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Related Rates - Worksheet 10 - 12th Grade

Hard

Use derivative signs and stated geometric constraints to interpret functions and rates.

hard: Use another set to practice explaining and checking your answers.

15 problems · 30-35 minutes · Skills: derivatives, related rates, calculus

About This Worksheet

This hard-level worksheet contains 15 problems designed for 12th Grade students practicing related rates. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Manipulate algebraic expressions and understand functions, limits and derivative notation.

  • Use derivative signs and stated geometric constraints to interpret functions and rates.
  • Check your result using the method in the example before moving to the next question.

How to Solve: Worked Example

Example Problem: A spherical balloon has radius 2 cm, increasing at 5 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
Solution Steps:
  1. Differentiate with respect to time: dV/dt = 4πr²(dr/dt).
  2. Substitute the instantaneous radius and its rate of change.
Answer: 80π cm³/s
Tip: Check your result using the method in the example before moving to the next question.
1
A spherical balloon has radius 1 cm, increasing at 1 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
2
A spherical balloon has radius 6 cm, increasing at 1 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
3
A spherical balloon has radius 4 cm, increasing at 2 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
4
A spherical balloon has radius 4 cm, increasing at 3 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
5
A spherical balloon has radius 9 cm, increasing at 1 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
6
A spherical balloon has radius 5 cm, increasing at 2 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
7
A spherical balloon has radius 4 cm, increasing at 5 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
8
A spherical balloon has radius 4 cm, increasing at 1 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
9
A spherical balloon has radius 3 cm, increasing at 1 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
10
A spherical balloon has radius 3 cm, increasing at 5 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
11
A spherical balloon has radius 1 cm, increasing at 5 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
12
A spherical balloon has radius 1 cm, increasing at 4 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
13
A spherical balloon has radius 9 cm, increasing at 5 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
14
A spherical balloon has radius 7 cm, increasing at 5 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.
15
A spherical balloon has radius 3 cm, increasing at 2 cm/s. Find dV/dt at this instant. Use V = (4/3)πr³ and leave π exact.

Answer Key

1. 4π cm³/s2. 144π cm³/s3. 128π cm³/s4. 192π cm³/s5. 324π cm³/s6. 200π cm³/s7. 320π cm³/s8. 64π cm³/s9. 36π cm³/s10. 180π cm³/s11. 20π cm³/s12. 16π cm³/s13. 1620π cm³/s14. 980π cm³/s15. 72π cm³/s

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What skills does this applications of derivatives worksheet cover?

This worksheet focuses on derivatives, related rates, calculus. It is designed for 12th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

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How many problems are on this worksheet?

This worksheet contains 15 problems. It typically takes 30-35 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.