A new software company wants to start selling DVDs with their
product. The manager notices that when the price for a DVD is 16
dollars, the company sells 134 units per week. When the price is
27 dollars, the number of DVDs sold decreases to 89 units per week.
Answer the following questions:
A. Assume that the demand curve is linear. Find the demand,
q
, as
a function of price,
p
.
Answer:
q
=
That is incorrect. The linear function is (-13/6)p + 476/3.
First, we need to find the slope of the demand curve. The slope is equal to the change in quantity divided by the change in price.
slope = (change in quantity)/(change in price)
= (89 - 134) / (27 - 16)
= -45 / 11
Now, we can use the point-slope form of a linear equation to find the demand function:
q - 134 = slope * (p - 16)
Substituting the values we have:
q - 134 = -(45/11) * (p - 16)
Simplifying the equation:
q - 134 = -(45/11) * p + (45/11) * 16
q = -(45/11) * p + (45/11) * 16 + 134
The demand function is:
q = -(45/11) * p + 198.18
To find the demand, q, as a function of price, p, we can use the given data points to determine the equation of a straight line (demand curve) that represents the relationship between price and quantity demanded.
Let's use the two data points provided:
When the price is $16, the company sells 134 units per week. This can be represented as the coordinates (16, 134).
When the price is $27, the company sells 89 units per week. This can be represented as the coordinates (27, 89).
We can use these two points to determine the equation of the demand curve using the two-point form of a linear equation:
(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)
Substituting the values:
( q - 134) / ( p - 16) = (89 - 134) / (27 - 16)
Simplifying the equation:
( q - 134) / ( p - 16) = -45 / 11
Cross-multiplying:
11( q - 134) = -45( p - 16)
Expanding:
11q - 1504 = -45p + 720
Rearranging the equation to represent demand as a function of price:
11q = -45p + 2224
Dividing both sides by 11:
q = (-45/11)p + 202.18
Therefore, the demand, q, as a function of price, p, is given by:
q = (-45/11)p + 202.18
If q is a linear function of p, then
q = mp + b
We know from the given data
134 = 16m + b
89 = 27m + b
45 = -11m
m = -0.2444
b = 137.9111
q = -0.2444p + 137.9111
or, roughly,
q = -1/4 p + 138