JAMB 2015 Mathematics · Question 15

The extension of a stretched string is directly proportional to its tension. If the extension produced by a…

Question

The extension of a stretched string is directly proportional to its tension. If the extension produced by a tension of 8 Newton’s is 2cm, find the extension produced by a tension of 12 newton’s.

Answer: D. 3

Explanation written with AI and checked against the answer key

Analysis

The problem states that the extension ( $e$ ) of a stretched string is directly proportional to the tension ( $T$ ) applied to it. This is a classic application of Hooke's Law principle in physics and variation in mathematics.

Concept

When two quantities are directly proportional, we express the relationship as:

e \propto T

To turn this into an equation, we introduce a constant of proportionality ( $k$ ):

e = kT

Where:

$e$ = extension
$T$ = tension
$k$ = constant

Step-by-Step Solution

Step 1: Find the constant ( $k$ ) using the given values. We are told that when $T = 8\text{ N}$ , $e = 2\text{ cm}$ .

2 = k \times 8

k = \frac{2}{8}

k = 0.25

Step 2: Use the constant to find the new extension. Now, we need to find the extension ( $e$ ) when the tension ( $T$ ) is $12\text{ N}$ .

e = k \times T

e = 0.25 \times 12

e = 3\text{ cm}

Alternative Method (Ratio Method): Since $e_1/T_1 = e_2/T_2$ :

\frac{2}{8} = \frac{e_2}{12}

Cross-multiply:

8 \times e_2 = 2 \times 12

8 \times e_2 = 24

e_2 = \frac{24}{8}

e_2 = 3\text{ cm}

Conclusion

Comparing our result to the provided options, the correct extension is 3 cm.

Exam Tip

When dealing with "directly proportional" problems, you can always use the ratio method ( $\frac{x_1}{y_1} = \frac{x_2}{y_2}$ ). It saves time during JAMB because you don't have to explicitly calculate the constant $k$ if you don't want to!

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