Related Rates — Practice Problems and Worked Solutions

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16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find dA/dt.

   $$A=\pi r^2,r=3,\frac{dr}{dt}=2$$

   Answer: $12\pi$

   Step-by-step solution

   1. $$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$$
   2. $$\text{Substitute the current measurements: }12\pi$$
2. 2.

   Find dA/dt.

   $$A=\pi r^2,r=5,\frac{dr}{dt}=1$$

   Answer: $10\pi$

   Step-by-step solution

   1. $$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$$
   2. $$\text{Substitute the current measurements: }10\pi$$
3. 3.

   Find dV/dt.

   $$V=\frac{4}{3}\pi r^3,r=2,\frac{dr}{dt}=3$$

   Answer: $48\pi$

   Step-by-step solution

   1. $$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$$
   2. $$\text{Substitute the current measurements: }48\pi$$
4. 4.

   Find dV/dt.

   $$V=\frac{4}{3}\pi r^3,r=4,\frac{dr}{dt}=\frac{1}{2}$$

   Answer: $32\pi$

   Step-by-step solution

   1. $$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$$
   2. $$\text{Substitute the current measurements: }32\pi$$
5. 5.

   Find dV/dt.

   $$V=s^3,s=4,\frac{ds}{dt}=2$$

   Answer: $96$

   Step-by-step solution

   1. $$\frac{dV}{dt}=3s^2\frac{ds}{dt}$$
   2. $$\text{Substitute the current measurements: }96$$
6. 6.

   Find dV/dt.

   $$V=s^3,s=6,\frac{ds}{dt}=\frac{1}{3}$$

   Answer: $36$

   Step-by-step solution

   1. $$\frac{dV}{dt}=3s^2\frac{ds}{dt}$$
   2. $$\text{Substitute the current measurements: }36$$
7. 7.

   Find dA/dt.

   $$A=\pi r^2,r=4,\frac{dr}{dt}=3$$

   Answer: $24\pi$

   Step-by-step solution

   1. $$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$$
   2. $$\text{Substitute the current measurements: }24\pi$$
8. 8.

   Find dA/dt.

   $$A=\pi r^2,r=7,\frac{dr}{dt}=2$$

   Answer: $28\pi$

   Step-by-step solution

   1. $$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$$
   2. $$\text{Substitute the current measurements: }28\pi$$
9. 9.

   Find dA/dt.

   $$A=\pi r^2,r=6,\frac{dr}{dt}=\frac{1}{2}$$

   Answer: $6\pi$

   Step-by-step solution

   1. $$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$$
   2. $$\text{Substitute the current measurements: }6\pi$$
10. 10.

    Find dV/dt.

    $$V=\frac{4}{3}\pi r^3,r=3,\frac{dr}{dt}=2$$

    Answer: $72\pi$

    Step-by-step solution

    1. $$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$$
    2. $$\text{Substitute the current measurements: }72\pi$$
11. 11.

    Find dV/dt.

    $$V=\frac{4}{3}\pi r^3,r=5,\frac{dr}{dt}=1$$

    Answer: $100\pi$

    Step-by-step solution

    1. $$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$$
    2. $$\text{Substitute the current measurements: }100\pi$$
12. 12.

    Find dV/dt.

    $$V=\frac{4}{3}\pi r^3,r=6,\frac{dr}{dt}=\frac{1}{3}$$

    Answer: $48\pi$

    Step-by-step solution

    1. $$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$$
    2. $$\text{Substitute the current measurements: }48\pi$$
13. 13.

    Find dV/dt.

    $$V=s^3,s=3,\frac{ds}{dt}=4$$

    Answer: $108$

    Step-by-step solution

    1. $$\frac{dV}{dt}=3s^2\frac{ds}{dt}$$
    2. $$\text{Substitute the current measurements: }108$$
14. 14.

    Find dV/dt.

    $$V=s^3,s=5,\frac{ds}{dt}=2$$

    Answer: $150$

    Step-by-step solution

    1. $$\frac{dV}{dt}=3s^2\frac{ds}{dt}$$
    2. $$\text{Substitute the current measurements: }150$$
15. 15.

    Find dV/dt.

    $$V=s^3,s=8,\frac{ds}{dt}=\frac{1}{4}$$

    Answer: $48$

    Step-by-step solution

    1. $$\frac{dV}{dt}=3s^2\frac{ds}{dt}$$
    2. $$\text{Substitute the current measurements: }48$$
16. 16.

    Find dV/dt.

    $$V=s^3,s=2,\frac{ds}{dt}=5$$

    Answer: $60$

    Step-by-step solution

    1. $$\frac{dV}{dt}=3s^2\frac{ds}{dt}$$
    2. $$\text{Substitute the current measurements: }60$$

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