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emmasim [6.3K]
3 years ago
12

the volume V of a right circular cylinder of height 3 feet and radius r feet is V=V(r)=3(pi)(r^2). find the instantaneous rate o

f change of the volume with respect to the radius at r=3.
Mathematics
1 answer:
Aleksandr-060686 [28]3 years ago
3 0

When I see the words "instantaneous rate of change", I have to assume that you're in some stage of pre-calculus in your math class.

The instantaneous rate of change of a function is just its first derivative.

We have the function

                                   V(r) = 3 π r²

and we need its first derivative with respect to ' r '.  That shouldn't be
too hard, because the ' 3 π ' is nothing but constants.

Watch me while I do it slowly for you:

-- The derivative of  ' r² ' with respect to ' r ' is ' 2r '.

-- The derivative of V(r) with respect to ' r ' is  (3 π) times the derivative of  ' r² '.

-- The derivative of V(r) with respect to ' r ' is  (3 π) times (2r) =  <u>6 π r</u> .

The value of the derivative when r=3 is (6 π 3) = 18π = about <em>56.5 feet³/foot .</em>


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\begin{gathered} A=\frac{\theta}{360\degree}\times\pi r^2 \\ Where\text{ r}=AB=6\text{ units} \\ A=9\pi\text{ square units} \end{gathered}

Substitute the values of the variables into the formula above to find the angle, θ, subtended by the sector.

\begin{gathered} 9\pi=\frac{\theta}{360\degree}\times\pi\times6^2 \\ Crossmultiply \\ 9\pi\times360=36\pi\times\theta \\ 3240\pi=36\pi\theta \\ Divide\text{ both sides by 36}\pi \\ \frac{3240\pi}{36\pi}=\frac{36\pi\theta}{36\pi} \\ 90\degree=\theta \\ \theta=90\degree \end{gathered}

To find the length of the arc, s, the formula is

\begin{gathered} s=\frac{\theta}{360\degree}\times2\pi r \\ Where \\ \theta=90\degree \\ r=6\text{ units} \end{gathered}

Substitute the variables into the formula to find the length of an arc, s above

\begin{gathered} s=\frac{\theta}{360}\times2\pi r \\ s=\frac{90\degree}{360\degree}\times2\times\pi\times6 \\ s=\frac{12\pi}{4}=3\pi\text{ units} \\ s=3\pi\text{ units} \end{gathered}

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Based on the calculations, the ratio of AD to DB ​ is equal to 0.6.

<h3>What is a ratio?</h3>

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