#1) In this problem, we will analyze the profit found for sales of decorative tiles. A demand equation (sometimes called a demand curve) shows how much money people would pay for a product depending on how much of that product is available on the open market. Often, the demand equation is found empirically (through experiment, or market research).
(a) Suppose that a market research company finds that at a price of p = $30, they would sell x = 36 tiles each month. If they lower the price to p = $15, then more people would purchase the tile, and they can expect to sell x = 41 tiles in a month’s time. Write two ordered pairs in the form (x, p) that represent the market research. (1 pt) Find the equation of the line for the demand equation using the two points in #1. Write your answer in the form P=mx+b. (Show all of your work) Hint: 1st, Find slope (m) using the two ordered pairs you created. Replace both x variables and both y variables in the formula (p-p_1 )=m(x-x_1 ) to get the value of m. 2nd, Find the intercept by replacing p and x in the equation y=mx+b with one ordered pair and m with the slope. Then solve for b. 3rd , Write the equation by replacing m and b in the equation y=mx+b with the values you found. You answer will have an x and a p variable. (This is a skill learned in MAT116)(2 pts) Slope: m= Intercept: b= Equation: P=
A company’s revenue is the amount of money that comes in from sales, before business costs are subtracted. For a single product, you can find the revenue by multiplying the quantity of the product sold, x, by the demand equation, p.
(b) Substitute the results you found from #2 into the equation R=xP to find the revenue equation. Provide your answer in simplified form. (DO NOT REPLACE THE X, only the P is Replaced with your equation from #2.) (2 pts)
The costs of doing business for a company can be found by adding fixed costs, such as rent, insurance, and wages, and variable costs, which are the costs to purchase the product you are selling. The portion of the company’s fixed costs allotted to this product is $600, and the supplier’s cost for a set of tile is $3 each. Let x represent the number of tile sets. If b represents a fixed cost, what value would represent b? (1 pt)
(c ) If m represents the cost per tile set, what value would represent m? (1 pt)
(d) Find the cost equation for the tile. Write your answer in the form C=mx+b. (2 pts)
The profit made from the sale of tiles is found by subtracting the costs from the revenue.
(e) Find the Profit Equation (P) by substituting your equations for R (#3) and C (#6) in the equation . Simplify the equation. (2 pts)
(f) What is the profit made from selling 16 tile sets per month? Show your work and state you answer as a complete sentence. (2 pts)
(g) What is the profit made from selling 20 tile sets each month? Show your work and state you answer as a complete sentence. (2 pts)
(h) What is the profit made from selling no tile sets each month? Show your work, and explain what this means for the business in a complete sentence. (2 pts)
(i) Use trial and error to find the quantity of tile sets per month that yields the highest profit. Show all your work for each number of tile sets you try. State your answer as a complete sentence including the number of tiles you should sell to make the most profit, and the amount of profit that you would make. Hint: Use your answers to #8 and #9 to start, then keep trying different numbers until the price starts to drop. (3 pts)
(j) What price would you sell the tile sets at to realize this profit? Show your work and state you answer as a complete sentence. (2 pts)
#2) The break even values for a profit model are the values for which you earn $0 in profit. 13. Use the profit equation you created in #7 to solve P = 0, and find your break even values. (Hint: there will be two break even values) Show all your work and state your answer as a complete sentence. (3 pts)
#3) In 2002, Home Depot’s sales amounted to $57,200,000,000. In 2006, its sales were $91,800,000,000. Write Home Depot’s 2002 sales in scientific notation. (1 pts)
(a) Write Home Depot’s 2006 sales in scientific notation. (1 pts)
(b) What was the percent of growth in Home Depot’s sales from 2002 to 2006? Show all of your work and state your answer as a complete sentence. (The entire problem should be done using scientific notation and your answer needs to be rounded to the nearest hundredth of a %. (3 pts) (Hint: %growth=difference in sales divided by sales to start with multiplied by 100%).
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