Question

1.The monthly revenue of a mobile mechanic business is given by R = 1,200p - 9p2​, where p is the hourly charge (in dollars) for the services the business provides. At what charge will the revenue be $14,400 if the price charged must be greater than​ $50? Provide your answer to the nearest dollar.

2. A company produces water bottles where the variable cost per bottle is $3.50 and fixed cost is $20,000 per year. Next year, the company wants the total cost to be $48,000. How many bottles should be made next year?

3. A company manufactures water filters that cost $15 for labour and material, plus $50,000 in fixed costs. If they sell the water filter for $20, how many filters must be sold to break even?

Answer 1

1. The monthly revenue is given by where p is the hourly charge(in dollars) for the services the business provides. Now, if the revenue is $14,400 then we have

Dividing both sides by 3 we get

Grouping common terms together and factorizing we get

This gives us the values of p as

Now, the price charged, p, must be greater $50, but we see that

Hence we can discard the first value. Thus, our required answer is

Therefore, for the revenue to be $14,400, if the price charged has to be greater than $50, then the price charged must be $120 per hour.

2. Let x bottles be made so that the total cost is $ 48,000. Now, the fixed costs are $20,000 and each bottle costs $3.50 to make. So, we must have

Hence, for the total cost to be $48,000 the company needs to make exactly 8000 bottles.

3. Let x water filters be sold by the company. Then, it's total cost is given by

and if water filters are sold at $20, then the total revenue is given by

Now, for the company to break even, revenue must be equal to costs, so, we have

Therefore, for the company to breakeven, they must sell 10000 water filters.