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{80000}{\sqrt{2p + 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 $300 each. question help: video submit question

Explanation:

Step1: Find the revenue function

Revenue \( R = p\times x\). Given \(x=\frac{80000}{\sqrt{2p + 1}}\), then \(R(p)=\frac{80000p}{\sqrt{2p + 1}}\).

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 \(u = 80000p\), \(u^\prime=80000\); \(v=(2p + 1)^{\frac{1}{2}}\), \(v^\prime=\frac{2}{2}(2p + 1)^{-\frac{1}{2}}=(2p + 1)^{-\frac{1}{2}}\).

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Step3: Substitute \(p = 300\) into the derivative

When \(p = 300\), \(R^\prime(300)=\frac{80000\times300+80000}{(2\times300 + 1)^{\frac{3}{2}}}=\frac{80000\times(300 + 1)}{(601)^{\frac{3}{2}}}\).

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Since \(\frac{dp}{dt}=2\) (price is increasing at a rate of \(\$2\) per month), by the chain rule \(\frac{dR}{dt}=R^\prime(p)\times\frac{dp}{dt}\).

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Answer:

\(3291\)