Non-linear regression with the calculator Problem set A

The height, h metres, of a ball is measured t seconds after it is thrown. Find a quadratic model \(h=at^2+bt+c\). Give each coefficient to 3 significant figures.

A theme park records its visitors in each of the first eight months after opening. V is the number of visitors in hundreds, and t is the time in months. Find a cubic model \(V=at^3+bt^2+ct+d\). Give each coefficient to 3 significant figures.

A culture of bacteria is counted at hourly intervals. N is the number of bacteria in thousands, t hours after the start. Find an exponential model \(N=ae^{bt}\). Give each coefficient to 3 significant figures.

The power output, P kW, of a wind turbine is recorded at wind speed v m/s. Find a power model \(P=av^{b}\). Give each coefficient to 3 significant figures.

The average temperature, T °C, in a city is recorded in eight months. Here t = 1 is January, t = 2 is February, and so on. Find a sine model \(T=a\sin(bt+c)+d\). Use radians, with a and b positive. Give each coefficient to 3 significant figures.

The concentration, C mg/L, of a drug in a patient's blood is measured t hours after an injection. Fit the model \(C=ae^{bt}\).

1. (a) Find a and b to 3 significant figures.
2. (b) Use your model to estimate the concentration 3.5 hours after the injection.
3. (c) Write down the value of a with its unit, and say what it represents.

The number of hours of daylight, D, in a northern city is recorded on the first day of each month from January (t = 0) to August (t = 7). Fit the model \(D=a\sin(bt+c)+d\), using radians with a and b positive.

1. (a) Find a, b, c and d to 3 significant figures.
2. (b) Find the period of the model, and say what it means in this context.
3. (c) Use the model to predict the hours of daylight at t = 3.5.

The mass, m grams, of a smooth pebble is measured for seven pebbles of different widths d cm. A power model \(m=ad^b\) is used.

1. (a) Find a and b to 3 significant figures.
2. (b) The width of a pebble is doubled. By what factor does the model say the mass is multiplied? Give your answer to 3 significant figures.

An online shop records its weekly sales, S hundred items, in each of eight consecutive weeks (t = 0 to 7). A cubic model is used.

1. (a) Find the cubic model \(S=at^3+bt^2+ct+d\), with each coefficient to 3 significant figures.
2. (b) Find the first time after t = 0 at which the model predicts sales below 10 hundred. Give t to 3 significant figures.