Suppose that for a company manufacturing calculators, the cost, revenue, and profit equations are given by
C = 70,000 + 20x
R = 300x - (x^2 / 20)
P = R - C
where production output in 1 week is x calculators. If production is increasing at a rate of 600 calculators per week when production output is 5,000 calculators. Find the rate of increase (decrease) in cost, revenue, and profit.
Well, well, well, it seems like we're diving into the world of numbers. Don't worry, I'll crunch them for you with a hint of humor.
Let's find the rate of increase (or decrease) in cost, revenue, and profit. To do that, we need to find the derivatives of the cost, revenue, and profit equations with respect to the production output (x).
For the cost equation, C = 70,000 + 20x, let's take the derivative:
dC/dx = 20
So, the rate of change in cost is a constant 20. It's like that friend who always costs you $20, no matter what.
Now, for the revenue equation, R = 300x - (x^2 / 20):
dR/dx = 300 - (x / 10)
When x = 5000, the rate of change in revenue is:
dR/dx = 300 - (5000 / 10) = 250
So, the rate of change in revenue is 250. It's like making $250 by picking up loose change on the street. Not bad, right?
Finally, let's find the rate of change in profit. We know that P = R - C:
dP/dx = dR/dx - dC/dx
Substituting the values we found earlier:
dP/dx = 250 - 20 = 230
So, the rate of change in profit is 230. It's like finding $230 in your pocket after doing laundry. A nice little bonus!
I hope this funny little calculation brings a smile to your face and helps you understand the problem.
To find the rate of increase or decrease in cost, revenue, and profit, we need to differentiate the cost, revenue, and profit equations with respect to the production output (x).
Given the equations:
C = 70,000 + 20x
R = 300x - (x^2 / 20)
P = R - C
Let's differentiate each equation with respect to x:
dC/dx = d(70,000 + 20x)/dx
dR/dx = d(300x - (x^2 / 20))/dx
dP/dx = d(R - C)/dx
Differentiating each equation, we get:
dC/dx = 20
dR/dx = 300 - (x / 10)
dP/dx = dR/dx - dC/dx
We are given that production is increasing at a rate of 600 calculators per week when production output is 5,000 calculators. So, x = 5,000 and dx/dt = 600.
To find the rate of increase (decrease) in cost, revenue, and profit, substitute x = 5,000 into the derivatives:
dC/dx = 20
dR/dx = 300 - (5,000 / 10) = 300 - 500 = -200
dP/dx = dR/dx - dC/dx = -200 - 20 = -220
Therefore, the rate of increase (decrease) in cost is 20 calculators per week, the rate of increase (decrease) in revenue is -200 calculators per week, and the rate of increase (decrease) in profit is -220 calculators per week.
To find the rate of increase or decrease in cost (C), revenue (R), and profit (P), we first need to calculate the derivatives of the cost, revenue, and profit equations with respect to x.
The cost equation is given by C = 70,000 + 20x. To find the rate of increase in cost, we differentiate this equation with respect to x:
dC/dx = d(70,000 + 20x)/dx
The derivative of a constant term (70,000) is zero, and the derivative of 20x is simply 20. Therefore, the rate of increase in cost is constant at 20.
The revenue equation is given by R = 300x - (x^2 / 20). To find the rate of increase in revenue, we differentiate this equation with respect to x:
dR/dx = d(300x - (x^2 / 20))/dx
The derivative of 300x is 300, and the derivative of (x^2 / 20) is (2x / 20) = (x / 10). Therefore, the rate of increase in revenue is constant at 300 - (x / 10).
The profit equation is given by P = R - C. To find the rate of increase in profit, we differentiate this equation with respect to x:
dP/dx = d(R - C)/dx
Since we already know the rates of increase in cost and revenue, we can substitute them into the profit equation:
dP/dx = dR/dx - dC/dx
Substituting the derivatives we found previously:
dP/dx = (300 - (x / 10)) - 20
Simplifying:
dP/dx = 280 - (x / 10)
Now, to find the specific rate of increase (decrease) in cost, revenue, and profit when production output is 5,000 calculators and production is increasing at a rate of 600 calculators per week, we substitute x = 5,000 and calculate the values:
Rate of increase in cost = dC/dx = 20
Rate of increase in revenue = dR/dx = 300 - (5,000 / 10) = 300 - 500 = -200
Rate of increase (decrease) in profit = dP/dx = 280 - (5,000 / 10) = 280 - 500 = -220
Therefore, the rate of increase in cost is 20 calculators per week, the rate of increase in revenue is -200 calculators per week (or a decrease of 200 calculators per week), and the rate of increase (decrease) in profit is -220 calculators per week.
just plug and chug:
dC/dt = 20 dx/dt = 20*600
dR/dt = 300 dx/dt - x/10 dx/dt = (300-500)(600)
dP/dt = dR/dt - dC/dt