(1 pt) A new software company wants to start selling DVDs with their
product. The manager notices that when the price for a DVD is 15
dollars, the company sells 132 units per week. When the price is
27 dollars, the number of DVDs sold decreases to 84 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=
B. Write the revenue function, as a function of price.
Answer: R(p)=
C. Find the price that maximizes revenue. Hint: you may sketch the
graph of the revenue function. Round your answer to the closest dollar.
Answer:
D. Find the maximum revenue, i.e., the revenue that corresponds to your answer for the preceding question.
Answer:
A. To find the demand, q, as a function of price, p, we can use the two given data points (price, quantity). The demand curve is assumed to be linear, so we can use the formula for a straight line: y = mx + b.
Using the two points (15, 132) and (27, 84), we can find the slope, m, of the line:
m = (change in y) / (change in x) = (84 - 132) / (27 - 15) = -48 / 12 = -4.
Now we have the slope, we can use one of the points to find the y-intercept, b. Let's use (15, 132):
132 = -4(15) + b
132 = -60 + b
132 + 60 = b
b = 192.
Therefore, the demand, q, as a function of price, p, is given by the equation:
q = -4p + 192.
Answer: q = -4p + 192.
B. The revenue function is given by the product of price and quantity: R(p) = p * q. Substituting the expression for q from part A, we have:
R(p) = p * (-4p + 192)
R(p) = -4p^2 + 192p.
Answer: R(p) = -4p^2 + 192p.
C. To find the price that maximizes revenue, we need to find the vertex of the parabolic revenue function. In this case, the coefficient of the quadratic term, -4p^2, is negative, which means the parabola opens downward and the vertex represents the maximum point.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a is the coefficient of p^2 and b is the coefficient of p in the revenue function.
In this case, a = -4 and b = 192, so:
p = -192 / (2 * -4)
p = -192 / -8
p = 24.
Rounded to the closest dollar, the price that maximizes revenue is $24.
Answer: $24.
D. To find the maximum revenue, we substitute the value of p = 24 into the revenue function:
R(p) = -4(24)^2 + 192(24)
R(p) = -4(576) + 4608
R(p) = -2304 + 4608
R(p) = 2304.
The maximum revenue is $2304.
Answer: $2304.