The weekly revenue from the production and sale of units of oil is given by thousand dollars. The cost function is given by
thousand dollars. Find the number of units of oil that are to be produced to maximize the profit if .

Ask by Ramirez Cervantes. in the United States

Mar 27,2025

Question

The weekly revenue from the production and sale of units of oil is given by thousand dollars. The cost function is given by
thousand dollars. Find the number of units of oil that are to be produced to maximize the profit if .

Ask by Ramirez Cervantes. in the United States

Mar 27,2025

UpStudy AI · Accepted Answer

Given the revenue function 
$$
R(x) = 267x - 3x^2 \quad \text{(in thousand dollars)},
$$
and the cost function 
$$
C(x) = x^2 + 195x + 4 \quad \text{(in thousand dollars)},
$$
the profit function is
$$
P(x) = R(x) - C(x).
$$

**Step 1.** Write the profit function:
$$
P(x) = (267x - 3x^2) - (x^2 + 195x + 4).
$$

**Step 2.** Simplify the profit function:
$$
P(x) = 267x - 3x^2 - x^2 - 195x - 4.
$$
$$
P(x) = (267x - 195x) + (-3x^2 - x^2) - 4.
$$
$$
P(x) = 72x - 4x^2 - 4.
$$
Rearrange as:
$$
P(x) = -4x^2 + 72x - 4.
$$

**Step 3.** Determine the value of $$ x $$ that maximizes $$ P(x) $$.  
Since $$ P(x) $$ is a quadratic function in the form
$$
P(x) = ax^2 + bx + c,
$$
with $$ a = -4 $$ and $$ b = 72 $$, the vertex (maximum point, since $$ a < 0 $$) is given by:
$$
x = -\frac{b}{2a}.
$$

Substitute the values:
$$
x = -\frac{72}{2(-4)} = \frac{72}{8} = 9.
$$

**Step 4.** Check the domain $$ 0 \leq x \leq 17 $$:  
Since $$ 9 $$ lies within the interval $$[0, 17]$$, the number of units that maximizes profit is $$ x = 9 $$.
To maximize profit, produce 9 units of oil.

The weekly revenue from the production and sale of units of oil is given by thousand dollars. The cost function is given by
thousand dollars. Find the number of units of oil that are to be produced to maximize the profit if .

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The weekly revenue from the production and sale of units of oil is given by thousand dollars. The cost function is given by thousand dollars. Find the number of units of oil that are to be produced to maximize the profit if .

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The weekly revenue from the production and sale of units of oil is given by thousand dollars. The cost function is given by
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