13. A round table cover has six equal designs as shown in Fig. 11.11. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹ 0.35 per . (Use )

We need to find the total area of the six designs and then calculate the cost. Step 1 — Find the area of one design The round table cover has a radius of 28 cm. There are six equal designs. These designs are segments of the circle. Let's find the angle of each sector. Consider one sector, say OAB. OA and OB are radii, so . The angle is . Since two sides are equal and the included angle is , triangle OAB is equilateral. Let's calculate the area of this sector. Now, let's find the area of the equilateral triangle OAB. The area of one design is the area of the sector minus the area of the triangle. > Step 2 — Calculate the total area of all designs There are 6 such designs. We will multiply the area of one design by 6. > Step 3 — Calculate the total cost The cost of making the designs is ₹ 0.35 per . We multiply the total area by the cost per square centimeter. > Answer > The cost of making the designs is ₹ 162.68.

Areas Related to Circles | Exercise 11.1

Question 13

A round table cover has six equal designs as shown in Fig. 11.11. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹ 0.35 per $\text{cm}^2$ . (Use $\sqrt{3} = 1.7$ )

Solution

We need to find the total area of the six designs and then calculate the cost.

Step 1 — Find the area of one design

The round table cover has a radius of 28 cm. There are six equal designs. These designs are segments of the circle. Let's find the angle of each sector.

$\text{Angle of each sector} = \frac{360^\circ}{6}$

$= 60^\circ$

Consider one sector, say OAB. OA and OB are radii, so $OA = OB = \mathbf{28 \text{ cm}}$ . The angle $\angle AOB$ is $\mathbf{60^\circ}$ . Since two sides are equal and the included angle is $60^\circ$ , triangle OAB is equilateral. Let's calculate the area of this sector.

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

$= \frac{60^\circ}{360^\circ} \times \frac{22}{7} \times (28)^2$

$= \frac{1}{6} \times \frac{22}{7} \times 28 \times 28$

$= \frac{1}{6} \times 22 \times 4 \times 28$

$= \frac{2464}{6}$

$= \frac{1232}{3} \text{ cm}^2$

Now, let's find the area of the equilateral triangle OAB.

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

$= \frac{1.7}{4} \times (28)^2$

$= \frac{1.7}{4} \times 784$

$= 1.7 \times 196$

$= 333.2 \text{ cm}^2$

The area of one design is the area of the sector minus the area of the triangle.

$\text{Area of one design} = \left(\frac{1232}{3} - 333.2\right) \text{ cm}^2$

$\boxed{\text{Area of one design} = \left(\frac{1232}{3} - 333.2\right) \text{ cm}^2}$

Diagram 1

Step 2 — Calculate the total area of all designs

There are 6 such designs. We will multiply the area of one design by 6.

$\text{Total area of designs} = 6 \times \left(\frac{1232}{3} - 333.2\right)$

$= \left(6 \times \frac{1232}{3}\right) - (6 \times 333.2)$

$= (2 \times 1232) - 1999.2$

$= 2464 - 1999.2$

$\boxed{\text{Total area of designs} = 464.8 \text{ cm}^2}$

Step 3 — Calculate the total cost

The cost of making the designs is ₹ 0.35 per $\text{cm}^2$ . We multiply the total area by the cost per square centimeter.

$\text{Total cost} = 464.8 \times 0.35$

$\boxed{\text{Total cost} = ₹ 162.68}$

Answer

The cost of making the designs is ₹ 162.68.

Question 13

Answer

More questions in Exercise 11.1