The volume of a sphere with radius $r$ is given by $\frac{4}{3}\pi {{r}^{3}}$ and the surface ar

Question

The volume of a sphere with radius $r$ is given by $\frac{4}{3}\pi {{r}^{3}}$ and the surface area of a sphere with radius $r$ is given by $4\pi {{r}^{2}}$. A meteorite enters the Earth’s atmosphere and starts burning up in such a way that its surface area is decreasing at a constant rate of $100$cm$^{2}$ per second. Assuming the meteorite is always spherical, find the rate at which the radius is changing when the radius is $5$ cm.[2] Another meteorite enters the Earth’s atmosphere and burns up at a rate such that its volume is decreasing at a rate proportional to its surface area. Assuming the meteorite is always spherical, show that the radius decreases at a constant rate.[4]

Accepted Answer

(a) $\frac{\mathrm dr}{\mathrm dt}=-\frac{5}{2\pi}\text{ cm/s}$ (b) $\frac{\mathrm dr}{\mathrm dt}=-k$, where $k>0$, so the radius decreases at a constant rate.

2023 RI P1 Q3

2023 RI P1 Q3

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