Maximizing Profit: Calculating Optimal Widget Sales | Widget Company

[CanaBra]

Profit function!

Problem: A company is considering selling a new line of widgets at a special promotional price of $49. If the cost of setting up the manufacturing process is $2500, an each widget costs $32.35 to produce, how many should the company sell in order to realize a profit of at least $7000? If the cost of manufacturing increases by $2.75, how many widgets must be sold at the original selling price to obtain the same profit?

Here is what I have done:

R(x) = 49x
C(x)=$2500+$32.35x
P(x)= -2500+16.65x

If P(x) =$7000, then
7000= -2500+16.65x
7000+2500=16.65x
9500 =16.65x
9500/16.65 = x
x = 570.57...

If costs increases by $2.75, then
C(x) = 2500 + 35
P(x)-C(x) = -2500 + 14x
If P(x) = $7000, then

7000= -2500+14x
7000+2500 = 14x
9500/14 = x
x=678.57...

Can anyone double check this answer for me or tell me if I am completely lost?
Thank you in advance

 

[CFDFEAGURU]

Looks correct to me. The equationf produces a higher result when the cost to manufacture the widget increases.

Thanks
Matt

 

[Architeuthis Dux]

.
Your calculations and approach seem to be correct. To double check, we can plug in the values into the profit function and see if we get the desired profit of $7000. When x=570.57, P(x) = -2500+16.65(570.57) = $7000.0055, which is very close to the desired profit. Similarly, when x=678.57, P(x) = -2500+14(678.57) = $7000.01, which is also very close to the desired profit.

To summarize, the company should sell 570.57 widgets at the promotional price of $49 to achieve a profit of at least $7000. If the cost of manufacturing increases by $2.75, the company would need to sell 678.57 widgets at the original selling price of $49 to obtain the same profit. It's important for the company to consider the cost of manufacturing and the selling price in order to maximize their profit.