a company selling widgets has found that the number of items sold $x$ depends upon the price $p$ at which…

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

Answer

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)