Mathematical models: exam review Exam review: second paper

The mass M grams of a radioactive isotope in a sample, t days after the sample is prepared, is modelled by \(M=M_0e^{-kt}\). [7]

1. (a) The half-life of the isotope is 8 days. Find the value of k. Give your answer to 3 significant figures. [2]
2. (b) After 12 days the mass is 35 g. Find \(M_0\). Use the unrounded value of k. [2]
3. (c) Find the time for the mass to fall to 5 g. Give your answer to 3 significant figures. [2]
4. (d) State one assumption made by this model. [1]

The height H cm of a seedling t days after planting is modelled by \(H=a+b\ln t\). After 3 days the height is 8. After 12 days the height is 15. [6]

1. (a) Use the information to find a and b. Give each value to 3 significant figures. [3]
2. (b) Use the model to predict H when t = 30. [1]
3. (c) Find the value of t when H = 20. [2]

A metal pan is taken out of an oven at 200°C into a kitchen at 24°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]
2. (b) After 10 minutes the temperature is 120°C. Find the value of k. Give your answer to 3 significant figures. [2]
3. (c) Find the time for the temperature to reach 50°C. Give your answer to 3 significant figures. [2]
4. (d) Find the half-life of the difference between the temperature and the surroundings. Give your answer to 3 significant figures. [1]

A pond is filled with water. The volume V litres after t minutes is modelled by \(V=pt\) for \(0\le t\le 5\), and \(V=qt^2\) for \(t>5\). After 4 minutes the volume is 180 litres. [5]

1. (a) Find the value of p. [1]
2. (b) The model is continuous at t = 5. Find the value of q. [2]
3. (c) Find the time at which the volume reaches 1600 litres. Give your answer to 3 significant figures. [2]

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

1. (a) \(f(x)=\begin{cases}2x^2+a & x\le 3\\ 4x+1 & x>3\end{cases}\). Find the value of a. [2]
2. (b) \(g(x)=\begin{cases}x^a & x\le 8\\ 3x-8 & x>8\end{cases}\). Find the value of a. Give your answer to 3 significant figures. [3]
3. (c) \(h(x)=\begin{cases}e^{ax} & x\le 3\\ 2x+6 & x>3\end{cases}\). Find the value of a. Give your answer to 3 significant figures. [3]

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

1. (a) The time t days taken to build a wall is inversely proportional to the number n of workers, so \(t=\dfrac{a}{n}\). With 5 workers the wall is built in 12 days. Find the value of a, and the time when n = 8. [3]
2. (b) Describe what the model predicts for the time as n gets close to 0, and say why this is not reasonable. [1]
3. (c) A metal rod is left to cool in a room that is kept at 20 °C. Its temperature T °C after t minutes is modelled by \(T=c+k(0.9)^t\), where c and k are constants. At t = 0 the temperature is 180 °C. Write down the equation of the horizontal asymptote of the graph of T, and find the value of k. [3]
4. (d) Find the time for the temperature to reach 50 °C. Give your answer to 3 significant figures. [2]

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

1. (a) The value of a car falls by 9% each year. Find the value of k. Give your answer to 3 significant figures. [2]
2. (b) A bone has 62% of the carbon-14 that a living animal has. The half-life of carbon-14 is 5730 years. Find the age of the bone. Give your answer to 3 significant figures. [3]

A planter is made from a rectangular sheet of plastic 40 cm by 24 cm. A square of side x cm is cut from each corner and the sides are folded up. The volume of the planter is \(V=x(40-2x)(24-2x)\) cm³. [3]

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

The temperature C°C at a weather station, t hours after midnight, can be modelled by a sinusoidal function. The greatest temperature is 17°C and the least temperature is 3°C. The temperature repeats every 24 hours. [12]

1. (a) Find the amplitude and the mean value of C. [2]
2. (b) The period is 24 hours. Find the value of b. Give your answer to 3 significant figures. [2]
3. (c) The graph of C first crosses its mean value, going upwards, at t = 8. Write down a model of the form \(C=a\sin(b(t-c))+d\). Use b to 3 significant figures. [1]
4. (d) Use your model to find C when t = 15. [2]
5. (e) Find the total time, in the first 24 hours, for which C is greater than 15. Give your answer to 3 significant figures. [3]
6. (f) Find the first time at which C is a maximum. [2]

The number P of members of a new online club, t months after it is launched, is modelled by \(P=\dfrac{L}{1+Ce^{-kt}}\). The club can have at most 5000 members, so L = 5000. At launch there are 400 members. [12]

1. (a) Find the value of C. [2]
2. (b) After 2 months the number of members is 1100. Find the value of k. Give your answer to 3 significant figures. [3]
3. (c) Find the number of members after 5 months. [2]
4. (d) Find the time at which the number of members is half of 5000. Give your answer to 3 significant figures. [2]
5. (e) Write down the limiting value of the number of members as t increases. [1]
6. (f) Give one reason why a logistic model is more realistic than an exponential model for this situation. [2]

A scientist studies two cultures. Culture X starts with 650 units and loses 5% of its amount each day. Culture Y starts with 25 units. [8]

1. (a) Write down an expression for the amount of culture X after n days. [1]
2. (b) Find the least whole number of days after which there is less than 80 units of culture X. [3]
3. (c) The amount of culture Y is modelled by \(Y=B_0e^{kt}\) with \(B_0=25\). It doubles every 9 days. Find the value of k. Give your answer to 3 significant figures. [2]
4. (d) Find the time for the amount of culture Y to reach 700 units. Give your answer to 3 significant figures. [2]

A cinema sells n tickets for a film when the ticket price is £p. When the price is £9, 270 tickets are sold. When the price is £15, 180 tickets are sold. The cinema uses a linear model for n in terms of p. [9]

1. (a) Find a model of the form \(n=mp+c\). [3]
2. (b) State what the value of m means in this context. [1]
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]
4. (d) Find the range of ticket prices for which the income is more than £2500. Give your answers to 3 significant figures. [2]
5. (e) State a reasonable domain for p, and say why the model should not be used for a price of £40. [1]

A water wheel has its lowest point 0.5 m above the stream and its highest point 6.5 m above the stream. It makes one complete turn every 10 seconds. A bucket is at the lowest point at t = 0, where t is in seconds. The height of the bucket above the stream 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]
2. (b) Find the height of the bucket at t = 4. Give your answer to 3 significant figures. [1]
3. (c) Find the total time in one turn for which the height is more than 5 m. Give your answer to 3 significant figures. [3]
4. (d) State what the value of |a| represents in this context. [1]
5. (e) Give one reason why the model may not be reasonable. [1]