Non-linear regression with the calculator Problem set B

The fuel use, F litres per 100 km, of a car is measured at six steady speeds. v is the speed in m/s. Find a quadratic model \(F=av^2+bv+c\). Give each coefficient to 3 significant figures.

The temperature, T °C, of a chemical mixture is recorded each minute for six minutes. t is the time in minutes. Find a cubic model \(T=at^3+bt^2+ct+d\). Give each coefficient to 3 significant figures.

The mass, M grams, of a radioactive sample is measured t days after the start of an experiment. Find an exponential model \(M=ae^{bt}\). Give each coefficient to 3 significant figures.

The braking distance, d metres, of a car is measured at speed v km/h. Find a power model \(d=av^{b}\). Give each coefficient to 3 significant figures.

A rider on a Ferris wheel is timed. h is the height of the rider above the ground in metres, t seconds after the ride starts. Find a sine model \(h=a\sin(bt+c)+d\). Use radians, with a and b positive. Give each coefficient to 3 significant figures.

The number of followers, N thousand, of a new social media account is recorded t months after it is created. Fit the model \(N=ae^{bt}\).

1. (a) Find a and b to 3 significant figures.
2. (b) Use your model to estimate the number of followers 3.5 months after the account was created.
3. (c) Write down the value of a with its unit, and say what it represents.

The height, h metres, of a buoy above the mean sea level is recorded at eight times, t seconds after the first reading, as waves pass. Fit the model \(h=a\sin(bt+c)+d\), 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 height of the buoy at t = 4.0.

The flow rate, Q litres per second, of water through a pipe is measured for six pipes of different radii r cm. A power model \(Q=ar^b\) is used.

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

The depth of water, W cm, in a tank is recorded in each of eight consecutive minutes (t = 0 to 7). A cubic model is used.

1. (a) Find the cubic model \(W=at^3+bt^2+ct+d\), with each coefficient to 3 significant figures.
2. (b) Find the first time at which the model predicts a depth of 15 cm. Give t to 3 significant figures.