1. In this problem, we 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 a market research company finds that at a price of p = $20, they would sell x = 42 tiles each month. If they lower the price to p = $10, then more people would purchase the tile, and they can expect to sell x = 52 tiles in a month’s time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. Hint: Write an equation using two points in the form (x,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 result you found from part a. into the equation R = xp to find the revenue equation. Provide your answer in simplified form.
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 $300, and the supplier’s cost for a set of tile is $6 each. Let x represent the number of tile sets.
c. If b represents a fixed cost, what value would represent b?
B would represent $300.
d. Find the cost equation for the tile. Write your answer in the form C = mx + b.
C= 52(6) + 300
C= 312 +300
C= 612
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.
P= R – C
P= 10(52) – 612
P= 520 – 612
P= -92
f. What is the profit made from selling 20 tile sets per month?
P=20(6)- 300
P= 120-300
P= -180
No profit would be made cause you still have to pay the rent, electric, etc.
g. What is the profit made from selling 25 tile sets each month?
P=25(6) – 300
P= 150 -300
P=-150
No profit would be made because u still have to pay fixed cost.
h. What is the profit made from selling no tile sets each month? Interpret your answer.
There would be no profit made from selling no tiles. The company would actually be in the red because they have to still pay the 300 dollars in fixed costs
i. Use trial and error to find the quantity of tile sets per month that yields the highest profit.
The highest profit would be yield by the more sets of tile that is sold. Selling anything more than 23 sets would yield a 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.
2. The break even values for a profit model are the values for which you earn $0 in profit. Use the equation you created in question one to solve P = 0, and find your break even values.
3. In 2002, Home Depot’s sales amounted to $58,200,000,000. In 2006, its sales were $90,800,000,000.
a. Write Home Depot’s 2002 sales and 2006 sales in scientific notation.
2002 sales- 5.82 x 10^10
2006 sales- 9.08 x 10^10
You can find the percent of growth in Home Depot’s sales from 2002 to 2006 by following these steps:
• Find the increase in sales from 2002 to 2006.
• Find what percent that increase is of the 2002 sales.
b. What was the percent growth in Home Depot’s sales from 2002 to 2006? Do all your work by using scientific notation.
I got some of the answers. Can u tell me if I am figuring them correctly? And help with the ones I don't have....PLEASE!?!?
Sure! I'd be happy to help you with the questions you have and check if your answers are correct. Let's go through each question one by one:
1a. To find the demand equation in the form p = mx + b, we need to use the two points given: (42, 20) and (52, 10). We can use the slope-intercept form of a linear equation:
p = mx + b
Using the formula for slope, m = (p2 - p1) / (x2 - x1), substituting the values:
m = (10 - 20) / (52 - 42) = -1
Then, substituting one of the points into the equation, we can solve for b:
20 = -1(42) + b
b = 20 + 42 = 62
So, the demand equation is p = -x + 62.
1b. To find the revenue equation, we substitute the demand equation (from 1a.) into the formula R = xp:
R = x(-x + 62)
R = -x^2 + 62x
So, the revenue equation is R = -x^2 + 62x.
c. You are correct, b represents the fixed cost, which in this case is $300.
d. To find the cost equation, we can use the formula C = mx + b:
C = 6x + 300
So, the cost equation is C = 6x + 300.
e. To find the profit equation, we subtract the cost equation (from d.) from the revenue equation (from 1b.):
P = R - C
P = (-x^2 + 62x) - (6x + 300)
P = -x^2 + 56x - 300
So, the profit equation is P = -x^2 + 56x - 300.
f. To find the profit made from selling 20 tile sets per month, we substitute x = 20 into the profit equation:
P = -(20)^2 + 56(20) - 300
P = -400 + 1120 - 300
P = 420 - 300
P = 120
g. To find the profit made from selling 25 tile sets each month, we substitute x = 25 into the profit equation:
P = -(25)^2 + 56(25) - 300
P = -625 + 1400 - 300
P = 475 - 300
P = 175
h. To find the profit made from selling no tile sets each month, we substitute x = 0 into the profit equation:
P = -(0)^2 + 56(0) - 300
P = 0 - 0 - 300
P = -300
i. To find the quantity of tile sets per month that yields the highest profit, we need to find the vertex of the parabolic profit equation. The x-value of the vertex gives the quantity of tile sets per month that yields the highest profit. We can use the formula x = -b / (2a), where a = -1 and b = 56:
x = -56 / (2(-1))
x = 56 / 2
x = 28
So, selling 28 tile sets per month would yield the highest profit.
j. To find the profit earned from selling 28 tile sets per month, we substitute x = 28 into the profit equation:
P = -(28)^2 + 56(28) - 300
P = -784 + 1568 - 300
P = 484 - 300
P = 184
k. To find the price to sell the tile sets at to realize a profit of 184, we substitute x = 28 into the demand equation (from 1a.):
p = -x + 62
p = -(28) + 62
p = 34
So, to realize a profit of 184, the tile sets should be sold at a price of $34.
2. To find the break-even values, we set the profit equation (from e.) to zero:
0 = -x^2 + 56x - 300
Solving this quadratic equation will give you the break-even values.
3a. You correctly wrote the sales amounts for Home Depot in scientific notation.
3b. To find the percent growth in Home Depot's sales from 2002 to 2006, we can use the formula:
Percent Growth = (New Value - Old Value) / Old Value * 100
Using scientific notation, we have:
Percent Growth = ((9.08 x 10^10) - (5.82 x 10^10)) / (5.82 x 10^10) * 100
Please let me know if you need further assistance with any specific questions or calculations!