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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 $\pi = 3.14$)
Solution :
Given Variables & Initial Setup
To determine the distance covered by the tip of the minute hand, we must first establish the geometric parameters of the system based on the provided values:
- Length of the minute hand (Radius, $r$) = $15$ cm
- Time elapsed ($t$) = $1$ hour
- Value of Pi ($\pi$) = $3.14$
Step 1: Analyzing the Kinematics of the Minute Hand
The minute hand of a clock is anchored at the center of the clock face. As time progresses, the tip of the minute hand traces a perfect circle. [Per the definition of a circle: the locus of all points in a plane that are at a given distance from a given point, the center].
In exactly $1$ hour ($60$ minutes), the minute hand completes one full revolution around the clock face. Therefore, the angular displacement is $360^\circ$ or $2\pi$ radians. The total linear distance moved by the tip of the minute hand is exactly equal to the circumference of the circle it traces.
Step 2: Formulating the Geometric Equation
The circumference ($C$) of a circle is given by the standard geometric formula:
$C = 2\pi r$
Where:
- $C$ represents the total distance moved by the tip in one revolution.
- $\pi$ is the mathematical constant (given as $3.14$).
- $r$ is the radius of the circular path, which corresponds to the length of the minute hand ($15$ cm).
Step 3: Visualizing the Circular Path
The following diagram illustrates the circular path traced by the tip of the minute hand over the course of $1$ hour.
Step 4: Execution and Algebraic Calculation
Substitute the given values into the circumference formula:
$C = 2 \times 3.14 \times 15$
To simplify the calculation, use the commutative property of multiplication to group the integers first:
$C = (2 \times 15) \times 3.14$
$C = 30 \times 3.14$
Perform the final multiplication:
$C = 94.2$ cm
Final Solution: The tip of the minute hand moves a total distance of $94.2$ cm in $1$ hour.
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
- 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 $\pi = \frac{22}{7}$)
- 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 $\pi = 3.14$)
- Q16: How many times a wheel of radius $28$ cm must rotate to go $352$ m? (Take $\pi = \frac{22}{7}$)
CBSE Solutions for Class 9 Mathematics Coordinate Geometry
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Coordinate Geometry
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