A company is increasing production at the rate of 25 units per day. The daily demand function is determined by the fact that the price (in dollars) is a linear function of q. At a price of $70, the demand is 0, and 100 items will be demanded at a price of $60. Find the rate of change of revenue with respect to time (in days) when the daily production (and sales) is 20 items.
To find the rate of change of revenue with respect to time, we need to first express the revenue as a function of time.
Revenue is calculated by multiplying the price, p, by the quantity, q, sold.
Let's start by finding the equation of the demand function. We are given that the demand is a linear function of q. At a price of $70, the demand is 0, and at a price of $60, the demand is 100 items.
Using the two points (price, quantity): (70, 0) and (60, 100), we can find the equation of the demand function.
The equation of a linear function can be written as: q = m*p + b, where m is the slope and b is the y-intercept.
Let's solve for m and b.
Using the point (70, 0):
0 = m*70 + b
Using the point (60, 100):
100 = m*60 + b
Solving these two equations simultaneously, we get:
m = -10/7
b = 700
Therefore, the demand function is:
q = (-10/7)*p + 700
Now, we need to find the revenue function.
Revenue (R) = price (p) * quantity (q)
R = p*q
Substituting the demand function into the revenue function, we get:
R = p * ((-10/7)*p + 700)
Now, to find the rate of change of revenue with respect to time (t), we need to differentiate the revenue function with respect to time.
dR/dt = dR/dp * dp/dt
The first part, dR/dp, is the derivative of the revenue function with respect to price.
dR/dp = (-10/7)*p + 700
The second part, dp/dt, is the rate of change of production with respect to time.
Given that the daily production (and sales) is 20 items, the rate of change of production with respect to time is 20 units per day.
So, dp/dt = 20
Finally, we can calculate the rate of change of revenue with respect to time:
dR/dt = dR/dp * dp/dt
= [(-10/7)*p + 700] * 20
Now, substitute the given daily production, q = 20, into the demand function to find the corresponding price, p.
20 = (-10/7)*p + 700
p = (20 - 700) / (-10/7)
p = 70
Now, substitute this price, p = 70, into the rate of change of revenue equation:
dR/dt = [(-10/7)*70 + 700] * 20
= [-100 + 700] * 20
= 600 * 20
= 12,000
Therefore, the rate of change of revenue with respect to time when the daily production is 20 items is 12,000 dollars per day.