MHB What is the Optimal Number of Birdhouses to Produce?

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Tom plans to sell birdhouses for $60 each and expects to sell at least 30 per week. If he produces more than 30, he incurs a loss of $2 for each additional house. The proposed revenue function is C(x) = (30 + x)(60 - 2x), where x represents the number of additional houses produced. This setup correctly reflects the total number of houses and the adjusted price per house. The next step is to optimize this revenue function for maximum profit.
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Hi everyone - this is my second post on here today!

Anyway, I have a word problem, and would just like to make sure I am setting it up correctly.

Tom expects to sell at least 30 birdhouses a week, at \$60 each. If he produces more than 30, he will lose \$2 per additional house. What is the ideal number of houses to produce?

I have an idea of how to set it up, but just want to make sure it is correct.

I have the equation as: C(x) = (30 + x)(60 - 2x)

Would this be correct to use in this case? The number of total houses to produce would be 30 + x, and the price per house would be 60 - 2x. Is my reasoning correct?

Thanks in advance!
 
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Yes, your revenue function $C(x)$ looks good to me. You have expressed the total revenue as the product of the number of units produced and the revenue per unit. (Yes)

Can you proceed to optimize the revenue function?

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