Mathematical models: exam review Exam review One question at a time

The amount M mg of a medicine in a patient's blood, t hours after it is taken, is modelled by \(M=M_0e^{-kt}\). [7]

1. (a) The half-life of the medicine is 6 hours. Find the value of k. Give your answer to 3 significant figures. [2]

   • M1 for e^(−6k) = 1/2, so k = ln 2 ÷ 6
   • A1 0.116

2. (b) After 10 hours the amount is 48 mg. Find \(M_0\). Use the unrounded value of k. [2]

   • M1 for 48 = M₀e^(−k×10), so M₀ = 48e^(10k)
   • A1 152

3. (c) Find the time for the amount to fall to 10 mg. Give your answer to 3 significant figures. [2]

   • M1 for solving 152e^(−kt) = 10, using the unrounded values
   • A1 23.6

4. (d) State one assumption made by this model. [1]

   • R1 The amount is lost at a rate proportional to the amount present, and no more medicine is taken.

A learner's typing speed S (words per minute) after t weeks of practice is modelled by \(S=a+b\ln t\). After 2 weeks the speed is 35. After 10 weeks the speed is 62. [6]

1. (a) Use the information to find a and b. Give each value to 3 significant figures. [3]

   • M1 for two equations: a + b ln 2 = 35 and a + b ln 10 = 62
   • M1 for solving the pair, using the calculator
   • A1 a = 23.4, b = 16.8

2. (b) Use the model to predict S when t = 20. [1]

   • A1 73.6

3. (c) Find the value of t when S = 80. [2]

   • M1 for solving 23.4 + 16.8 ln t = 80, or t = e^((80 − a)/b)
   • A1 29.2

A cup of tea is made at 92°C in a room at 21°C. Its temperature T°C, t minutes later, is modelled by \(T=B+Ae^{-kt}\). [7]

1. (a) Write down the value of B and the value of A. [2]

   • A1 B = 21 (the temperature of the surroundings)
   • A1 A = 71 (92 − 21)

2. (b) After 5 minutes the temperature is 70°C. Find the value of k. Give your answer to 3 significant figures. [2]

   • M1 for 70 = 21 + 71e^(−5k), so e^(−5k) = 49/71
   • A1 0.0742

3. (c) Find the time for the temperature to reach 40°C. Give your answer to 3 significant figures. [2]

   • M1 for solving 21 + 71e^(−kt) = 40
   • A1 17.8

4. (d) Find the half-life of the difference between the temperature and the surroundings. Give your answer to 3 significant figures. [1]

   • A1 9.35 (ln 2 ÷ k)

A tank is filled with water. The volume V litres after t minutes is modelled by \(V=pt\) for \(0\le t\le 6\), and \(V=qt^2\) for \(t>6\). After 3 minutes the volume is 120 litres. [5]

1. (a) Find the value of p. [1]

   • A1 p = 120 ÷ 3 = 40.0

2. (b) The model is continuous at t = 6. Find the value of q. [2]

   • M1 for pt = qt² at t = 6: 240 = q×6²
   • A1 q = 6.67

3. (c) Find the time at which the volume reaches 1000 litres. Give your answer to 3 significant figures. [2]

   • M1 for qt² = 1000, using t > 6
   • A1 12.2

In each part, the function is continuous at the point where the rule changes. [8]

1. (a) \(f(x)=\begin{cases}3x^2+a & x\le 2\\ 5x-1 & x>2\end{cases}\). Find the value of a. [2]

   • M1 for 3×2² + a = 5×2 − 1
   • A1 a = −3

2. (b) \(g(x)=\begin{cases}x^a & x\le 9\\ 2x+9 & x>9\end{cases}\). Find the value of a. Give your answer to 3 significant figures. [3]

   • M1 for 9^a = 2×9 + 9 = 27
   • M1 for a ln 9 = ln 27
   • A1 a = 1.50

3. (c) \(h(x)=\begin{cases}e^{ax} & x\le 2\\ 4x+2 & x>2\end{cases}\). Find the value of a. Give your answer to 3 significant figures. [3]

   • M1 for e^(2a) = 4×2 + 2 = 10
   • M1 for 2a = ln 10
   • A1 a = 1.15

In each part, give answers that are not exact to 3 significant figures. [9]

1. (a) The time t hours taken to empty a tank is inversely proportional to the number n of pumps used, so \(t=\dfrac{a}{n}\). With 4 pumps the tank is emptied in 9 hours. Find the value of a, and the time when n = 6. [3]

