Given:

1. The brooch is circular in shape.
2. The diameter of the circle ($d$) = $35\text{ mm}$.
3. The number of diameters used to divide the circle = $5$.
4. The value of $\pi = \frac{22}{7}$.

To Find:

The total length of the silver wire required to make the brooch.

Step 1: Calculate the circumference of the circle.

The silver wire forms the outer boundary (circumference) of the circle. The formula for the circumference ($C$) of a circle is given by:
$C = \pi \times d$
Substituting the given values:
$C = \frac{22}{7} \times 35\text{ mm}$
$C = 22 \times 5\text{ mm}$ [Since $35 \div 7 = 5$]
$C = 110\text{ mm}$

Step 2: Calculate the length of the 5 diameters.

The wire is also used to make 5 diameters. The length of one diameter is $35\text{ mm}$.
Total length of 5 diameters = $5 \times d$
Total length of 5 diameters = $5 \times 35\text{ mm}$
Total length of 5 diameters = $175\text{ mm}$

Step 3: Calculate the total length of the silver wire.

The total length of the wire required is the sum of the circumference of the circle and the length of the 5 diameters.
Total Length = Circumference + (5 $\times$ Diameter)
Total Length = $110\text{ mm} + 175\text{ mm}$
Total Length = $285\text{ mm}$

Final Answer: The total length of the silver wire required is 285 mm.

Given:

• Radius of the circle ($r$) = $21 \text{ cm}$
• Angle subtended by the arc at the centre ($\theta$) = $60^\circ$
• Value of $\pi = \frac{22}{7}$

To Find:

• The length of the arc ($l$).

Step 1: Stating the Formula

The length of an arc ($l$) of a circle that subtends an angle $\theta$ at the centre is given by the formula:

$l = \frac{\theta}{360^\circ} \times 2\pi r$

[Where $\theta$ is the central angle in degrees, and $r$ is the radius of the circle.]

Step 2: Substituting the Given Values

Substitute $\theta = 60^\circ$, $r = 21 \text{ cm}$, and $\pi = \frac{22}{7}$ into the formula:

$l = \frac{60^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 21$

Step 3: Simplifying the Expression

First, simplify the fraction representing the portion of the circumference:

$\frac{60}{360} = \frac{1}{6}$

Now, substitute this back into the equation:

$l = \frac{1}{6} \times 2 \times \frac{22}{7} \times 21$

Step 4: Performing Arithmetic Calculations

Calculate the product of the constants:

$l = \frac{1}{6} \times 44 \times \frac{21}{7}$

[Since $2 \times 22 = 44$]

$l = \frac{1}{6} \times 44 \times 3$

[Since $21 \div 7 = 3$]

$l = \frac{132}{6}$

[Since $44 \times 3 = 132$]

$l = 22 \text{ cm}$

Final Answer: The length of the arc is 22 cm.

Given:

1. A circular table cover with radius $r = 28$ cm.

2. There are six equal designs on the table cover, which are segments of the circle.

3. The rate of making the designs is ₹ $0.35$ per cm$^2$.

4. The value of $\sqrt{3} = 1.7$.

To Find:

The total cost of making the six designs.

Step 1: Determine the central angle of each sector.

Since there are 6 equal designs, the circle is divided into 6 equal sectors. The central angle $\theta$ for each sector is:

$\theta = \frac{360^\circ}{6} = 60^\circ$

Step 2: Calculate the area of one design (segment).

The area of a segment is given by the formula: Area of sector - Area of the triangle formed by the two radii and the chord.

Area of one sector = $\frac{\theta}{360^\circ} \times \pi r^2$

Using $\pi = \frac{22}{7}$ and $r = 28$:

Area of one sector = $\frac{60}{360} \times \frac{22}{7} \times 28 \times 28$

Area of one sector = $\frac{1}{6} \times 22 \times 4 \times 28 = \frac{2464}{6} = \frac{1232}{3} \text{ cm}^2$

Since the central angle is $60^\circ$ and the two sides are equal (radii), the triangle is equilateral. The area of an equilateral triangle is $\frac{\sqrt{3}}{4} \times \text{side}^2$.

Area of triangle = $\frac{\sqrt{3}}{4} \times 28 \times 28 = \sqrt{3} \times 7 \times 28 = 196\sqrt{3}$

Given $\sqrt{3} = 1.7$, Area of triangle = $196 \times 1.7 = 333.2 \text{ cm}^2$

Area of one design = $\frac{1232}{3} - 333.2 = 410.67 - 333.2 = 77.47 \text{ cm}^2$

Step 3: Calculate the total area of the six designs.

Total Area = $6 \times (\text{Area of one design})$

Total Area = $6 \times (\frac{1232}{3} - 333.2) = 2 \times 1232 - 6 \times 333.2$

Total Area = $2464 - 1999.2 = 464.8 \text{ cm}^2$

Step 4: Calculate the total cost.

Cost = Total Area $\times$ Rate

Cost = $464.8 \times 0.35$

Cost = $162.68$

Final Answer: The total cost of making the designs is ₹ 162.68.

Given:

A circle with radius $R$ and a sector subtending an angle $p$ (in degrees) at the center.

To Find:

The area of the sector of the circle.

Step 1: Understanding the relationship between the angle and the area of a circle.

The total angle subtended by a circle at its center is $360^\circ$. The area of a full circle with radius $R$ is given by the formula:

$\text{Area of circle} = \pi R^2$

Step 2: Applying the Unitary Method.

We know that an angle of $360^\circ$ corresponds to an area of $\pi R^2$.

Therefore, an angle of $1^\circ$ corresponds to an area of:

$\text{Area for } 1^\circ = \frac{\pi R^2}{360^\circ}$

Step 3: Calculating the area for angle $p$.

To find the area of a sector subtending an angle $p$ at the center, we multiply the area per degree by $p$:

$\text{Area of sector} = \left( \frac{\pi R^2}{360^\circ} \right) \times p$

$\text{Area of sector} = \frac{p}{360^\circ} \times \pi R^2$

Step 4: Verification with standard options.

Often, this formula is represented by multiplying the numerator and denominator by $2$ to relate it to the circumference or specific sector properties, resulting in:

$\text{Area of sector} = \frac{p}{720^\circ} \times 2\pi R^2$

However, the standard derived formula remains $\frac{p}{360} \times \pi R^2$.

Final Answer: The area of the sector is $\frac{p}{360} \times \pi R^2$.

Given:

Radius of the circle ($r$) = $6\text{ cm}$

Angle of the sector ($\theta$) = $60^\circ$

Value of $\pi = \frac{22}{7}$

To find:

The area of the sector of the circle.

Step 1: State the formula for the area of a sector.

The area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is given by the formula:

$\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2$

[Justification: The area of a sector is a fraction of the total area of the circle ($\pi r^2$), where the fraction is determined by the ratio of the sector's angle to the total angle of a circle ($360^\circ$)]

Step 2: Substitute the given values into the formula.

Substituting $r = 6$, $\theta = 60^\circ$, and $\pi = \frac{22}{7}$:

$\text{Area} = \frac{60}{360} \times \frac{22}{7} \times (6)^2$

Step 3: Perform the arithmetic calculations.

First, simplify the fraction $\frac{60}{360}$:

$\frac{60}{360} = \frac{1}{6}$

[Since $60 \times 6 = 360$]

Next, calculate the square of the radius:

$6^2 = 36$

Now, substitute these back into the expression:

$\text{Area} = \frac{1}{6} \times \frac{22}{7} \times 36$

Step 4: Simplify the final expression.

$\text{Area} = \frac{1 \times 22 \times 36}{6 \times 7}$

Divide $36$ by $6$:

$\text{Area} = \frac{22 \times 6}{7}$

[Since $36 \div 6 = 6$]

$\text{Area} = \frac{132}{7}$

[Since $22 \times 6 = 132$]

Converting to decimal form (optional, but standard for verification):

$\text{Area} \approx 18.857\text{ cm}^2$

Final Answer: The area of the sector is $\frac{132}{7}\text{ cm}^2$ (or approximately $18.86\text{ cm}^2$).

Given:

Radius of the circle ($r$) = $15\text{ cm}$

Angle subtended by the chord at the centre ($\theta$) = $60^\circ$

Constants: $\pi = 3.14$, $\sqrt{3} = 1.73$

To Find:

1. Area of the minor segment.

2. Area of the major segment.

Step 1: Calculate the Area of the Sector

The formula for the area of a sector is: $A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2$

$A_{sector} = \frac{60}{360} \times 3.14 \times (15)^2$

$A_{sector} = \frac{1}{6} \times 3.14 \times 225$

$A_{sector} = \frac{706.5}{6} = 117.75\text{ cm}^2$

Step 2: Calculate the Area of the Triangle (OAB)

Since the angle at the centre is $60^\circ$ and the two sides are radii ($OA = OB = 15\text{ cm}$), the triangle is equilateral. The area of an equilateral triangle is given by: $A_{triangle} = \frac{\sqrt{3}}{4} \times r^2$

$A_{triangle} = \frac{1.73}{4} \times (15)^2$

$A_{triangle} = \frac{1.73}{4} \times 225$

$A_{triangle} = \frac{389.25}{4} = 97.3125\text{ cm}^2$

Step 3: Calculate the Area of the Minor Segment

The area of the minor segment is the difference between the area of the sector and the area of the triangle.

$A_{minor\_segment} = A_{sector} - A_{triangle}$

$A_{minor\_segment} = 117.75 - 97.3125 = 20.4375\text{ cm}^2$

Step 4: Calculate the Area of the Major Segment

The area of the major segment is the total area of the circle minus the area of the minor segment.

$A_{circle} = \pi r^2 = 3.14 \times (15)^2 = 3.14 \times 225 = 706.5\text{ cm}^2$

$A_{major\_segment} = A_{circle} - A_{minor\_segment}$

$A_{major\_segment} = 706.5 - 20.4375 = 686.0625\text{ cm}^2$

Final Answer: The area of the minor segment is 20.4375 cm² and the area of the major segment is 686.0625 cm².

Given:

Radius of the circle ($r$) = $10\text{ cm}$

Angle subtended by the chord at the centre ($\theta$) = $90^\circ$

Value of $\pi$ = $3.14$

To Find:

Area of the major sector.

Step 1: Calculate the area of the entire circle.

The formula for the area of a circle is $A = \pi r^2$.

$A = 3.14 \times (10)^2$

$A = 3.14 \times 100$

$A = 314\text{ cm}^2$

Step 2: Calculate the area of the minor sector.

The formula for the area of a sector with angle $\theta$ is $\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2$.

$\text{Area}_{\text{minor}} = \frac{90^\circ}{360^\circ} \times 314$

$\text{Area}_{\text{minor}} = \frac{1}{4} \times 314$

$\text{Area}_{\text{minor}} = 78.5\text{ cm}^2$

Step 3: Calculate the area of the major sector.

The area of the major sector is the difference between the area of the circle and the area of the minor sector.

$\text{Area}_{\text{major}} = \text{Area}_{\text{circle}} - \text{Area}_{\text{minor}}$

$\text{Area}_{\text{major}} = 314 - 78.5$

$\text{Area}_{\text{major}} = 235.5\text{ cm}^2$

Alternative Method (Using the reflex angle):

The angle of the major sector is $360^\circ - 90^\circ = 270^\circ$.

$\text{Area}_{\text{major}} = \frac{270^\circ}{360^\circ} \times \pi r^2$

$\text{Area}_{\text{major}} = \frac{3}{4} \times 314$

$\text{Area}_{\text{major}} = 3 \times 78.5 = 235.5\text{ cm}^2$

Final Answer: The area of the major sector is 235.5 cm².

Given:

1. The brooch is circular in shape.
2. The diameter of the circle ($d$) = $35\text{ mm}$.
3. The wire is used to form the circumference and 5 diameters.
4. The 5 diameters divide the circle into 10 equal sectors.

To Find:

The area of each sector of the brooch.

Step 1: Determine the radius of the circle.

The radius ($r$) is half of the diameter ($d$).
$r = \frac{d}{2}$
$r = \frac{35}{2}\text{ mm} = 17.5\text{ mm}$

Step 2: Determine the area of the entire circle.

The formula for the area of a circle is $A = \pi r^2$.
Using $\pi = \frac{22}{7}$:
$A = \frac{22}{7} \times \left(\frac{35}{2}\right) \times \left(\frac{35}{2}\right)$
$A = \frac{22}{7} \times \frac{1225}{4}$
$A = \frac{22 \times 175}{4}$ [Since $1225 \div 7 = 175$]
$A = \frac{3850}{4} = 962.5\text{ mm}^2$

Step 3: Calculate the area of each sector.

Since the 5 diameters divide the circle into 10 equal sectors, the area of each sector is $\frac{1}{10}$ of the total area of the circle.
Area of each sector = $\frac{\text{Total Area}}{10}$
Area of each sector = $\frac{962.5}{10}$
Area of each sector = $96.25\text{ mm}^2$

Alternative Method (Using Central Angle):

The total angle of a circle is $360^\circ$.
Since there are 10 equal sectors, the central angle ($\theta$) of each sector is:
$\theta = \frac{360^\circ}{10} = 36^\circ$
Area of sector = $\frac{\theta}{360^\circ} \times \pi r^2$
Area of sector = $\frac{36}{360} \times \frac{22}{7} \times \frac{35}{2} \times \frac{35}{2}$
Area of sector = $\frac{1}{10} \times \frac{22}{7} \times \frac{1225}{4}$
Area of sector = $\frac{1}{10} \times 962.5 = 96.25\text{ mm}^2$

Final Answer: The area of each sector of the brooch is 96.25 mm².

Given:

• Radius of the circle ($r$) = $10\text{ cm}$
• Angle subtended by the chord at the center ($\theta$) = $90^\circ$
• Value of $\pi$ = $3.14$

To find:

The area of the minor segment.

Step 1: Formula for the Area of a Minor Segment

The area of a minor segment is calculated by subtracting the area of the triangle formed by the chord and the radii from the area of the corresponding sector.

Formula: $\text{Area of minor segment} = \text{Area of sector} - \text{Area of } \triangle OAB$

Step 2: Calculate the Area of the Sector

The formula for the area of a sector is given by: $\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2$

Substituting the given values:

$\text{Area of sector} = \frac{90}{360} \times 3.14 \times (10)^2$

$\text{Area of sector} = \frac{1}{4} \times 3.14 \times 100$

$\text{Area of sector} = 0.25 \times 314 = 78.5\text{ cm}^2$

Step 3: Calculate the Area of the Triangle ($\triangle OAB$)

Since the angle at the center is $90^\circ$, $\triangle OAB$ is a right-angled triangle with base $OA = 10\text{ cm}$ and height $OB = 10\text{ cm}$.

Formula: $\text{Area of } \triangle OAB = \frac{1}{2} \times \text{base} \times \text{height}$

$\text{Area of } \triangle OAB = \frac{1}{2} \times 10 \times 10$

$\text{Area of } \triangle OAB = \frac{1}{2} \times 100 = 50\text{ cm}^2$

Step 4: Calculate the Area of the Minor Segment

$\text{Area of minor segment} = \text{Area of sector} - \text{Area of } \triangle OAB$

$\text{Area of minor segment} = 78.5\text{ cm}^2 - 50\text{ cm}^2$

$\text{Area of minor segment} = 28.5\text{ cm}^2$

Final Answer: The area of the minor segment is 28.5 cm²

Given:

• The angle of the sector, $\theta = 80^\circ$.
• The radius of the sector (distance to which the light spreads), $r = 16.5 \text{ km}$.
• The value of $\pi = 3.14$.

To Find:

The area of the sea over which the ships are warned, which is equivalent to the area of the sector formed by the lighthouse light.

Step 1: State the Formula for the Area of a Sector

The area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is given by the formula:

$\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$

Step 2: Substitute the Given Values into the Formula

Substitute $\theta = 80^\circ$, $r = 16.5 \text{ km}$, and $\pi = 3.14$ into the equation:

$\text{Area} = \frac{80}{360} \times 3.14 \times (16.5)^2$

Step 3: Simplify the Fraction and Calculate the Square

First, simplify the fraction $\frac{80}{360}$:

$\frac{80}{360} = \frac{8}{36} = \frac{2}{9}$

Next, calculate $(16.5)^2$:

$16.5 \times 16.5 = 272.25$

Step 4: Perform the Final Multiplication

Now, substitute these values back into the expression:

$\text{Area} = \frac{2}{9} \times 3.14 \times 272.25$

Multiply $3.14$ by $272.25$:

$3.14 \times 272.25 = 854.865$

Now, multiply by $\frac{2}{9}$:

$\text{Area} = \frac{2 \times 854.865}{9}$

$\text{Area} = \frac{1709.73}{9}$

$\text{Area} = 189.97 \text{ km}^2$

Final Answer: The area of the sea over which the ships are warned is 189.97 km².

Given: The circumference of a circle is $C = 22 \text{ cm}$.

To find: The area of a quadrant of the circle.

Step 1: Finding the radius of the circle.

The formula for the circumference of a circle is given by:

$C = 2\pi r$

Substituting the given value $C = 22 \text{ cm}$ and $\pi = \frac{22}{7}$:

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

To isolate $r$, multiply both sides by $7$ and divide by $(2 \times 22)$:

$r = \frac{22 \times 7}{2 \times 22}$

$r = \frac{7}{2} \text{ cm}$

$r = 3.5 \text{ cm}$

Step 2: Defining the area of a quadrant.

A quadrant is one-fourth of a circle. Therefore, the area of a quadrant ($A_q$) is given by:

$A_q = \frac{1}{4} \times \text{Area of the circle}$

$A_q = \frac{1}{4} \times \pi r^2$

Step 3: Calculating the area.

Substitute $r = \frac{7}{2}$ and $\pi = \frac{22}{7}$ into the formula:

$A_q = \frac{1}{4} \times \frac{22}{7} \times \left(\frac{7}{2}\right)^2$

$A_q = \frac{1}{4} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2}$

Cancel the common factors ($7$ in the numerator and denominator):

$A_q = \frac{1}{4} \times \frac{22 \times 7}{2 \times 2}$

$A_q = \frac{1}{4} \times \frac{154}{4}$

$A_q = \frac{154}{16}$

Simplify the fraction by dividing both numerator and denominator by $2$:

$A_q = \frac{77}{8} \text{ cm}^2$

Converting to decimal form:

$A_q = 9.625 \text{ cm}^2$

Final Answer: The area of the quadrant is 9.625 cm² (or $\frac{77}{8}$ cm²).

Given:

• A square-shaped grass field with side length $s = 15\text{ m}$.
• A horse is tied to a peg at one corner.
• Initial rope length $r_1 = 5\text{ m}$.
• Increased rope length $r_2 = 10\text{ m}$.
• Value of $\pi = 3.14$.

To Find:

The increase in the grazing area if the rope length is increased from $5\text{ m}$ to $10\text{ m}$.

Step 1: Understanding the Geometry of Grazing

Since the horse is tied to a corner of a square field, the area it can graze is a sector of a circle with radius $r$ and a central angle $\theta = 90^\circ$ (the angle of a square corner). The formula for the area of a sector is:

$\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2$

[Since $\theta = 90^\circ$, the area simplifies to $\frac{90}{360} \times \pi r^2 = \frac{1}{4} \pi r^2$]

Step 2: Calculating the Initial Grazing Area ($A_1$)

Given $r_1 = 5\text{ m}$:

$A_1 = \frac{1}{4} \times 3.14 \times (5)^2$

$A_1 = \frac{1}{4} \times 3.14 \times 25$

$A_1 = \frac{78.5}{4} = 19.625\text{ m}^2$

Step 3: Calculating the New Grazing Area ($A_2$)

Given $r_2 = 10\text{ m}$:

$A_2 = \frac{1}{4} \times 3.14 \times (10)^2$

$A_2 = \frac{1}{4} \times 3.14 \times 100$

$A_2 = \frac{314}{4} = 78.5\text{ m}^2$

Step 4: Calculating the Increase in Grazing Area

The increase in area is the difference between the new area and the initial area:

$\text{Increase} = A_2 - A_1$

$\text{Increase} = 78.5 - 19.625$

$\text{Increase} = 58.875\text{ m}^2$

Final Answer: The increase in the grazing area is 58.875 m².

Given:

The length of the minute hand of the clock, which acts as the radius ($r$) of the circular path, is $14\text{ cm}$.

The time duration for which the area is swept is $5\text{ minutes}$.

To Find:

The area swept by the minute hand in $5\text{ minutes}$.

Step 1: Determine the angle swept by the minute hand in 60 minutes.

A minute hand completes one full rotation in $60\text{ minutes}$. A full rotation corresponds to an angle of $360^\circ$.

Angle swept in $60\text{ minutes} = 360^\circ$.

Step 2: Calculate the angle swept by the minute hand in 1 minute.

Using the unitary method:

Angle swept in $1\text{ minute} = \frac{360^\circ}{60} = 6^\circ$.

Step 3: Calculate the angle swept ($\theta$) in 5 minutes.

$\theta = 5 \times 6^\circ = 30^\circ$.

Step 4: Apply the formula for the area of a sector.

The area swept by the minute hand is the area of a sector of a circle with radius $r$ and central angle $\theta$.

Formula: $\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2$

[Where $\theta = 30^\circ$, $r = 14\text{ cm}$, and $\pi = \frac{22}{7}$]

Step 5: Perform the calculation.

$\text{Area} = \frac{30}{360} \times \frac{22}{7} \times 14 \times 14$

Simplify the fraction $\frac{30}{360}$:

$\text{Area} = \frac{1}{12} \times \frac{22}{7} \times 14 \times 14$

Cancel terms ($14$ divided by $7$ is $2$):

$\text{Area} = \frac{1}{12} \times 22 \times 2 \times 14$

$\text{Area} = \frac{1}{12} \times 616$

$\text{Area} = \frac{616}{12} = \frac{154}{3}\text{ cm}^2$

Converting to decimal form:

$\text{Area} \approx 51.33\text{ cm}^2$

Final Answer: The area swept by the minute hand in 5 minutes is $\frac{154}{3}\text{ cm}^2$ or approximately $51.33\text{ cm}^2$.

Given:

Radius of the circle ($r$) = $12\text{ cm}$

Angle subtended by the chord at the centre ($\theta$) = $120^\circ$

Constants: $\pi = 3.14$, $\sqrt{3} = 1.73$

To Find:

Area of the corresponding segment of the circle.

Step 1: Formula for the Area of a Segment

The area of a segment of a circle is given by the formula:

$\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}$

$\text{Area of Segment} = \left( \frac{\theta}{360^\circ} \times \pi r^2 \right) - \left( \frac{1}{2} r^2 \sin \theta \right)$

Step 2: Calculating the Area of the Sector

Substitute the given values into the sector area formula:

$\text{Area of Sector} = \frac{120}{360} \times 3.14 \times (12)^2$

$\text{Area of Sector} = \frac{1}{3} \times 3.14 \times 144$

$\text{Area of Sector} = 3.14 \times 48$

$\text{Area of Sector} = 150.72\text{ cm}^2$

Step 3: Calculating the Area of the Triangle

The area of the triangle formed by two radii and the chord is $\frac{1}{2} r^2 \sin \theta$.

$\text{Area of Triangle} = \frac{1}{2} \times (12)^2 \times \sin(120^\circ)$

Since $\sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$:

$\text{Area of Triangle} = \frac{1}{2} \times 144 \times \frac{\sqrt{3}}{2}$

$\text{Area of Triangle} = 72 \times \frac{1.73}{2}$

$\text{Area of Triangle} = 36 \times 1.73$

$\text{Area of Triangle} = 62.28\text{ cm}^2$

Step 4: Calculating the Area of the Segment

Subtract the area of the triangle from the area of the sector:

$\text{Area of Segment} = 150.72 - 62.28$

$\text{Area of Segment} = 88.44\text{ cm}^2$

Final Answer: The area of the corresponding segment of the circle is 88.44 cm².

Given:

Number of wipers = $2$

Length of each wiper blade ($r$) = $25\text{ cm}$

Angle of sweep ($\theta$) = $115^\circ$

Value of $\pi = \frac{22}{7}$

To Find:

The total area cleaned by the two wipers at each sweep.

Step 1: Understanding the Geometry

Each wiper sweeps out a sector of a circle. The area of a sector of a circle with radius $r$ and central angle $\theta$ is given by the formula:

$\text{Area of a sector} = \frac{\theta}{360^\circ} \times \pi r^2$

Step 2: Calculating the area cleaned by one wiper

Substitute the given values into the formula:

$\text{Area of one sector} = \frac{115}{360} \times \frac{22}{7} \times (25)^2$

$\text{Area of one sector} = \frac{115}{360} \times \frac{22}{7} \times 625$

Simplifying the fraction $\frac{115}{360}$ by dividing by 5: $\frac{23}{72}$

$\text{Area of one sector} = \frac{23}{72} \times \frac{22}{7} \times 625$

$\text{Area of one sector} = \frac{23 \times 11 \times 625}{36 \times 7}$ [Dividing 22 and 72 by 2]

$\text{Area of one sector} = \frac{158125}{252} \text{ cm}^2$

Step 3: Calculating the total area cleaned by two wipers

Since there are two wipers that do not overlap, the total area is twice the area of one sector:

$\text{Total Area} = 2 \times \left( \frac{158125}{252} \right)$

$\text{Total Area} = \frac{158125}{126} \text{ cm}^2$

Step 4: Final Calculation

Performing the division:

$158125 \div 126 \approx 1254.96 \text{ cm}^2$

Final Answer: The total area cleaned at each sweep of the blades is $\frac{158125}{126} \text{ cm}^2$ or approximately $1254.96 \text{ cm}^2$.

Given:

• A square-shaped grass field with side length $s = 15\text{ m}$.
• A horse is tied to one corner of the square field.
• The length of the rope, which acts as the radius $r$ of the grazing area, is $r = 5\text{ m}$.

To Find:

The area of the part of the field in which the horse can graze.

Step 1: Understanding the Geometry of the Grazing Area

Since the horse is tied to a corner of a square field, the angle at the corner of the square is $90^\circ$. The horse can move within a circular sector defined by the length of the rope. Therefore, the grazing area is a sector of a circle with radius $r = 5\text{ m}$ and central angle $\theta = 90^\circ$.

Step 2: Formula for the Area of a Sector

The formula for the area of a sector of a circle is given by:

$\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$

[Where $\theta$ is the central angle in degrees and $r$ is the radius of the circle.]

Step 3: Substituting the Given Values

Given values: $\theta = 90^\circ$, $r = 5\text{ m}$, and $\pi = 3.14$.

$\text{Area} = \frac{90^\circ}{360^\circ} \times 3.14 \times (5)^2$

Step 4: Performing the Calculation

First, simplify the fraction:

$\frac{90}{360} = \frac{1}{4}$

Next, calculate the square of the radius:

$(5)^2 = 25$

Now, substitute these back into the equation:

$\text{Area} = \frac{1}{4} \times 3.14 \times 25$

$\text{Area} = \frac{78.5}{4}$

$\text{Area} = 19.625\text{ m}^2$

Final Answer: The area of the part of the field in which the horse can graze is 19.625 m².

Given:

1. The umbrella is assumed to be a flat circle.
2. The radius of the circle ($r$) = $45\text{ cm}$.
3. The number of ribs in the umbrella ($n$) = $8$.
4. The ribs are equally spaced, meaning the circle is divided into $8$ equal sectors.

To Find:

The area between two consecutive ribs of the umbrella.

Step 1: Determine the central angle of each sector.

Since the umbrella is a full circle, the total angle at the center is $360^\circ$. Because there are $8$ equally spaced ribs, the circle is divided into $8$ equal sectors.

Let $\theta$ be the central angle of the sector between two consecutive ribs.

$\theta = \frac{360^\circ}{n}$

$\theta = \frac{360^\circ}{8}$

$\theta = 45^\circ$

Step 2: State the formula for the area of a sector.

The area of a sector of a circle with radius $r$ and central angle $\theta$ is given by the formula:

$\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2$

Step 3: Substitute the given values into the formula.

Given $r = 45\text{ cm}$, $\theta = 45^\circ$, and $\pi = \frac{22}{7}$:

$\text{Area} = \frac{45^\circ}{360^\circ} \times \frac{22}{7} \times (45)^2$

Step 4: Perform the arithmetic calculations.

Simplify the fraction $\frac{45}{360}$:

$\frac{45}{360} = \frac{1}{8}$

Now, calculate $45^2$:

$45 \times 45 = 2025$

Substitute these back into the area equation:

$\text{Area} = \frac{1}{8} \times \frac{22}{7} \times 2025$

$\text{Area} = \frac{1 \times 22 \times 2025}{8 \times 7}$

$\text{Area} = \frac{44550}{56}$

Divide both numerator and denominator by $2$:

$\text{Area} = \frac{22275}{28}\text{ cm}^2$

Converting to decimal form:

$\text{Area} \approx 795.5357\text{ cm}^2$

Final Answer: The area between two consecutive ribs of the umbrella is $\frac{22275}{28}\text{ cm}^2$ or approximately $795.54\text{ cm}^2$.

Given:

Radius of the circle ($r$) = $21\text{ cm}$

Angle subtended by the arc at the centre ($\theta$) = $60^\circ$

Value of $\pi$ = $\frac{22}{7}$

To Find:

Area of the sector formed by the arc.

Step 1: Stating the Formula

The area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is given by the formula:

$\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$

[Where $\theta$ is the angle subtended at the centre and $r$ is the radius of the circle.]

Step 2: Substituting the Given Values

Substitute $\theta = 60^\circ$, $r = 21\text{ cm}$, and $\pi = \frac{22}{7}$ into the formula:

$\text{Area} = \frac{60}{360} \times \frac{22}{7} \times (21)^2$

Step 3: Simplifying the Expression

First, simplify the fraction $\frac{60}{360}$:

$\frac{60}{360} = \frac{1}{6}$

Now, expand $(21)^2$:

$(21)^2 = 21 \times 21 = 441$

Substitute these back into the equation:

$\text{Area} = \frac{1}{6} \times \frac{22}{7} \times 441$

Step 4: Performing the Arithmetic Calculation

Divide $441$ by $7$:

$441 \div 7 = 63$

Now the expression is:

$\text{Area} = \frac{1}{6} \times 22 \times 63$

Simplify $\frac{63}{6}$ by dividing both by $3$:

$\frac{63}{6} = \frac{21}{2} = 10.5$

Now multiply the remaining terms:

$\text{Area} = 22 \times 10.5$

$22 \times 10.5 = 231$

Final Answer:

The area of the sector formed by the arc is $231\text{ cm}^2$.

Given:

• Radius of the circle ($r$) = $21\text{ cm}$
• Angle subtended by the arc at the centre ($\theta$) = $60^\circ$
• Value of $\pi = \frac{22}{7}$

To Find:

The area of the segment formed by the corresponding chord.

Step 1: Understanding the Formula

The area of a segment of a circle is calculated by subtracting the area of the triangle formed by the two radii and the chord from the area of the corresponding sector.

Formula: $\text{Area of Segment} = \text{Area of Sector} - \text{Area of } \triangle OAB$

Step 2: Calculating the Area of the Sector

The formula for the area of a sector is $\frac{\theta}{360^\circ} \times \pi r^2$.

Substituting the given values:

$\text{Area of Sector} = \frac{60}{360} \times \frac{22}{7} \times 21 \times 21$

$\text{Area of Sector} = \frac{1}{6} \times 22 \times 3 \times 21$ [Since $\frac{21}{7} = 3$]

$\text{Area of Sector} = \frac{1}{2} \times 22 \times 3 \times 7$ [Simplifying $\frac{3}{6} = \frac{1}{2}$]

$\text{Area of Sector} = 11 \times 3 \times 7 = 231\text{ cm}^2$

Step 3: Calculating the Area of $\triangle OAB$

Since the central angle $\theta = 60^\circ$ and the two sides $OA = OB = r = 21\text{ cm}$, the triangle is an equilateral triangle.

The formula for the area of an equilateral triangle is $\frac{\sqrt{3}}{4} \times (\text{side})^2$.

$\text{Area of } \triangle OAB = \frac{\sqrt{3}}{4} \times (21)^2$

$\text{Area of } \triangle OAB = \frac{\sqrt{3}}{4} \times 441$

$\text{Area of } \triangle OAB = 110.25\sqrt{3}\text{ cm}^2$

Step 4: Calculating the Area of the Segment

$\text{Area of Segment} = \text{Area of Sector} - \text{Area of } \triangle OAB$

$\text{Area of Segment} = 231 - 110.25\sqrt{3}$

Using $\sqrt{3} \approx 1.732$:

$\text{Area of Segment} = 231 - 110.25(1.732)$

$\text{Area of Segment} = 231 - 190.953 = 40.047\text{ cm}^2$

Final Answer: The area of the segment is $(231 - 110.25\sqrt{3})\text{ cm}^2$ or approximately $40.05\text{ cm}^2$.