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

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.

• Enter the t values and the h values in two lists
• From the calculator: Quadratic regression: a = −5.01, b = 14.2, c = 0.845
• Model: \(h = -5.01t^2 + 14.2t + 0.845\)

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.

• Enter the t values and the V values in two lists
• From the calculator: Cubic regression: a = 0.365, b = −4.26, c = 10.9, d = 20.1
• Model: \(V = 0.365t^3 - 4.26t^2 + 10.9t + 20.1\)

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.

• Enter the t values and the N values in two lists
• From the calculator: Exponential regression, \(y=ae^{bx}\): a = 3.08, b = 0.451
• Model: \(N = 3.08e^{0.451t}\)

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.

• Enter the v values and the P values in two lists
• From the calculator: Power regression, \(y=ax^b\): a = 0.501, b = 2.92
• Model: \(P = 0.501v^{2.92}\)

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.

• Enter the t values and the T values in two lists
• From the calculator: Sine regression, \(y=a\sin(bx+c)+d\), radians: a = 9.34, b = 0.502, c = −1.98, d = 12.7
• Model: \(T = 9.34\sin(0.502t - 1.98) + 12.7\)
• 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 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.

   • From the calculator: Exponential regression, \(y=ae^{bx}\): a = 12.5, b = −0.314
   • Model: \(C = 12.5e^{-0.314t}\)

2. (b) Use your model to estimate the concentration 3.5 hours after the injection.

   • C(3.5) from the stored model = 4.17 mg/L
   • 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 = 12.5 is the value of C when t = 0
   • It is the concentration at the moment of the injection

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.

   • From the calculator: Sine regression, \(y=a\sin(bx+c)+d\), radians: a = 3.18, b = 0.530, c = −1.46, d = 12.2
   • Model: \(D = 3.18\sin(0.530t - 1.46) + 12.2\)

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

   • Period = 2π ÷ b = 2π ÷ 0.530 = 11.9 months
   • The pattern repeats every 11.9 months

3. (c) Use the model to predict the hours of daylight at t = 3.5.

   • D(3.5) from the stored model = 13.5

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.

   • From the calculator: Power regression, \(y=ax^b\): a = 2.71, b = 2.85
   • Model: \(m = 2.71d^{2.85}\)

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.

   • Doubling d: \(m=a(2d)^b=2^b\times ad^b\)
   • Factor = \(2^b\) with the stored b = 7.19

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.

   • From the calculator: Cubic regression: a = 0.346, b = −3.86, c = 10.2, d = 11.3
   • Model: \(S = 0.346t^3 - 3.86t^2 + 10.2t + 11.3\)

2. (b) Find the first time after t = 0 at which the model predicts sales below 10 hundred. Give t to 3 significant figures.

   • Solve \(S=10\) with the equation solver, or find where the curve meets the line \(S=10\)
   • The first solution is t = 4.68