Questions: The total profit (in dollars) from the sale of x answering machines is P(x) = 20x - 0.6x^2 - 280. (A) Find the exact profit from the sale of the 26th machine. Exact Profit on 26th machine = (B) Use the marginal profit to approximate the profit from the sale of the 26th machine. Approx. profit on 26th machine = -10

The total profit (in dollars) from the sale of x answering machines is
P(x) = 20x - 0.6x^2 - 280.
(A) Find the exact profit from the sale of the 26th machine.

Exact Profit on 26th machine =

(B) Use the marginal profit to approximate the profit from the sale of the 26th machine.

Approx. profit on 26th machine = -10

Transcript text: The total profit (in dollars) from the sale of $x$ answering machines is \[ P(x)=20 x-0.6 x^{2}-280 . \] (A) Find the exact profit from the sale of the 26th machine. Exact Profit on 26th machine $=$ (B) Use the marginal profit to approximate the profit from the sale of the 26th machine. Approx. profit on 26th machine $=$ $-10$

Solution

Solution Steps

Step 1: Finding the Exact Profit for the $x$th Item

To find the exact profit from the sale of the $x$th item, we calculate $P(x) - P(x-1)$: $P(x) = -0.6x^2 + 20x - 280$ $P(x-1) = -0.6(x-1)^2 + 20(x-1) - 280$ Substituting $x$ into the equations and simplifying gives us the exact profit for the $x$th item. The exact profit for selling the $x$th item is: $P(x) - P(x-1) = -10.6$

Step 2: Approximating the Profit for the $x$th Item Using Marginal Profit

The marginal profit is the derivative of the profit function $P(x)$ with respect to $x$, which is $P'(x) = 2ax + b$. Substituting $x$ into the derivative gives us the marginal profit for the $x$th item. The approximate profit for selling the $x$th item, using marginal profit, is: $P'(x) = -11.2$