Non-linear regression with the calculator Problem set B Worked solutions

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.

• Enter the v values and the F values in two lists
• From the calculator: Quadratic regression: a = 0.0116, b = −0.539, c = 10.9
• Model: \(F = 0.0116v^2 - 0.539v + 10.9\)

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.

• Enter the t values and the T values in two lists
• From the calculator: Cubic regression: a = −0.439, b = 4.54, c = −8.25, d = 11.9
• Model: \(T = -0.439t^3 + 4.54t^2 - 8.25t + 11.9\)

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.

• Enter the t values and the M values in two lists
• From the calculator: Exponential regression, \(y=ae^{bx}\): a = 23.8, b = −0.167
• Model: \(M = 23.8e^{-0.167t}\)

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.

• Enter the v values and the d values in two lists
• From the calculator: Power regression, \(y=ax^b\): a = 0.00614, b = 2.06
• Model: \(d = 0.00614v^{2.06}\)

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.

• Enter the t values and the h values in two lists
• From the calculator: Sine regression, \(y=a\sin(bx+c)+d\), radians: a = 11.9, b = 0.204, c = −1.50, d = 13.7
• Model: \(h = 11.9\sin(0.204t - 1.50) + 13.7\)
• Plot the curve over the points to check it follows them. (c may differ from another calculator's value by a multiple of 2π.)

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.

   • From the calculator: Exponential regression, \(y=ae^{bx}\): a = 4.27, b = 0.376
   • Model: \(N = 4.27e^{0.376t}\)

2. (b) Use your model to estimate the number of followers 3.5 months after the account was created.

   • N(3.5) from the stored model = 15.9 thousand
   • 3.5 is inside the data range, so this is an interpolation

3. (c) Write down the value of a with its unit, and say what it represents.

   • a = 4.27 is the value of N when t = 0
   • It is the number of followers when the account was created

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.

   • From the calculator: Sine regression, \(y=a\sin(bx+c)+d\), radians: a = 0.563, b = 0.790, c = 0.503, d = 0.121
   • Model: \(h = 0.563\sin(0.790t + 0.503) + 0.121\)

2. (b) Find the period of the model, and say what it means in this context.

   • Period = 2π ÷ b = 2π ÷ 0.790 = 7.95 seconds
   • The pattern repeats every 7.95 seconds

3. (c) Use the model to predict the height of the buoy at t = 4.0.

   • h(4.0) from the stored model = -0.160

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.

   • From the calculator: Power regression, \(y=ax^b\): a = 0.816, b = 3.96
   • Model: \(Q = 0.816r^{3.96}\)

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.

   • Doubling r: \(Q=a(2r)^b=2^b\times ar^b\)
   • Factor = \(2^b\) with the stored b = 15.6

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.

   • From the calculator: Cubic regression: a = −0.354, b = 3.26, c = −5.18, d = 8.21
   • Model: \(W = -0.354t^3 + 3.26t^2 - 5.18t + 8.21\)

2. (b) Find the first time at which the model predicts a depth of 15 cm. Give t to 3 significant figures.

   • Solve \(W=15\) with the equation solver, or find where the curve meets the line \(W=15\)
   • The first solution is t = 3.55