Question

a company selling widgets has found that the number of items sold ( x ) depends upon the price ( p ) at which theyre sold, according the equation ( x=\frac{90000}{sqrt{4 p + 1}} ).
due to inflation and increasing health benefit costs, the company has been increasing the price by ( $ 2 ) per month. find the rate at which revenue is changing when the company is selling widgets at ( $ 110 ) each.
( square ) dollars per month

Explanation:

Step1: Find the revenue function

Revenue \( R = p\times x\). Given \(x=\frac{90000}{\sqrt{4p + 1}}\), then \(R(p)=\frac{90000p}{\sqrt{4p + 1}}\).

Step2: Differentiate the revenue function using the quotient rule

The quotient rule is \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\). Let \(u = 90000p\), \(u^\prime=90000\) and \(v=(4p + 1)^{\frac{1}{2}}\), \(v^\prime=\frac{4}{2}(4p + 1)^{-\frac{1}{2}}=2(4p + 1)^{-\frac{1}{2}}\).

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Step3: Use the chain - rule

We know that \(\frac{dR}{dt}=\frac{dR}{dp}\times\frac{dp}{dt}\). Given \(\frac{dp}{dt} = 2\).

When \(p = 110\), first find \(4p+1=4\times110 + 1=441\).

Then \(\frac{dR}{dp}=\frac{180000\times110+90000}{441^{\frac{3}{2}}}=\frac{19800000 + 90000}{(21)^{3}}=\frac{19890000}{9261}\)

\(\frac{dR}{dt}=\frac{19890000}{9261}\times2\)

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Answer:

\(4295.43\)