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Q16:
How many times a wheel of radius $28$ cm must rotate to go $352$ m? (Take $\pi = \frac{22}{7}$)

Solution :

Step 1: Given Variables & Initial Setup

To determine the number of rotations a wheel must make to cover a specific distance, we must first establish the geometric properties of the wheel and the total distance to be traversed. The fundamental principle governing this motion is that the linear distance covered by a wheel in exactly one full rotation is equal to its perimeter, known as the circumference.

  • Radius of the wheel, $r = 28 \text{ cm}$
  • Total distance to be covered, $D = 352 \text{ m}$
  • Constant, $\pi = \frac{22}{7}$
r = 28 cm 1 Rotation = 2Ď€r Total Distance (D) = 352 m

Step 2: Dimensional Homogeneity (Unit Conversion)

To perform valid algebraic operations, all physical quantities must be expressed in the same units. The radius is given in centimeters ($\text{cm}$), while the total distance is given in meters ($\text{m}$). We will convert the total distance into centimeters.

[Per the standard metric conversion: $1 \text{ m} = 100 \text{ cm}$]

$D = 352 \text{ m}$
$D = 352 \times 100 \text{ cm}$
$D = 352,000 \text{ cm}$

Step 3: Calculating the Circumference of the Wheel

The distance covered by the wheel in one complete revolution is exactly equal to its circumference ($C$).

[By the geometric definition of a circle's perimeter: $C = 2\pi r$]

Substituting the given values into the formula:

$C = 2 \times \left(\frac{22}{7}\right) \times 28$

We simplify the expression by dividing $28$ by $7$:

$C = 2 \times 22 \times 4$
$C = 44 \times 4$
$C = 176 \text{ cm}$

Thus, the wheel covers a linear distance of $176 \text{ cm}$ in exactly one rotation.

Step 4: Determining the Number of Rotations

Let $n$ represent the total number of rotations required to cover the distance $D$. The relationship between total distance, circumference, and the number of rotations is given by the linear equation:

$D = n \times C$

Isolating $n$, we obtain:

$n = \frac{D}{C}$

Substituting the calculated values into the equation:

$n = \frac{352,000 \text{ cm}}{176 \text{ cm}}$

To simplify the division, observe the relationship between the significant digits: $176 \times 2 = 352$. Therefore:

$n = 2000$

(Self-Correction/Refinement: Wait, $352 \times 100 = 35,200$, not $352,000$. Let us rigorously re-evaluate the arithmetic in Step 2.)

Correction in arithmetic:
$D = 352 \text{ m} \times 100 \text{ cm/m} = 35,200 \text{ cm}$.

Recalculating $n$ with the corrected magnitude:

$n = \frac{35,200}{176}$
$n = 200$

Final Solution: The wheel must rotate exactly 200 times to cover a distance of 352 meters.


More Questions from Class 9 Mathematics Coordinate Geometry EXERCISE 9.2


CBSE Solutions for Class 9 Mathematics Coordinate Geometry


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