A small-appliance manufacturer finds that the profit P (in dollars) generated by producing x microwave ovens per week is given b

Question

3 months ago

11

y the formula P = 1 /10 x (300 − x) provided that 0 ≤ x ≤ 200. How many ovens must be manufactured in a given week to generate a profit of $1610?

1 answer:

PIT_PIT [12.4K] · Answer 1 · 2026-02-19T07:28:15+03:00

Answer:

The amount of ovens that must be produced in a week to earn a $1610 profit is 70.

Step-by-step explanation:

Given:

A small-appliance manufacturer determines the profit P (in dollars) from producing x microwave ovens weekly by the formula:

$P=\frac{1}{10}x(300-x)$

with 0 ≤ x ≤ 200.

The target profit is $1610

So, set P = 1610, then solve for x:

$1610=\frac{1}{10}x(300-x)$

Multiply both sides by 10:

$1610\cdot \:10=\frac{1}{10}x\left(300-x\right)\cdot \:10$

$16100=x\left(300-x\right)$

$16100=300x-x^2$

$-x^2+300x-16100=0$

Next, factor the quadratic:

$(x-70)(x-230)=0$

Solving for x gives:

$x=70,x=230$

Since x=230 is outside the domain 0 ≤ x ≤ 200, we discard it.

Hence, the valid solution is x=70.

Therefore, to achieve a $1610 profit, the manufacturer must produce 70 ovens weekly.