The total profit (in dollars) from the sale of sweatshirts is . Evaluate the marginal profit at the given values of x , and interpret the results.
(A) 450
(B) 750
C. The marginal profit at is . At a production level of sweatshirts the profit is $ per sweatshirt.
D. The marginal profit at is . At a production level of sweatshirts a unit increase in production will increase the total profit by $ per sweatshirt.
(B) Select the correct choice and fill in the answer boxes to complete your choice.
(Type integers or decimals.)
A. The marginal profit at is At a production level of sweatshirts the profit is increasing at a rate of $ per sweatshirt.
B. The marginal profit at is . At a production level of sweatshirts the profit is per sweatshirt.
C. The marginal profit at is At a production level of sweatshirts a unit decrease in production will decrease the total profit by per sweatshirt.
D. The marginal profit at is . At a production level of sweatshirts the profit is decreasing at a rate of $ per sweatshirt.

Ask by Griffin Cruz. in the United States

Mar 29,2025

Question

The total profit (in dollars) from the sale of sweatshirts is . Evaluate the marginal profit at the given values of x , and interpret the results.
(A) 450
(B) 750
C. The marginal profit at is . At a production level of sweatshirts the profit is $ per sweatshirt.
D. The marginal profit at is . At a production level of sweatshirts a unit increase in production will increase the total profit by $ per sweatshirt.
(B) Select the correct choice and fill in the answer boxes to complete your choice.
(Type integers or decimals.)
A. The marginal profit at is At a production level of sweatshirts the profit is increasing at a rate of $ per sweatshirt.
B. The marginal profit at is . At a production level of sweatshirts the profit is per sweatshirt.
C. The marginal profit at is At a production level of sweatshirts a unit decrease in production will decrease the total profit by per sweatshirt.
D. The marginal profit at is . At a production level of sweatshirts the profit is decreasing at a rate of $ per sweatshirt.

Ask by Griffin Cruz. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

We start with the profit function

$$
P(x) = 6x - 0.006x^2 - 650.
$$

**Step 1. Compute the marginal profit function.**

The marginal profit is the derivative $$P'(x)$$:

$$
P'(x) = \frac{d}{dx}\Bigl(6x - 0.006x^2 - 650\Bigr) = 6 - 0.012x.
$$

---

**Step 2. Evaluate the marginal profit at $$x=450$$.**

Substitute $$x=450$$ into $$P'(x)$$:

$$
P'(450) = 6 - 0.012(450) = 6 - 5.4 = 0.6.
$$

*Interpretation:* At a production level of 450 sweatshirts, a unit (one sweatshirt) increase in production will increase the total profit by about \$0.60 per sweatshirt.

Thus, for $$x=450$$, the answer matching one of the choices is:

- The marginal profit at $$x=450$$ is $$0.6$$.
- At a production level of $$450$$ sweatshirts a unit increase in production will increase the total profit by \$$$0.6$$ per sweatshirt.

---

**Step 3. Evaluate the marginal profit at $$x=750$$.**

Substitute $$x=750$$ into $$P'(x)$$:

$$
P'(750) = 6 - 0.012(750) = 6 - 9 = -3.
$$

*Interpretation:* At a production level of 750 sweatshirts, a unit increase in production will decrease the total profit by about \$3 per sweatshirt.

Thus, for $$x=750$$, the answer matching the correct interpretation is:

- The marginal profit at $$x=750$$ is $$-3$$.
- At a production level of $$750$$ sweatshirts the profit is decreasing at a rate of \$$$3$$ per sweatshirt.
At $$ x=450 $$, the marginal profit is \$0.60 per sweatshirt. At $$ x=750 $$, the marginal profit is $$-\$3.00$$ per sweatshirt, indicating a decrease in profit with each additional sweatshirt produced.

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