Rates of Change and Differential Equations — Topic Review

← Rates of Change and Differential Equations

This review covers all four lessons: Related Rates of Change, Separable Differential Equations, First-Order Linear Differential Equations, and Applications of Differential Equations. Questions are exam-style and increase in difficulty.

Mixed Review Questions

1. Fluency

   Q1 — Related Rates

   A spherical balloon is being inflated. When the radius is 6 cm, the radius is increasing at 0.5 cm/s. Find the rate of increase of the volume at that instant. (V = (4/3)πr³)

2. Fluency

   Q2 — Related Rates

   A 5-metre ladder leans against a vertical wall. The base slides away at 0.3 m/s. Find the rate at which the top slides down when the base is 3 m from the wall.

3. Fluency

   Q3 — Separable DE

   Solve the separable DE dy/dx = 2xy, given y(0) = 3.

4. Fluency

   Q4 — Separable DE

   Solve dy/dx = y² sin x, given y(0) = 1.

5. Fluency

   Q5 — First-Order Linear DE

   Solve dy/dx + 5y = 10, finding the general solution.

6. Understanding

   Q6 — Related Rates

   Water drains from a conical tank (apex down) of half-angle 30° at a rate of 2 m³/min. Find the rate at which the water depth decreases when the depth is 3 m. (V = (1/3)πr²h, and r = h tan 30° = h/√3.)

7. Understanding

   Q7 — Separable DE with Initial Condition

   Solve dP/dt = 0.04P(200 − P). Find P(t) given P(0) = 20. What is lim_t→∞ P(t)?

8. Understanding

   Q8 — First-Order Linear DE

   Solve dy/dx + (1/x)y = x² + 1, for x > 0. Give the general solution.

9. Understanding

   Q9 — Newton’s Law of Cooling

   A metal bar at 800°C is plunged into water at 20°C. After 30 seconds the bar is at 400°C. How long until the bar reaches 50°C?

10. Understanding

    Q10 — Mixing Problem

    A 100 L tank initially contains 20 g of salt dissolved in water. Brine with 0.3 g/L flows in at 5 L/min, and the well-mixed solution flows out at 5 L/min. Find Q(t), the amount of salt at time t.

11. Problem Solving

    Q11 — Related Rates: Angle

    A camera films a rocket launched vertically from a point 500 m away. When the rocket is 1200 m high and climbing at 80 m/s, find the rate at which the camera elevation angle θ is increasing.

12. Problem Solving

    Q12 — First-Order Linear DE (Variable Forcing)

    Solve dy/dx − (2/x)y = x³e^x, for x > 0, given y(1) = 0.

13. Problem Solving

    Q13 — Radioactive Decay Chain

    Substance A decays into substance B at rate k_A = 0.02 per year. Substance B decays at rate k_B = 0.05 per year. Initially, A(0) = 100 g and B(0) = 0. Write the DE for B and solve to find B(t).

14. Problem Solving

    Q14 — RL Circuit with Sinusoidal Input

    An RL circuit has L = 1 H, R = 1 Ω, and V(t) = sin t V. Find I(t) given I(0) = 0.

15. Problem Solving

    Q15 — Logistic Growth: Maximum Rate

    A population grows logistically: dP/dt = 0.1P(1 − P/400). Show that the growth rate dP/dt is maximised when P = 200, and find the maximum rate.