Exercise · ZW-RELATED-RATES-002 · ★★☆ Intermediate
The radius of a sphere grows at \(\frac{1}{2}\) cm/s. How fast is the volume growing when \(r = 3\) cm?
Choose one option.
Show answer
- D. \(18 \pi\)
Explanation
From \(V=\dfrac{4}{3}\pi r^{3}\) we get \(\dfrac{dV}{dt}=4\pi r^{2}\dfrac{dr}{dt}=4\pi(3)^{2}\cdot\frac{1}{2}=18 \pi\) cm³/s.
Review the concept: Related rates
In a related rates problem several quantities change with time \(t\) and are linked by an equation. The equation is differentiated with respect to \(t\) (chain rule) and the known values are substituted.
\[A=\pi r^{2}\ \Rightarrow\ \frac{dA}{dt}=2\pi r\,\frac{dr}{dt}\]
More practice problems: related rates
- ZW-RELATED-RATES-001 The radius of a circle grows at 2 cm/s. How fast is the area growing when r = 5 cm?
- ZW-RELATED-RATES-003 A 13 m ladder leans against a wall. Its base slides away from the wall at 2 m/s. How fast is the to…
- ZW-RELATED-RATES-004 The edge of a cube grows at 1 cm/s. How fast is the volume growing when the edge is 4 cm?
- ZW-RELATED-RATES-005 The side of a square grows at 3 cm/s. How fast is its area growing when s = 2 cm?
- ZW-RELATED-RATES-006 The radius of a circle grows at 1 cm/s. How fast is its circumference growing? (in cm/s)
- ZW-RELATED-RATES-007 A rectangle has length x=4 m and width y=3 m. The length grows at 1 m/s and the width shrinks at 2…
- ZW-RELATED-RATES-008 The radius of a sphere is 2 cm and grows at (1)/(2) cm/s. How fast is its surface area growing? (in…
- ZW-RELATED-RATES-009 The side of a square grows at 2 cm/s. How fast is its perimeter growing? (in cm/s)