A-LevelMathematicsDifferential equations9709/33 · Oct/Nov 2022 · Q109 Marks

A gardener is filling an ornamental pool with water, using a hose that delivers 30 litres of water per minute. Initially the pool is empty. At time t minutes after filling begins the volume of water in the pool is V litres. The pool has a small leak and loses water at a rate of 0.01V litres per minute. The differential equation satisfied by V and t is of the form dV/dt = a - bV.

(a)Write down the values of the constants a and b.[1]
(b)Solve the differential equation and find the value of t when V = 1000.[6]
(c)Obtain an expression for V in terms of t and hence state what happens to V as t becomes large.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

B1 a = 30 and b = 0.01
M1 Separate variables and integrate one side
A1 FT Obtain terms -100ln(30–0.01V) and t, or equivalent FT their a and b. + A1 FT
M1 Evaluate a constant, or use t = 0, V = 0 as limits, in a solution containing terms c ln(30-0.01V) and dt where cd ≠ 0
A1 Obtain solution 100ln30-100ln(30−0.01V)=t, or equivalent
A1 Substitute V = 1000 and obtain answer t = 40.5
B1 OE Obtain V = 3000(1-e⁻⁰˙⁰¹ᵗ)
B1 State that V approaches 3000

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.