The revenue obtained from the sale of q units of a product is given by :
R= 30q-(0.3q^2)
A) How fast does R change with respect to q when:
q=10
B) Find the relative rate of change of R, and to the nearest percent, find the percentage rate of change of R.
dR/dq = 30 - .6q
so when q = 10
dR/dq = 30 - .6(10) = 24
Oh! So, then the relative rate is: R1/R = 30-.6q/30q-(0.3q^2) ?
Which is 9%?
To find the rate of change of R with respect to q, we need to take the derivative of the revenue function R with respect to q. Let's do that step by step:
A) How fast does R change with respect to q when q = 10?
Step 1: Take the derivative of R with respect to q:
dR/dq = d/dq (30q - 0.3q^2)
Step 2: Apply the power rule: For a term of the form ax^n, the derivative is n * ax^(n-1):
dR/dq = 30 - 0.6q
Step 3: Substitute q = 10 into the derivative to find the rate of change:
dR/dq = 30 - 0.6(10)
= 30 - 6
= 24
Therefore, when q = 10, the rate of change of R with respect to q is 24 units.
B) To find the relative rate of change of R, we need to determine the percentage rate of change of R. Here's how:
Step 1: Compute the relative rate of change as the absolute value of the derivative divided by the value of R:
Relative rate of change = |(dR/dq) / R|
Step 2: Substitute q = 10 into the revenue function R:
R = 30(10) - 0.3(10)^2
= 300 - 0.3(100)
= 300 - 30
= 270
Step 3: Plug the values into the formula:
Relative rate of change = |(24/270)|
Step 4: Determine the percentage rate of change by multiplying the relative rate of change by 100:
Percentage rate of change = (|24/270|) * 100
To the nearest percent, the percentage rate of change of R is 9%.