Question
upstudy study bank question image url

The volume of a right circular cylinder of radius \( r \) and height \( h \) is \( V=\pi r^{2} h \). \( \begin{array}{ll}\text { (a) Assume that } r \text { and } h \text { are functions of } t \text {. Find } V^{\prime}(t) \text {. } \\ \text { (b) Suppose that } r=e^{6 t} \text { and } h=e^{-6 t} \text {. Use part (a) to find } V^{\prime}(t) \text {. } \\ \begin{array}{ll}\text { (c) Does the volume of the cylinder of part (b) increase or decrease as } t \text { increases? } \\ \text { (a) Find } V^{\prime}(t) \text {. Choose the correct answer below. } \\ \text { A. } V^{\prime}(t)=2 \pi r(t) h(t) r^{\prime}(t) & \text { B. } V^{\prime}(t)=2 \pi r(t) h(t) r^{\prime}(t)+\pi(r(t))^{2} h^{\prime}(t)\end{array} \\ \text { C. } V^{\prime}(t)=\pi(r(t))^{2} h^{\prime}(t) & \text { D. } V^{\prime}(t)=2 \pi r(t) h(t) h^{\prime}(t)+\pi(r(t))^{2} r^{\prime}(t)\end{array} \)

Ask by Daniel Harper. in the United States
Mar 17,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

**(a) Find \( V'(t) \):** \[ V'(t) = 2\pi r(t) h(t) r'(t) + \pi r(t)^2 h'(t) \] **(b) Substitute \( r(t) = e^{6t} \) and \( h(t) = e^{-6t} \):** \[ V'(t) = 6\pi e^{6t} \] **(c) The volume increases as \( t \) increases.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To find \( V^{\prime}(t) \), we can apply the product rule since the volume \( V = \pi r^2 h \) is a product of the functions \( r(t) \) and \( h(t) \). This leads us to the formula: \[ V^{\prime}(t) = \pi \left(2r(t) h(t) r^{\prime}(t) + r(t)^2 h^{\prime}(t)\right) \] So, the correct answer is B: \( V^{\prime}(t) = 2 \pi r(t) h(t) r^{\prime}(t) + \pi (r(t))^2 h^{\prime}(t) \). Next, using the given functions \( r = e^{6t} \) and \( h = e^{-6t} \), we find their derivatives: \( r^{\prime}(t) = 6e^{6t} \) and \( h^{\prime}(t) = -6e^{-6t} \). Plugging these into our expression for \( V^{\prime}(t) \), we have: \[ V^{\prime}(t) = 2\pi e^{6t} e^{-6t} (6e^{6t}) + \pi (e^{6t})^2 (-6e^{-6t}) \] This simplifies to: \[ V^{\prime}(t) = 12\pi (6) + \pi (36e^{6t} e^{-6t})(-6) \] The overall behavior of the volume changes based on the values of \( \frac{6}{36} \) and \( -6 \), revealing that it increases when combining the growth of the base and the height. Ultimately, as \( t \) increases, the volume indeed increases!

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy