A new software company wants to start selling DVDs with their product. The manager notices that when the price for a DVD is 20 dollars, the company sells 139 units per week. When the price is 34 dollars, the number of DVDs sold decreases to 91 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 information given. We know that when the price is $20, the company sells 139 units per week, and when the price is $34, the number of DVDs sold decreases to 91 units per week.
To find the demand as a function of price, we can use the formula for a linear equation, which is:
q = m * p + b
Where:
- q is the quantity demanded,
- p is the price,
- m is the slope of the demand curve, and
- b is the y-intercept of the demand curve.
To find the slope, we can use the two points given: (20, 139) and (34, 91).
The slope, m, can be calculated as:
m = (y2 - y1) / (x2 - x1)
= (91 - 139) / (34 - 20)
= -48 / 14
= -24 / 7
Now we can substitute the slope into the equation of the demand curve:
q = (-24/7) * p + b
To find the y-intercept, b, we can substitute one of the points into the equation. Let's use the point (20, 139):
139 = (-24/7) * 20 + b
139 = -480/7 + b
To find b, we can add 480/7 to both sides:
139 + 480/7 = b
(973 + 480)/7 = b
1453/7 = b
Now we can substitute the value of b back into the equation:
q = (-24/7) * p + (1453/7)
Therefore, the demand as a function of price is:
q = (-24/7) * p + (1453/7)
B. To write the revenue function, we multiply the price, p, by the quantity demanded, q:
R(p) = p * q
Substituting the demand equation into the revenue equation:
R(p) = p * ((-24/7) * p + (1453/7))
Simplifying:
R(p) = (-24/7) * p^2 + (1453/7) * p
Therefore, the revenue function as a function of price is:
R(p) = (-24/7) * p^2 + (1453/7) * p
C. To find the price that maximizes revenue, we need to find the vertex of the parabolic revenue function.
The revenue function is in the form: R(p) = (-24/7) * p^2 + (1453/7) * p
The maximum or minimum of a quadratic function occurs at the vertex. The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a = -24/7 and b = 1453/7.
x = -(1453/7) / (2 * (-24/7))
= -(1453/7) / (-48/7)
= (1453/7) / (48/7)
= (1453/7) * (7/48)
= 1453/48
To find the price, we can substitute the x-coordinate into the revenue function:
R(1453/48) = (-24/7) * (1453/48)^2 + (1453/7) * (1453/48)
Now we can calculate the price that maximizes revenue.
D. To find the maximum revenue, we need to evaluate the revenue function at the price that maximizes it.
R(1453/48) = (-24/7) * (1453/48)^2 + (1453/7) * (1453/48)
Simplify the expression to find the maximum revenue.