The profit made from the sale of tiles is found by subtracting the costs from the revenue.
e. Find the Profit Equation by substituting your equations for R and C in the equation . Simplify the equation.
f. What is the profit made from selling 20 tile sets per month?
g. What is the profit made from selling 25 tile sets each month?
h. What is the profit made from selling no tile sets each month? Interpret your answer.
i. Use trial and error to find the quantity of tile sets per month that yields the highest profit.
j. How much profit would you earn from the number you found in part i?
k. What price would you sell the tile sets at to realize this profit? Hint: Use the demand equation from part a.
To answer these questions, we need to start by understanding the profit equation and the given information about revenue (R) and costs (C).
a. The revenue equation given in part a needs to be substituted into the profit equation. The profit equation is:
Profit = Revenue - Costs
Substituting the equations for R and C, we get:
Profit = (p - 0.02q)q - (10000 + 300q)
Simplifying this equation will give us the profit equation:
Profit = -0.02q^2 + pq - 10000
Now we can proceed to answering each part of the question:
b. To find the profit from selling 20 tile sets per month, we substitute q = 20 into the profit equation:
Profit = -0.02(20)^2 + p(20) - 10000
c. To find the profit from selling 25 tile sets per month, we substitute q = 25 into the profit equation:
Profit = -0.02(25)^2 + p(25) - 10000
d. To find the profit from selling no tile sets per month, we substitute q = 0 into the profit equation:
Profit = -0.02(0)^2 + p(0) - 10000
Interpreting the answer for selling no tile sets per month means we would have a negative profit equal to $10,000. This indicates a loss of $10,000, which makes sense since there are costs involved even if no items are sold.
e. To find the quantity of tile sets per month that yields the highest profit, we can use trial and error or calculus techniques such as finding the vertex of the quadratic equation.
For trial and error method, you can try substituting different values for q into the profit equation and calculate the profit each time. Record the profits obtained for different values of q and identify the value of q that yields the highest profit.
f. Once you have determined the quantity (q) that yields the highest profit, substitute this value back into the profit equation to find out how much profit you would earn from selling that quantity of tile sets per month.
Profit = -0.02q^2 + pq - 10000
g. To determine the price at which you should sell the tile sets to realize the profit calculated in part f, you need to use the demand equation given in part a:
p = 60 - 0.04q
Substitute the value of q obtained in part i into the demand equation to get the corresponding price.
It is worth noting that the profit, quantity, and price values obtained in answering these questions are dependent on the equations and information given. If there are any changes to the values or equations, the results could also change. Always ensure to double-check the calculations and equations used.