Rates of change question (1 Viewer)

em_angel23 · Jun 28, 2009

from grove ext1 exercise 6.2

Q12) The volume of a balloon being inflated is increasing at a constant rate of 115cm3/s. find the rate of increase in its radius when the radius is 3cm.

i realize it isn't that hard but i'm having trouble figuring out the variables. Thanks!

answer: 1.02cm/s

Aquawhite · Jun 28, 2009

We shall assume the balloon is spherical in shape and is totally constant throughout.

thus, Volume of a sphere = $\frac{4}{3}\pi r^{3}$

Now, you are told that the Volume is increasing at a rate of 115cm^3/s = dV/dt. You can figure dV/dr through differentiation.

dV/dt = dV/dr . dr/dt

$115 = 4\pi r^{2}$ . dr/dt -> Where r=3 -> $115 = 36\pi$ . dr/dt

$\therefore\frac{dr}{dt} = \frac{115}{36\pi }\approx 1.017cm/s$

em_angel23 · Jun 28, 2009

Thank you so much

Aquawhite · Jun 28, 2009

You are very welcome.

Drongoski · Jun 28, 2009

$\100dpi Given:\ \ \frac{\mathrm{d} v}{\mathrm{d} t}\ =\ 115\ cm^{3}/s\\ \\ Assuming\ baloon\ a\ sphere:\ v\ =\ \frac{4}{3}\pi r^{3}\\ \\ Now:\ \ \frac{\mathrm{d} v}{\mathrm{d} t}\ =\ \frac{\mathrm{d} v}{\mathrm{d} r}\times \frac{\mathrm{d} r}{\mathrm{d} t}\ \ \ \Rightarrow \ \ \ 115\ =\ 4\pi r^{2}\times \frac{\mathrm{d} r}{\mathrm{d} t}\\ \\ \therefore \ \ \frac{\mathrm{d} r}{\mathrm{d} t}\ =\ \frac{115}{4\pi r^{2}}\ =\ \frac{115}{4\pi \times 3^{2}}\ \ cm/s$

Oh . . . way too late.

em_angel23 · Jun 28, 2009

thank you though! I am always impressed with how helpful everyone is!

Aquawhite · Jun 28, 2009

Drongoski said:
$\100dpi Given:\ \ \frac{\mathrm{d} v}{\mathrm{d} t}\ =\ 115\ cm^{3}/s\\ \\ Assuming\ baloon\ a\ sphere:\ v\ =\ \frac{4}{3}\pi r^{3}\\ \\ Now:\ \ \frac{\mathrm{d} v}{\mathrm{d} t}\ =\ \frac{\mathrm{d} v}{\mathrm{d} r}\times \frac{\mathrm{d} r}{\mathrm{d} t}\ \ \ \Rightarrow \ \ \ 115\ =\ 4\pi r^{2}\times \frac{\mathrm{d} r}{\mathrm{d} t}\\ \\ \therefore \ \ \frac{\mathrm{d} r}{\mathrm{d} t}\ =\ \frac{115}{4\pi r^{2}}\ =\ \frac{115}{4\pi \times 3^{2}}\ \ cm/s$

Oh . . . way too late.

Hehe, I had to have this one! I love the practical applications of calculus.. well anything calculus related I am good at and love to solve...

You can have the next one, promise

em_angel23 said:
thank you though! I am always impressed with how helpful everyone is!

Most people are very helpful on BoS. The reason is that we also get practice in answering there questions as well ^_^

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