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 $x = \frac{70000}{\sqrt{4p + 1}}$. due to inflation and increasing health benefit costs, the company has been increasing the price by $\\$3$ per month. find the rate at which revenue is changing when the company is selling widgets at $\\$120$ each. dollars per month

Explanation:

Step1: Write the revenue function

Revenue \( R = p\times x\). Given \(x=\frac{70000}{\sqrt{4p + 1}}\), then \(R(p)=\frac{70000p}{\sqrt{4p + 1}}\).

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 \(u = 70000p\) and \(v=(4p + 1)^{\frac{1}{2}}\).

• \(u^\prime=70000\)
• \(v^\prime=\frac{1}{2}(4p + 1)^{-\frac{1}{2}}\times4 = \frac{2}{\sqrt{4p+1}}\)

Then \(R^\prime(p)=\frac{70000\sqrt{4p + 1}-70000p\times\frac{2}{\sqrt{4p + 1}}}{4p + 1}\).

Simplify \(R^\prime(p)=\frac{70000(4p + 1)-140000p}{(4p + 1)^{\frac{3}{2}}}=\frac{280000p+70000 - 140000p}{(4p + 1)^{\frac{3}{2}}}=\frac{140000p + 70000}{(4p + 1)^{\frac{3}{2}}}\).

Step3: Use the chain - rule

We know that \(\frac{dR}{dt}=\frac{dR}{dp}\times\frac{dp}{dt}\). Given \(\frac{dp}{dt}=3\).

When \(p = 120\), first find \(4p+1=4\times120 + 1=481\).

Then \(R^\prime(p)=\frac{140000\times120+70000}{481^{\frac{3}{2}}}=\frac{16800000+70000}{481^{\frac{3}{2}}}=\frac{16870000}{481\sqrt{481}}\).

\(\frac{dR}{dt}=\frac{16870000}{481\sqrt{481}}\times3\).

Calculate \(481\sqrt{481}=481\times21.9317=10559.14\).

\(\frac{16870000}{10559.14}\times3\approx1597.6\times3 = 4792.8\).

Answer:

\(4792.8\)