Q 11. If the volume and equator of a s... [FREE SOLUTION]

Chapter 3: Q 11. (page 299)

If the volume and equator of a sphere are functions of time, what is the relationship between the rate of change of the volume of the sphere and the rate of change of the equator of the sphere?

Short Answer

Expert verified

The required relationship between the rate of change of the volume of the sphere and the rate of change of the equator of the sphere is $V^{'} (t) = 2 E^{'} (t) {(r (t))}^{2}$ .

Step by step solution

Step 1. Given information and formulas

The volume of sphere is $V = \frac{4}{3} 蟺 r^{3}$

And equator of sphere is $E = 2 蟺 r$

Therefore, $V = V (t)$ and $E = E (t)$

Step 2. Explanation

Now $V (t) = \frac{4}{3} 蟺 {(r (t))}^{3}$

Differentiate both sides with respect to $t$

$鈬� V^{'} (t) = 4 蟺 {(r (t))}^{2} r^{'} (t) 鈰� 鈰� (i)$

Also $E (t) = 2 蟺 r (t)$

Differentiate both sides with respect to $t$

$鈬� E^{'} (t) = 2 蟺 r^{'} (t) 鈰� 鈰� (i i)$

From $(i)$ and $(i i)$

$鈬� \frac{1}{4 蟺 {(r (t))}^{2}} V^{'} (t) = \frac{1}{2 蟺} E^{'} (t)$

$鈬� V^{'} (t) = 2 E^{'} (t) {(r (t))}^{2}$

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91影视

If the volume and equator of a sphere are functions of time, what is the relationship between the rate of change of the volume of the sphere and the rate of change of the equator of the sphere?

Short Answer

Step by step solution

Step 1. Given information and formulas

Step 2. Explanation

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