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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, $\pi = \frac{22}{7}$
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
- Q1(a): Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (a) $14$ cm
- Q1(b): Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (b) $28$ mm
- Q1(c): Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (c) $21$ cm
- Q10: From a circular card sheet of radius $14$ cm, two circles of radius $3.5$ cm and a rectangle of length $3$ cm and breadth $1$cm are removed. (as shown in the adjoining figure). Find the area of the remaining sheet. (Take $\pi = \frac{22}{7}$)
- Q11: A circle of radius $2$ cm is cut out from a square piece of an aluminium sheet of side $6$ cm. What is the area of the left over aluminium sheet? (Take $\pi = 3.14$)
- Q12: The circumference of a circle is $31.4$ cm. Find the radius and the area of the circle? (Take $\pi = 3.14$)
- Q13: A circular flower bed is surrounded by a path $4$ m wide. The diameter of the flower bed is $66$ m. What is the area of this path? ($\pi = 3.14$)
- Q14: A circular flower garden has an area of $314$ m$^2$. A sprinkler at the centre of the garden can cover an area that has a radius of $12$ m. Will the sprinkler water the entire garden? (Take $\pi = 3.14$)
- Q15: Find the circumference of the inner and the outer circles, shown in the adjoining figure? (Take $\pi = 3.14$)
- Q17: The minute hand of a circular clock is $15$ cm long. How far does the tip of the minute hand move in $1$ hour. (Take $\pi = 3.14$)
- Q2(a): Find the area of the following circles, given that: (a) radius = $14$ mm (Take $\pi = \frac{22}{7}$)
- Q2(b): Find the area of the following circles, given that: (b) diameter = $49$ m
- Q2(c): Find the area of the following circles, given that: (c) radius = $5$ cm
- Q3: If the circumference of a circular sheet is $154$ m, find its radius. Also find the area of the sheet. (Take $\pi = \frac{22}{7}$)
- Q4: A gardener wants to fence a circular garden of diameter $21$ m. Find the length of the rope he needs to purchase, if he makes $2$ rounds of fence. Also find the cost of the rope, if it costs ₹ $4$ per meter. (Take $\pi = \frac{22}{7}$)
- Q5: From a circular sheet of radius $4$ cm, a circle of radius $3$ cm is removed. Find the area of the remaining sheet. (Take $\pi = 3.14$)
- Q6: Saima wants to put a lace on the edge of a circular table cover of diameter $1.5$ m. Find the length of the lace required and also find its cost if one meter of the lace costs ₹ $15$. (Take $\pi = 3.14$)
- Q7: Find the perimeter of the adjoining figure, which is a semicircle including its diameter.
- Q8: Find the cost of polishing a circular table-top of diameter $1.6$ m, if the rate of polishing is ₹ $15/m^2$. (Take $\pi = 3.14$)
- Q9: Shazli took a wire of length $44$ cm and bent it into the shape of a circle. Find the radius of that circle. Also find its area. If the same wire is bent into the shape of a square, what will be the length of each of its sides? Which figure encloses more area, the circle or the square? (Take $\pi = \frac{22}{7}$)
CBSE Solutions for Class 9 Mathematics Coordinate Geometry
Chapters in CBSE - Class 9 Mathematics
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