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.
I think I know how to do the actual math, but I'm unsure how to set up the problem.
Thanks for your help!
To solve this problem, we need to follow a few steps to set up the problem correctly:
1. Express the demand function: Based on the given information, we know that the price is a linear function of q (the quantity). Let's denote the price as P and the quantity as q. We have two data points: when the price is $70, the demand is 0, and when the price is $60, the demand is 100. We can use these points to find the equation of the demand function.
Using the slope-intercept form, the equation becomes:
P(q) = mx + b,
where m represents the slope of the line and b represents the y-intercept.
Given two data points (q1, P1) = (0, 70) and (q2, P2) = (100, 60), we can calculate the slope:
m = (P2 - P1) / (q2 - q1)
= (60 - 70) / (100 - 0)
= -10/100
= -0.1
Now, plug in one of the points into the equation to find the y-intercept (b):
70 = -0.1(0) + b
70 = b
So, the demand function equation is:
P(q) = -0.1q + 70.
2. Express revenue as a function of q: Revenue is calculated by multiplying the quantity produced and sold (q) by the price per unit (P). So, the revenue function (R) can be expressed as:
R(q) = P(q) * q
= (-0.1q + 70) * q
= -0.1q^2 + 70q.
3. Determine the rate of change of revenue with respect to time: To find the rate of change of revenue with respect to time (t), we need to relate q, the quantity produced per day, to t, the time in days.
Given that the daily production and sales is 20 items, we have q = 20. To find the rate of change of revenue with respect to time, we need to differentiate R(q) with respect to t (using chain rule). Since R(q) is a function of q, we need to use the chain rule as follows:
(dR/dt) = (dR/dq) * (dq/dt)
The first term, (dR/dq), represents the rate of change of revenue with respect to quantity (which we already found to be R(q) = -0.1q^2 + 70q). The second term, (dq/dt), represents the rate of change of quantity with respect to time, which is the given production rate of 25 units per day.
So, we have:
(dR/dt) = (dR/dq) * (dq/dt)
= (-0.1q^2 + 70q) * 25
= -0.1(20)^2 + 70(20)
= -0.1(400) + 1400
= -40 + 1400
= 1360.
Therefore, the rate of change of revenue with respect to time is 1360 (dollars per day).