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Q3:
If the circumference of a circular sheet is $154$ m, find its radius. Also find the area of the sheet. (Take $\pi = \frac{22}{7}$)
Solution :
Given Variables & Initial Setup
We are analyzing a two-dimensional circular sheet with the following given parameters:
- Circumference ($C$) = $154$ m
- Mathematical constant ($\pi$) $\approx \frac{22}{7}$
Our objective is to determine the radius ($r$) and the total enclosed area ($A$) of the circular sheet.
Step 1: Calculating the Radius of the Circular Sheet
The circumference of a circle is defined as the continuous linear distance forming the boundary of the closed geometric figure. [Per standard Euclidean geometry, the formula relating circumference to radius is $C = 2\pi r$].
Substituting the given values into the equation:
$154 = 2 \times \left(\frac{22}{7}\right) \times r$
Multiplying the constants on the right side of the equation:
$154 = \frac{44}{7} \times r$
To isolate the variable $r$, we multiply both sides of the equation by the reciprocal of $\frac{44}{7}$, which is $\frac{7}{44}$:
$r = 154 \times \frac{7}{44}$
We can simplify the fraction by recognizing that both $154$ and $44$ are divisible by $22$:
$r = \left(\frac{154}{22}\right) \times \left(\frac{7}{2}\right)$
$r = 7 \times \frac{7}{2}$
$r = \frac{49}{2} = 24.5 \text{ m}$
Step 2: Calculating the Area of the Circular Sheet
The area of a circle represents the total two-dimensional space enclosed within its circumference. [The fundamental formula for the area of a circle is $A = \pi r^2$].
Substituting $r = \frac{49}{2}$ m and $\pi = \frac{22}{7}$ into the area formula to maintain absolute precision before the final decimal conversion:
$A = \frac{22}{7} \times \left(\frac{49}{2}\right)^2$
Expanding the squared term:
$A = \frac{22}{7} \times \frac{49}{2} \times \frac{49}{2}$
Executing the arithmetic simplification by cross-canceling the terms in the numerator and denominator:
- Divide $22$ by $2$ to get $11$.
- Divide $49$ by $7$ to get $7$.
$A = 11 \times 7 \times \frac{49}{2}$
$A = 77 \times \frac{49}{2}$
$A = \frac{3773}{2}$
Converting the improper fraction to a precise decimal:
$A = 1886.5 \text{ m}^2$
Final Solution: The radius of the circular sheet is $24.5$ m, and its total enclosed area is $1886.5$ m$^2$.
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 $\pi = 3.14$)
- Q16: How many times a wheel of radius $28$ cm must rotate to go $352$ m? (Take $\pi = \frac{22}{7}$)
- 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
- 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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