Question

the price received for a bicycle is given by the equation \\(p = 100 - 10x^2\\), where \\(x\\) is the number of bicycles produced, in millions.

it costs the company \\(\\$60\\) to make each bicycle.

what is the profit function for the company?

\\(p = -10x^2 - 60x + 100\\)
\\(p = -10x^3 + 40x\\)
\\(p = -10x^2 + 40\\)

Explanation:

🆕 New Concept Discovered: Revenue, Cost, and Profit Functions
How businesses calculate total money made, spent, and kept.

Step 1: Find the Revenue Function

Revenue is the total money brought in from selling a product. It is calculated by multiplying the price per item by the number of items sold:

$$ \text{Revenue } (R) = \text{Price } (p) \times \text{Quantity } (x) $$

Given the price equation \( p = 100 - 10x \):

$$ R = x(100 - 10x) $$

$$ R = 100x - 10x^2 $$

Step 2: Find the Cost Function

The cost to make each bicycle is \( \$60 \). For \( x \) million bicycles, the total cost function \( C \) is:

$$ C = 60x $$

Step 3: Find the Profit Function

Profit is the money left over after subtracting total costs from total revenue:

$$ \text{Profit } (P) = \text{Revenue } (R) - \text{Cost } (C) $$

Substitute the expressions we found:

$$ P = (100x - 10x^2) - 60x $$

Combine like terms:

$$ P = -10x^2 + 40x $$

Answer:

\( P = -10x^2 + 40x \)