4)
x2 = (-7 - √113) / (-4)
Thus, the break-even points are x1 ≈ 1.44 and x2 ≈ 3.06. Remember that x represents the number of dirt bikes produced in thousands, so the break-even points in this case are approximately 1440 and 3060 dirt bikes produced respectively.
To find the break-even points of a profit function, we need to find the values of x for which the profit function is equal to zero. In other words, we need to solve the equation P(x) = 0.
Given the profit function P(x) = -2x^2 + 7x + 8, we can set it equal to zero:
-2x^2 + 7x + 8 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = -2, b = 7, and c = 8. Plugging 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)
Thus, the two possible values for x are:
x1 = (-7 + √113) / (-4)
x2 = (-7 - √113) / (-4)
These are the break-even points of the profit function P(x).
break-even means profit is zero.
So, solve P(x) = 0