MHB Rational function equation problem solving

[laura3827]

Hi I've been tying to figure this out for days. The answer has to be in the form of a rational equation.

John wants to start a padlock company. The production cost of one padlock is 12$. John plans to spend 15 000$ on advertising. He decides to sell each padlock for 20$.

1.Which equation gives the average profit per padlock sold depending on the number of padlocks sold?

2.How many padlocks should John sell if he wants to make an average profit of 5$/padlock?

please help, I'm completely stuck
thanks

 

[MarkFL]

Hello and welcome to MHB, laura3827! (Wave)

1.) Let's first compute the total profit $P$, where $x$ is the number of padlocks sold. Profit is define as revenue $R$ minus costs $C$. If John sells $x$ padlocks each for \$20, then the revenue is:

$$R(x)=20x$$

The cost is made up of a fixed cost of \$15000 for advertising, and a marginal cost of \$12 per padlock, so the cost function is:

$$C(x)=12x+15000$$

Therefore, the profit function is:

$$P(x)=R(x)-C(x)=20x-(12x+15000)=8x-15000$$

And so the profit per padlock sold is the total profit divided by the number of padlocks sold:

$$\frac{P(x)}{x}=\frac{8x-15000}{x}=8-\frac{15000}{x}$$

2.) If there is to be a profit per padlock of \$5, then we need to solve the equation:

$$\frac{P(x)}{x}=5$$

$$8-\frac{15000}{x}=5$$

Can you proceed?

 

[MarkFL]

Just to follow up, we left off with:

$$8-\frac{15000}{x}=5$$

Arrange as:

$$3=\frac{15000}{x}$$

Multiply through by $$\frac{x}{3}\ne0$$:

$$x=5000$$

Thus, John must sell 5,000 padlocks in order to realize a profit of \$5 per padlock.