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A company knows that the unit cost C and the unit revenue R from the production and sale of x

units are related by 𝐢 =(𝑅^2/206 000)+ 11 849. Find the rate of change of unit revenue when the unit cost is changing at $8/unit and the unit revenue is $4 000

can you explain it more? hehe Thank you

To find the rate of change of unit revenue when the unit cost is changing at $8/unit and the unit revenue is $4,000, we need to take the derivative of the cost equation with respect to the revenue.

Given: C = (R^2 / 206,000) + 11,849
We need to find dR/dC.

First, let's differentiate the equation with respect to R:
dC/dR = (1 / 206,000) * 2R

Now we can calculate the rate of change of unit revenue:

Given:
dC/dR = (1 / 206,000) * 2R = 8 (since the unit cost is changing at $8/unit)
C = 4,000

Now, we can substitute the given values into the equation to find dR/dC:

8 = (1 / 206,000) * 2R

Rearranging the equation:
R = (8 * 206,000) / 2
R = 824,000

Now, to find dR/dC, we can substitute the value of R into the equation dC/dR:

dR/dC = (1 / 206,000) * 2 * 824,000
dR/dC = 16 / 206,000
dR/dC β‰ˆ 0.00007767

Therefore, the rate of change of unit revenue when the unit cost is changing at $8/unit and the unit revenue is $4,000 is approximately 0.00007767.

To find the rate of change of unit revenue, we need to find the derivative of the unit cost function with respect to the unit revenue.

1. Start by finding the derivative of the unit cost function 𝐢 with respect to the unit revenue 𝑅. Let's denote the derivative as d𝐢/d𝑅.

d𝐢/d𝑅 = (1/206000) * 2𝑅

2. Since we're given that the unit cost is changing at $8/unit (d𝐢/d𝑅 = $8/unit), we can set the derivative equal to 8.

8 = (1/206000) * 2𝑅

3. Solve the equation for 𝑅 by multiplying both sides by 206000/2.

8 * (206000/2) = 𝑅

𝑅 = 824000/2 = 412000

4. Now that we have the value of 𝑅, we can find the rate of change of unit revenue (d𝑅/d𝐢) when the unit revenue is $4000.

The rate of change of unit revenue (d𝑅/d𝐢) is the reciprocal of the derivative of the unit cost function with respect to unit revenue.

d𝑅/d𝐢 = 1 / (d𝐢/d𝑅)

= 1 / ((1/206000) * 2𝑅)

= (206000/2) / 2𝑅

= 103000 / 𝑅

d𝑅/d𝐢 = 103000 / 412000

Simplifying further, we get:

d𝑅/d𝐢 = 0.25

Therefore, when the unit cost is changing at $8/unit and the unit revenue is $4000, the rate of change of unit revenue is 0.25.

dC/du = R/103000 dR/du

so plug in your numbers and solve for dR/du