Answer:
The price per wind chime that will maximize revenue = $ 315
Step-by-step explanation:
Given - A gift shop sells 160 wind chimes per month at $150 each. the owners estimate that for each $15 increase in price, they will sell 5 fewer wind chimes per month.
To find - Find the price per wind chime that will maximize revenue.
Proof -
Given that,
Total Wind chimes selling = 160
Price of each Wind chime = $150
Now,
Given that, for each $15 increase in price, they will sell 5 fewer wind chimes per month.
So,
Let the price = 150 + 15x
So,
Number of wind Chimes sold per month = 160 - 5x
So,
Total Revenue, R = (150 + 15x)(160 - 5x)
= 24000 - 750x + 2400x - 75x²
= 24000 + 1650x - 75x²
⇒R(x) = 24000 + 1650x - 75x²
Differentiate R with respect to x , we get
R'(x) = 1650 - 150x
Now,
For Maximize Revenue, Put R'(x) = 0
⇒1650 - 150x = 0
⇒150x = 1650
⇒x = 1650/150
⇒x = 11
∴ we get
Price per Wind chime = $ 150 + 15(11)
= $ 150 + 165
= $ 315
So,
The price per wind chime that will maximize revenue = $ 315
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