   • M1 for substituting: 9 = a/4
   • A1 a = 36
   • A1 time = 36 ÷ 6 = 6

2. (b) Describe what the model predicts for the time as n gets close to 0, and say why this is not reasonable. [1]

   • A1 t increases without limit as n approaches 0; there cannot be 0 pumps, or part of a pump

3. (c) A drink is put into a fridge that is kept at 4 °C. Its temperature T °C after t minutes is modelled by \(T=c+k(0.92)^t\), where c and k are constants. At t = 0 the temperature is 25 °C. Write down the equation of the horizontal asymptote of the graph of T, and find the value of k. [3]

   • A1 T = 4 (the curve levels off at c)
   • M1 for substituting t = 0: 25 = 4 + k
   • A1 k = 21

4. (d) Find the time for the temperature to reach 8 °C. Give your answer to 3 significant figures. [2]

   • M1 for solving 4 + 21(0.92)^t = 8
   • A1 19.9

In each part, the amount is modelled by \(y=y_0e^{kt}\), where t is in years. [5]

1. (a) The value of a machine falls by 12% each year. Find the value of k. Give your answer to 3 significant figures. [2]

   • M1 for \(e^k=1-0.12\), or k = ln 0.88
   • A1 k = -0.128

2. (b) A fossil has 35% of the carbon-14 that a living organism has. The half-life of carbon-14 is 5730 years. Find the age of the fossil. Give your answer to 3 significant figures. [3]

   • M1 for k = −ln 2 ÷ 5730 = -0.000121 (half-life)
   • M1 for solving \(e^{kt}=0.35\)
   • A1 8680 years

A tray is made from a rectangular sheet of metal 50 cm by 30 cm. A square of side x cm is cut from each corner and the sides are folded up. The volume of the tray is \(V=x(50-2x)(30-2x)\) cm³. [3]

1. (a) Write down the domain of x for which V makes sense. [1]

   • A1 the shorter side gives 30 − 2x > 0, so 0 < x < 15

2. (b) Use your calculator to find the greatest volume, and the value of x that gives it. Give your answers to 3 significant figures. [2]

   • M1 for graphing V and finding the maximum with the calculator
   • A1 4100 cm³ at x = 6.07

The height h metres of the sea at a harbour, t hours after midnight, can be modelled by a sinusoidal function. The greatest height is 5.8 m and the least height is 1.2 m. The height repeats every 12.4 hours. [12]

1. (a) Find the amplitude and the mean value of h. [2]

   • A1 amplitude = (5.8 − 1.2) ÷ 2 = 2.30
   • A1 mean = (5.8 + 1.2) ÷ 2 = 3.50

2. (b) The period is 12.4 hours. Find the value of b. Give your answer to 3 significant figures. [2]

   • M1 for b = 2π ÷ 12.4
   • A1 0.507

3. (c) The graph of h first crosses its mean value, going upwards, at t = 2.1. Write down a model of the form \(h=a\sin(b(t-c))+d\). Use b to 3 significant figures. [1]

   • A1 h = 2.30 sin(0.507(t − 2.1)) + 3.50

4. (d) Use your model to find h when t = 5. [2]

   • M1 for substituting t = 5, with the calculator in radians
   • A1 5.79

5. (e) Find the total time, in the first 12.4 hours, for which h is greater than 4. Give your answer to 3 significant figures. [3]

   • M1 for solving h = 4
   • A1 t = 2.53 and t = 7.87
   • A1 5.34 (7.87 − 2.53)

6. (f) Find the first time at which h is a maximum. [2]

   • M1 for a quarter of a period after t = 2.1: 2.1 + 12.4/4
   • A1 5.20

The number P of fish in a lake, t years after fish are introduced, is modelled by \(P=\dfrac{L}{1+Ce^{-kt}}\). The lake can support at most 1200 fish, so L = 1200. Initially there are 150 fish. [12]

1. (a) Find the value of C. [2]

   • M1 for L ÷ (1 + C) = 150 at t = 0
   • A1 C = 7.00

2. (b) After 3 years the number of fish is 420. Find the value of k. Give your answer to 3 significant figures. [3]

   • M1 for 1200 ÷ (1 + 7.00e^(−3k)) = 420
   • M1 for solving for k using the calculator, or by rearranging
   • A1 k = 0.442

3. (c) Find the number of fish after 6 years. [2]

