The problem presented in the image involves profit calculations based on a revenue function $R(x)$ and cost function $C(x)$, where:

• $R(x) = 1.6x^2$
• $C(x) = 0.0002x^3 + 8x + 100$

We are asked to:

• (a) Find the profit function $P(x) = R(x) - C(x)$.
• (b) Find the profit when 14 hundred cellphones are sold (which means $x = 14$).
• (c) Interpret $P(14)$.

(a) Find the profit function $P(x)$:

The profit function $P(x)$ is defined as the revenue function minus the cost function:

$P(x) = R(x) - C(x)$

Substitute the given expressions for $R(x)$ and $C(x)$:

$P(x) = 1.6x^2 - \left(0.0002x^3 + 8x + 100\right)$

Simplifying:

$P(x) = 1.6x^2 - 0.0002x^3 - 8x - 100$

(b) Find the profit when $x = 14$:

To find the profit when 14 hundred cellphones are sold, substitute $x = 14$ into the profit function $P(x)$.

$P(14) = 1.6(14)^2 - 0.0002(14)^3 - 8(14) - 100$

I'll compute this value for you:

$P(14) = 1.6(196) - 0.0002(2744) - 112 - 100$

Now, calculating the numerical values:

$P(14) = 313.6 - 0.5488 - 112 - 100$

$P(14) \approx 101.0512$

(c) Interpretation of $P(14)$:

The value $P(14) \approx 101.05$ represents the profit (in hundreds of dollars) when 14 hundred cellphones (or 1,400 cellphones) are sold. This means that the profit is approximately $10,105.

Let me know if you'd like further details or have questions!

Here are 5 questions to expand your knowledge:

1. How does the profit function change if the cost function has a higher degree term?
2. How would the revenue function impact profit if it had a cubic term instead of a quadratic one?
3. What is the significance of the coefficient of the cubic term in the cost function?
4. How do you interpret a negative profit value?
5. Can you calculate the break-even point for this scenario?

Tip: When interpreting results from word problems, always remember to include the units and context—like how many items are sold and the total dollar value of profit.