Question

Hi can I need some help for the number 1

Answer 1

Profit is the difference between the revenue/income and the cost.

`Profit=Income-Cost`

In the problem, the following costs are:

Design and fixed production cost = $260,000

Advertising Campaign = $490,000

Manufacturing cost per pair = $25

Royalty payment per pair sold = $5

If x = the number of pairs sold, we can say that the total royalty cost would be 5x and the total manufacturing cost is 25x.

Hence, the total cost of all sunglasses is:

`\begin{gathered} TC=260,000+490,000+25x+5x \\ TC=750,000+30x \end{gathered}`

On the other hand, the income per sunglasses sold is only $105 since it is sold at a 30% discount. See the solution below.

`\begin{gathered} 150*0.30=45 \\ 150-45=105 \end{gathered}`

This means that if x = the number of sunglasses sold, the total income is 105x.

`TI=105x`

So, combining the two functions: function for total cost and function for total income in the profit formula, we get:

`\begin{gathered} Profit=Income-Cost \\ Profit=105x-(750,000+30x) \\ Profit=105x-30x-750,000 \\ Profit=75x-750,000 \end{gathered}`

1. To summarize, we have two functions:

Total Cost = 750,000 + 30x

Total Income = 105x

2. For the company to hit break-even, the total cost and income must be equal.

`\begin{gathered} TC=TI \\ 750,000+30x=105x \end{gathered}`

From this, we can now solve for x pairs of sunglasses sold.

`\begin{gathered} Subtract\text{ }30x\text{ }on\text{ }both\text{ }sides\text{ }of\text{ }the\text{ }equation. \\ 750,000+30x-30x=105x-30x \\ 750,000=75x \\ Divide\text{ }both\text{ }sides\text{ }by\text{ }75. \\ (750,000)/(75)=(75x)/(75) \\ 10,000=x \end{gathered}`

Therefore, 10,000 pairs of sunglasses must be sold for the company to break even.

3. As shown above, the formula for profit is 75x - 750,000.

If we want to have a profit of more than 500,000, then we can write the profit equation as:

`75x-750,000>500,000`

From this, we can solve for x pairs of sunglasses.

a. Add 750,000 on both sides of the equation.

`75x>1,250,000`

b. Divide both sides by 75.

`\begin{gathered} x>16,666.67 \\ x>16,667pairs\text{ }of\text{ }sunglasses \end{gathered}`

Therefore, at least 16, 667 pairs of sunglasses must be sold for the company to make a profit of more than 500,000.