   • M1 for substituting t = 6 with the unrounded k and C
   • A1 804

4. (d) Find the time at which the number of fish is half of 1200. Give your answer to 3 significant figures. [2]

   • M1 for 1200 ÷ (1 + Ce^(−kt)) = 600, or Ce^(−kt) = 1
   • A1 t = ln C ÷ k = 4.40

5. (e) Write down the limiting value of the number of fish as t increases. [1]

   • A1 1200 (L, the carrying capacity)

6. (f) Give one reason why a logistic model is more realistic than an exponential model for this situation. [2]

   • R1 the logistic model levels off at the carrying capacity
   • R1 an exponential model grows without limit, which cannot continue (limited space, food or users)

A scientist studies two cultures. Culture X starts with 900 units and loses 7% of its amount each day. Culture Y starts with 40 units. [8]

1. (a) Write down an expression for the amount of culture X after n days. [1]

   • A1 900 × 0.93^n

2. (b) Find the least whole number of days after which there is less than 100 units of culture X. [3]

   • M1 for 900×0.93^n < 100
   • M1 for n > ln(100/900) ÷ ln 0.93; the inequality turns round because ln 0.93 is negative
   • A1 n = 31 (n > 30.3)

3. (c) The amount of culture Y is modelled by \(Y=B_0e^{kt}\) with \(B_0=40\). It doubles every 12 days. Find the value of k. Give your answer to 3 significant figures. [2]

   • M1 for e^(12k) = 2
   • A1 k = ln 2 ÷ 12 = 0.0578

4. (d) Find the time for the amount of culture Y to reach 900 units. Give your answer to 3 significant figures. [2]

   • M1 for solving 40e^(kt) = 900
   • A1 53.9

A theatre sells n tickets for a show when the ticket price is £p. When the price is £10, 200 tickets are sold. When the price is £16, 128 tickets are sold. The theatre uses a linear model for n in terms of p. [9]

1. (a) Find a model of the form \(n=mp+c\). [3]

   • M1 for the gradient: (128 − 200) ÷ (16 − 10) = -12
   • M1 for substituting a point: 200 = -12(10) + c
   • A1 n = -12p + 320

2. (b) State what the value of m means in this context. [1]

   • A1 Each £1 increase in the ticket price reduces the number of tickets sold by 12

3. (c) The income from ticket sales is \(R=p\,n\) pounds. Find the ticket price that gives the greatest income, and that income. Give your answers to 3 significant figures. [2]

   • M1 for R = p(-12p + 320) and finding the maximum with the calculator
   • A1 p = 13.3, R = 2130

4. (d) Find the range of ticket prices for which the income is more than £2000. Give your answers to 3 significant figures. [2]

   • M1 for solving -12p² + 320p = 2000 (graph R and the line R = 2000)
   • A1 10.0 < p < 16.7

5. (e) State a reasonable domain for p, and say why the model should not be used for a price of £30. [1]

   • A1 n ≥ 0 gives p ≤ 320/12 = 26.7, so 0 ≤ p ≤ 26.7; at £30 n would be negative

A Ferris wheel has its lowest point 2 m above the ground and its highest point 32 m above the ground. It makes one complete turn every 8 minutes. A rider gets on at the lowest point at t = 0, where t is in minutes. The height of the rider above the ground is modelled by \(h=a\cos(bt)+d\) metres. [9]

1. (a) Find the values of a, b and d in the model \(h=a\cos(bt)+d\). Give b to 3 significant figures. [3]

   • M1 for a = −(greatest − least) ÷ 2 = -15, negative because it starts at the lowest point, and d = (32 + 2) ÷ 2 = 17
   • M1 for b = 2π ÷ 8
   • A1 a = -15, b = 0.785, d = 17

2. (b) Find the height of the rider at t = 3. Give your answer to 3 significant figures. [1]

   • A1 h = 17 − 15cos(0.785 × 3) = 27.6

3. (c) Find the total time in one turn for which the height is more than 25 m. Give your answer to 3 significant figures. [3]

   • M1 for solving -15cos(0.785t) + 17 = 25 (graph the model and the line h = 25)
   • A1 t = 2.72 and t = 5.28
   • A1 2.57

4. (d) State what the value of |a| represents in this context. [1]

   • A1 |a| = 15 is the radius of the wheel (half the difference between the greatest and least heights)

5. (e) Give one reason why the model may not be reasonable. [1]

   • A1 The wheel is assumed to turn at a constant speed with no stops; in reality it stops to load riders.