Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
so the sand pile looks more or less like the one in the image below.
let's check its circumference first. We know that dC/dt = 5π
[tex]C=2\pi r\implies \cfrac{dC}{dt}=2\pi \cfrac{dr}{dt}\implies 5\pi =2\pi \cfrac{dr}{dt}\implies \cfrac{5\pi }{2\pi }=\cfrac{dr}{dt}\implies \boxed{\cfrac{5}{2}=\cfrac{dr}{dt}}[/tex]
hmmm when the circumference C = 8π, what's the radius "r"?
[tex]8\pi =2\pi r\implies \cfrac{8\pi }{2\pi }=r\implies 4=r[/tex]
now let's take the derivative to the volume equation, and check what dV/dt is
[tex]V=\cfrac{r^3}{3}\implies V=\cfrac{1}{3}r^3\implies \cfrac{dV}{dt}=\stackrel{\textit{chain rule}}{\cfrac{3}{3}r^2\cdot \cfrac{dr}{dt}}\implies \cfrac{dV}{dt}=r^2\cdot \cfrac{5}{2}\implies \cfrac{dV}{dt}=\cfrac{5r^2}{2} \\\\\\ \stackrel{\textit{when }C=8\pi \textit{ we know that }r=4}{\cfrac{dV}{dt}=\left. \cfrac{5r^2}{2}\right|_{r=4}} \implies \cfrac{dV}{dt}=\cfrac{5(4)^2}{2}\implies \cfrac{dV}{dt}=40[/tex]
The rate of change of the volume is mathematically given as
dV/dt=40
What is the rate of change in the volume of the pile, in cubic feet per hour?
Question Parameter(s):
The volume V of the pile is given by V = r^3/3,
at a constant rate of 5 pi feet per hour
Generally, the equation for the Circumference is mathematically given as
[tex]C=2\pi r\\\\\ \frac{5}{2}=\frac{dr}{dt}}[/tex]
Therefore
8π=2πr
r=8π/2π
r=4
In conclusion
[tex]V=\cfrac{r^3}{3}[/tex]
Hence
[tex]\frac{dV}{dt}=\cfrac{5r^2}{2} \\\\\\ \frac{dV}{dt}=\cfrac{5(4)^2}{2} \frac{dV}{dt}[/tex]
dV/dt=40
Read more about volume
https://brainly.com/question/1578538
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.