KindiakMath

Problem 1. The period T seconds of a simple pendulum of length L cm satisfies the equation

for some positive constant α. Experimental measurements yielded the following results.

L (cm)	20	30	40	50	60	70
T (s)	0.89	1.10	1.26	1.41	1.55	1.67

Draw a straight-line graph of T² against L. Hence, estimate the value of α.

(Click for Solution)

Solution. We first plot the values as follows, correct to the nearest 0.025 for the T²-axis.

L	20	30	40	50	60	70
T2	0.800	1.200	1.600	2.000	2.400	2.800

Squaring both sides, T² = α² · L. Hence, we graph as follows.

We estimate the gradient of the straight-line graph by

Taking square roots, α ≈ 0.2.

Problem 2. For a thin converging lens with focal length f cm, the object distance u cm and the image distance v cm are related by

The following values were obtained in an experiment

u (cm)	20	25	30	40	50	60
v (cm)	60.0	37.5	30.0	24.0	21.4	20.0

Draw a straight-line graph of and . Hence, estimate the value of f correct to 3 significant figures.

(Click for Solution)

Solution. We first plot the values as follows.

1/u	0.0500	0.0400	0.0335	0.0250	0.0200	0.0165
1/v	0.0165	0.0265	0.0335	0.0415	0.0465	0.0500

Shifting 1/u to the right,

Hence, we graph as follows.

We estimate the (1/v)-intercept of the straight-line graph by

Problem 3. The population P of a city is modelled by

where t is the time in years after a reference year and A, k are positive constants. The following data was recorded.

t (years)	5	10	15	20
P	1778	3162	5623	10 000

Draw a straight-line graph of lg P against t. Hence, estimate the values of A and k.

(Click for Solution)

Solution. We first plot the values as follows.

t	5	10	15	20
lg P	3.250	3.500	3.750	4.000

Taking logarithms on both sides,

Hence, we graph as follows.

We estimate the gradient of the straight-line graph by

Taking exponentials, A ≈ 10³ = 1000.

Problem 4. In a second-order chemical reaction, the concentration x units of a reactant at time t seconds satisfies the equation

where k is the rate constant and x₀ is the initial concentration. The following data was recorded.

t (s)	2	4	6	8	10
x (units)	0.455	0.417	0.385	0.357	0.333

Draw a straight-line graph of against t. Hence, estimate the values of k and x₀.

(Click for Solution)

Solution. We first plot the values as follows.

t	2	4	6	8	10
1/x	2.200	2.400	2.600	2.800	3.000

Hence, we graph as follows.

We estimate the gradient of the straight-line graph by

Taking reciprocals, x₀ ≈ 1/2.0 = 0.5.

—Joel Kindiak, 7 Apr 26, 1808H