a company selling widgets has found that the number of items sold ( x ) depends upon the price ( p ) at…
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}} ).\ndue 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.\n( square ) dollars per month
Answer
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}}).
[ \begin{align*} R^\prime(p)&=\frac{90000\sqrt{4p + 1}-90000p\times2(4p + 1)^{-\frac{1}{2}}}{4p + 1}\ &=\frac{90000(4p + 1)-180000p}{(4p + 1)^{\frac{3}{2}}}\ &=\frac{360000p+90000 - 180000p}{(4p + 1)^{\frac{3}{2}}}\ &=\frac{180000p + 90000}{(4p + 1)^{\frac{3}{2}}} \end{align*} ]
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)
[ \begin{align*} \frac{dR}{dt}&=\frac{39780000}{9261}\ & = 4295.43 \end{align*} ]
Answer:
(4295.43)