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

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savvycarroll savvycarroll

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Mathematics

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

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rejkjavik rejkjavik · Answer 1 · 2025-04-06T04:01:40-04:00

Answer:

Number of ovens must be manufactured in a given week to generate a profit of $1610 is 70

Step-by-step explanation:

We are given

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

[tex]P=\frac{1}{10}x(300-x)[/tex]

provided that 0 ≤ x ≤ 200

We are given

profit is $1610

so, we can set P=1610

and then we can solve for x

[tex]1610=\frac{1}{10}x(300-x)[/tex]

Multiply both sides by 10

[tex]1610\cdot \:10=\frac{1}{10}x\left(300-x\right)\cdot \:10[/tex]

[tex]16100=x\left(300-x\right)[/tex]

[tex]16100=300x-x^2[/tex]

[tex]-x^2+300x-16100=0[/tex]

now, we can factor it

[tex](x-70)(x-230)=0[/tex]

we can solve for x

and we get

[tex]x=70,x=230[/tex]

Since, x=230 does not lie on 0 ≤ x ≤ 200

So, we will consider only x=70

So,

Number of ovens must be manufactured in a given week to generate a profit of $1610 is 70