An inverted cone has a height of 15 mm and a radius of 16mm. The volum

f480forever2rz 2021-11-29 Answered
An inverted cone has a height of 15 mm and a radius of 16mm. The volume of the inverted cone is decreasing a rate of 534 cubic mm per second, with the height begin held constant. What is the rate of change of the radius, in mm per second, when the radius is 6 mm?
Round your answer to the nearest hundreth. (Do not include any units in your answer
Remember that the volume of a cone is \(\displaystyle={V}=\frac{{1}}{{3}}\pi{r}^{{{2}}}{h}\)

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Expert Answer

Tamara Donohue
Answered 2021-11-30 Author has 9044 answers
Step 1
The volume of a cone is\(\displaystyle={V}=\frac{{1}}{{3}}\pi{r}^{{{2}}}{h}\)
The volume of the inverted cone is decreasing at a rate of 534 cubic mm per second, with the height begin held constant.
\(\displaystyle={d}\frac{{V}}{{\left.{d}{t}\right.}}=\frac{{1}}{{3}}/\pi{h}.{d}\frac{{{r}^{{{2}}}}}{{\left.{d}{t}\right.}}\) as h is held constant and hence \(\displaystyle={d}\frac{{h}}{{\left.{d}{t}\right.}}={0}\)
Step 2
Find the formula for the rate of change of the volume, in mm per second.
\(\displaystyle={\frac{{{d}{V}}}{{{\left.{d}{t}\right.}}}}={\frac{{{1}}}{{{3}}}}\pi{h}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({r}^{{{2}}}\right)}\)
\(\displaystyle={\frac{{{1}}}{{{3}}}}\pi{h}{\left({2}{r}\right)}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({r}\right)}\)
\(\displaystyle={\frac{{{2}}}{{{3}}}}\pi{r}{h}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({r}\right)}\)
Step 3
To find the rate of changt of the radius, in mm per second, when the radius is 6 mm.
Here \(\displaystyle={d}\frac{{V}}{{\left.{d}{t}\right.}}={534}\) cubic mm per second, \(\displaystyle={h}={15}{m}{m}\) and \(\displaystyle={r}={6}{m}{m}\).
\(\displaystyle={\frac{{{d}{r}}}{{{\left.{d}{t}\right.}}}}={\frac{{{2}}}{{{3}}}}\pi{r}{h}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({r}\right)}\)
\(\displaystyle=\Rightarrow{\frac{{{d}{r}}}{{{\left.{d}{t}\right.}}}}={\frac{{{3}}}{{{2}\pi{r}{h}}}}{\frac{{{d}{V}}}{{{\left.{d}{t}\right.}}}}{\left({r}\right)}\)
\(\displaystyle={\frac{{{3}}}{{{2}\pi\times{6}\times{15}}}}\times{534}\)
\(\displaystyle={2.8329}\)
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