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Question 476556: 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).
b. 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?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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).
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this is a little difficult to model, but it can be done as follows:
the demand equation tells you the demand for a given value of the price.
the independent variable is the price.
the dependent variable is the demand.
the facts that you are given are as follows:
if we allow p to be equal to the price and d to be equal to the demand, then:
when p = 20, d = 42
when p = 10, d = 52
the equation we will be dealing with will be a straight line equation in the form of p = mx + b.
actually, this is technically correct.
the equation we will be dealing with is:
d = mp + b
the demand is equal to m times the price + b
m is the slope
b is the d intercept which means that b is the value of d when p is equal to 0.
they actually threw you off when they gave you p = mx + b, because that would be a price equation rather than a demand equation.
anyway, our equation is:
d = mp + b
since this is a straight line equation, m is the slope and b is the d intercept.
to find the slope we need to find the change in d divided by the change in p.
you are given 2 points on the line.
they are:
when p1 = 20, d1 = 42
when p2 = 10, d2 = 52
the change in demand would be equal to d2 - d1 which would be equal to 10.
the corresponding change in price would be equal to p2 - p1 which would be equal to -10.
the slope of the line is 10/-10 which is equal to -1.
your demand equation becomes:
d = -p + b
now you need to find b which is the d intercept which is the value of d when p is equal to 0.
you find this by replacing d and p with their values for one of the points in your line.
let's take d = 42 when p = 20
d = -p + b becomes:
42 = -20 + b
add 20 to both sides of this equation and you get:
b = 62
that the value of d when p = 0.
you now have your complete equation.
it is:
d = -p + 62
if you want to graph this equation, then you need to make d = y and p = x and you get:
y = -x + 62
a graph of this equation looks like this:
now:
x is the price and y is the demand.
when the price is equal to 0, then the demand is equal to 62.
when the price is equal to 10, then the demand is equal to 52.
when the price is equal to 20, then the demand is equal to 42.
when the price is equal to 62, then the demand is equal to 0.
i drew horizontal lines at y = 42 and y = 52 to show you that the value of x will be equal to 20 when y = 42, and the value of x will be equal to 10 when y = 52.
remember that y is equal to the demand and x is equal to the price.
this occurred when we made y equal to d and x equal to p.
here's the graph again with the y = 42 and 52 lines showing. just drop a perpendicular from each intersection to find the value of x for that intersection.
it's hard to see on the graph, but you can easily solve it using the equation.
the eqution is:
y = -x + 62
when x = 10, y = 52
when x = 20, y = 42
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b. 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.
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not sure what you're asking here.
the equation for the total cost of a product would be the fixed cost plus the incremental cost.
the fixed cost is an amount of money that has to be paid regardless of the number of units that are produced.
the incremental cost is the additional cost per unit.
if the company allocates $300 to the fixed cost of a product, and the incremental cost for that product is $7.00 per tile, then the total cost for producing x amounts of tile would be:
y = 7x + 300
if the company builds 0 tiles, it will cost 300.
if the company builds 1 tile, it will cost 307.
if the company builds 2 tiles, it will cost 314.
etc.
Assuming that $300 is the fixed cost, then the equation is what I showed above, which is:
y = 7x + 300.
the total cost for building x products is 300 + 7*x.
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c. If b represents a fixed cost, what value would represent b?
not sure which b you're talking about here.
in the general slope intercept form of the equation of a straight line, the equation is:
y = mx + b
b would represent the y intercept which is the value of y when x is equal to 0.
if you are talking about the value of the fixed cost in question b, then the value of b would be equal to $300 which is the fixed cost.
it is also the value of y when x is equal to 0 in the equation of y = 7x + 300.
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hope this helps.
the hardest part of this was determine the demand equation. that's because the equation that was shown as an example threw me off.
here's a reference on demand questions from the tutors at algebra.com so you can see for yourself what the demand equation is all about.
http://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.187579.html
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