Measuring Space: Perimeter and Area | Exercise 6.1

Question 6

If the diameter of a car tyre is $56\text{ cm}$ , then:

(i) How far does the car need to travel for the tyre to complete one revolution?

(ii) How many revolutions does the tyre make if the car travels $10\text{ km}$ ?

Solution

Understand the Question

The distance covered by a wheel or tyre in one complete revolution is equal to its circumference, $C = 2\pi r = \pi d$ .
To find the number of revolutions over a given distance, divide the total distance by the distance covered in one revolution: $\text{Number of revolutions} = \dfrac{\text{Total distance}}{\text{Circumference}}$
Ensure all measurements are converted to the same unit (centimetres) before calculating.

(i) How far does the car need to travel for the tyre to complete one revolution?

Step 1 · Calculate Distance Covered in One Revolution

Given diameter $d = 56\text{ cm}$ . Diagram 1

Radius of the tyre:

\begin{aligned} r &= \dfrac{\text{diameter}}{2} \\[0.6em] &= \dfrac{56}{2} \\[0.6em] &= 28\text{ cm} \end{aligned}

Distance covered in one revolution is the circumference of the tyre ( $C = 2\pi r$ ):

\begin{aligned} C &= 2 \times \dfrac{22}{7} \times 28 \\[0.6em] &= 2 \times 22 \times 4 \\[0.6em] &= 44 \times 4 \\[0.6em] &= 176\text{ cm} \end{aligned}

Answer

(i) $176\text{ cm}$

(ii) How many revolutions does the tyre make if the car travels $10\text{ km}$ ?

Step 1 · Convert Distance and Calculate Number of Revolutions

Convert total distance from $\text{km}$ to $\text{cm}$ ( $1\text{ km} = 1000\text{ m} = 100{,}000\text{ cm}$ ):

\begin{aligned} \text{Total distance} &= 10 \times 1000 \times 100\text{ cm} \\[0.6em] &= 1{,}000{,}000\text{ cm} \end{aligned}

Calculate the number of revolutions:

\begin{aligned} \text{Number of revolutions} &= \dfrac{\text{Total distance}}{\text{Distance per revolution}} \\[0.6em] &= \dfrac{1{,}000{,}000}{176} \\[0.6em] &\approx 5681.82 \end{aligned}

Rounding to the nearest whole number gives $5682$ revolutions.

Answer

(ii) Approximately $5682\text{ revolutions}$

Common Mistakes

Unit Inconsistency: Failing to convert $10\text{ km}$ into centimetres ( $1\text{ km} = 100{,}000\text{ cm}$ ), leading to an incorrect result.
Diameter vs. Radius: Using the diameter ( $56\text{ cm}$ ) instead of the radius ( $28\text{ cm}$ ) when applying the formula $2\pi r$ .

More questions in Exercise 6.1