A company produces portable cassette players estimates that the profit P (in dollars) for selling a particular model is P=-76x^3+4830x^2-320,000, 0 ≤ x ≤ 50, where x is the advertising expense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of $800,000.
This is just algebra I. You want x where
-76x^3+4830x^2-320000 = 800000
I suggest a graphing utility, as this has no rational roots.
To find the smaller advertising amount that will yield a profit of $800,000, we need to set the profit equation equal to $800,000 and solve for x.
The profit equation is:
P = -76x^3 + 4830x^2 - 320,000
Setting P equal to $800,000, we get:
800,000 = -76x^3 + 4830x^2 - 320,000
Rearranging the equation, we get:
76x^3 - 4830x^2 + 320,000 + 800,000 = 0
Combining like terms, we have:
76x^3 - 4830x^2 + 1,120,000 = 0
Now we can solve this cubic equation using numerical methods or a graphing calculator. One approach is to use a graphing calculator to find the points where the function graph intersects the line y = $800,000.
Alternatively, we can use a numerical method called the Newton-Raphson method to approximate the roots of the equation. This method requires an initial guess for the root, and it iteratively improves the approximation until it reaches a desired level of accuracy.
Let's use a graphing calculator to find the smaller advertising amount that will yield a profit of $800,000.