a. Suppose that a market research company finds that at a price of p = $31, they would sell x = 74 boxes of product each month. If they lower the price to p = $25, then more people would purchase the product, and they can expect to sell x = 80 boxes 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)).
To find the equation of the line for the demand equation in the form p = mx + b, we need to determine the values of m and b.
Given that at a price of p = $31, they sell x = 74 boxes per month, we have the point (74, 31).
Given that at a price of p = $25, they sell x = 80 boxes per month, we have the point (80, 25).
We can use these two points to find the equation of the line.
First, let's calculate the slope, m, using the formula:
m = (p2 - p1) / (x2 - x1)
Substituting the values from the two points, we have:
m = (25 - 31) / (80 - 74)
m = -6 / 6
m = -1
Now that we have the slope, we can use the point-slope form of the equation of a line to find the equation.
The point-slope form is:
y - y1 = m(x - x1)
Substituting the values from one of the points, let's use the first point (74, 31), so x1 = 74 and y1 = 31:
p - 31 = -1(x - 74)
Expanding the equation:
p - 31 = -x + 74
Rearranging the equation and isolating p:
p = -x + 74 + 31
p = -x + 105
Therefore, the equation of the line for the demand equation is:
p = -x + 105.