Apc.2.8.1 ap vertical circular cylinder related rates

In summary, the vertical circular cylinder's lateral surface area increases at a rate of 4π(r+h) ft/sec when both the height and radius increase at a constant rate of 2 ft/sec.
  • #1
karush
Gold Member
MHB
3,269
5
$\tiny{2.8.1}$

The vertical circular cylinder has radius r ft and height h ft.
If the height and radius both increase at the constant rate of 2 ft/sec,
Then what is the rate at which the lateral surface area increases?
\een
$\begin{array}{ll}
a&4\pi r\\
b&2\pi(r+h)\\
c&4\pi(r+h)\\
d&4\pi rh\\
e&4\pi h
\end{array}$
ok here is my setup
\begin{array}{lll}
\textit{given rates}
&\dfrac{dr}{dt}=2 \quad \dfrac{dh}{dt}=2
&(1)\\ \\
\textit{surface area eq}
&2\pi rh
&(2)\\ \\
\end{array}
so far
 
Last edited:
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  • #2
$\dfrac{dA}{dt} = 2\pi\left(r \cdot \dfrac{dh}{dt} + h \cdot \dfrac{dr}{dt} \right)$
 
  • #3
$\dfrac{dA}{dt} = 2\pi\left(r \cdot \dfrac{dh}{dt} + h \cdot \dfrac{dr}{dt} \right)
=2\pi(2r+2h)=4\pi(r+h)$
 

FAQ: Apc.2.8.1 ap vertical circular cylinder related rates

1. What is the equation for the related rates problem involving an AP vertical circular cylinder?

The equation for this problem is dV/dt = πr²dh/dt, where dV/dt represents the rate of change of volume, r represents the radius of the cylinder, and dh/dt represents the rate of change of height.

2. How do you set up the related rates problem for an AP vertical circular cylinder?

To set up the problem, you need to identify the variables given and the ones that need to be found. Then, use the equation dV/dt = πr²dh/dt and substitute in the given values. Finally, solve for the unknown variable using algebraic manipulation.

3. What does the AP vertical circular cylinder related rates problem involve?

This problem involves finding the rate of change of either the volume or the height of a vertical circular cylinder, given the rate of change of the other variable and the radius of the cylinder.

4. What are some real-life applications of the AP vertical circular cylinder related rates problem?

This problem can be applied in situations where a cylindrical tank is being filled or drained, and the rate of change of volume or height needs to be determined. It can also be used in problems involving the expansion or contraction of cylindrical objects due to temperature changes.

5. What are some tips for solving the AP vertical circular cylinder related rates problem?

Some tips include drawing a diagram to visualize the problem, carefully labeling all the given and unknown variables, and using the correct formula for the problem. It is also important to pay attention to units and convert them if necessary. Additionally, double-checking the solution and ensuring it makes sense in the given context is crucial.

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