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 = $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)).
p – 20 = (10 – 20)/ (52 – 42) x – 42
p = (-1) x (x-42) + 20
p = -1x + 62
Is this right?
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.
How would I do this?
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 = 300
d.Find the cost equation for the tile. Write your answer in the form C = mx + b.
C = m(6) + 300
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.
a) First let's look at the demand equation. They want it in a form
p = mx + b, where p is price in $ and x is the number sold. The facts they give you are
20 = 42 m + b
10 = 52 m + b
10 = -10 m
m = -1
20 = -42 + b
b = 62
p = 62 - x
(or x = 62 - p)
Yes, your equation is right, but you don't need the -1 in front of x.
b), For the Revenue (p*x)
R = (62-x)*x = 62x - x^2
c)yes, in the cost equation the constant "b" term is 300
d)C = 6x + 300
(Remember that x is the number sold).
e) Profit = R - C = 62x - x^2 -6x -300
= 56x -300 -x^2
This equation can be used to determine the amount of sales that will provide maximum possible profit (x = 28 sold). The unit price that will achieve that maximum profit is 62-28 = $34
Thanks. I have just one more question. How would I find the profit made from selling 20 tile sets per month?
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 P = R-C . Simplify the equation.
f.What is the profit made from selling 20 tile sets per month?
Can you just help me set up the equation?
I think he did part e for you above.
For part f, plug in the 20 for x .
f. What is the profit made from selling 20 tile sets per month?
i. Use trial and error to find the quantity of tile sets per month that yields the highest profit.
this is hard
Brain cramp!
Cheaters
4. A customer wants to make a teepee in his backyard for his children. He plans to use lengths of PVC plumbing pipe for the supports on the teepee, and he wants the teepee to be 12 feet across and 8 feet tall (see figure). How long should the pieces of PVC plumbing pipe be?
To find the Profit Equation, we need to substitute the revenue equation (R = xp) and the cost equation (C = mx + b) into the equation Profit = Revenue - Cost.
Given that the revenue equation is R = -x + 62 and the cost equation is C = 6x + 300, we can substitute them into the Profit Equation.
Profit = R - C
Profit = (-x + 62) - (6x + 300)
Profit = -x + 62 - 6x - 300
Profit = -7x - 238
Therefore, the Profit Equation is P = -7x - 238.