Answer:
$990 per unit.
Step-by-step explanation:
The Unit Cost (C) and Unit Revenue R from the production and sale of x units are related by the function:
[tex]C=\dfrac{R^2}{220000} +3221[/tex]
We are required to find the rate of change of revenue per unit when the cost per unit is changing by $9 and the revenue is ​$1,000.
Rate of Change of C,
[tex]\dfrac{dC}{dt}=\dfrac{R}{110000} \dfrac{dR}{dt} \\\dfrac{dC}{dt}=\$9, R=\$1000\\9=\dfrac{1000}{110000} \dfrac{dR}{dt}\\\dfrac{dR}{dt}=9 \div \dfrac{1000}{110000}\\=9 X \dfrac{110000}{1000}\\\dfrac{dR}{dt}=\$990 $ per unit$[/tex]
The Revenue is changing at a rate of $990 per unit.
Cost and revenue can be represented as functions
The rate of change of revenue per unit is 990
The cost function from revenue is given as:
[tex]\mathbf{C(R) = \frac{R^2}{220000} + 3221}[/tex]
Differentiate the function with respect to time
[tex]\mathbf{C' = 2 \times \frac{R}{220000} \times R'}[/tex]
[tex]\mathbf{C' = \frac{R}{110000} \times R'}[/tex]
When the change in cost per time is $9, the revenue is 1,000.
So, we have:
[tex]\mathbf{9 = \frac{1000}{110000} \times R'}[/tex]
Make R' the subject
[tex]\mathbf{R' = \frac{110000}{1000} \times 9}[/tex]
Divide
[tex]\mathbf{R' = 110 \times 9}[/tex]
Multiply
[tex]\mathbf{R' = 990}[/tex]
Hence, the rate of change of revenue per unit is 990
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