# Arc Length Homework

#### Solution:

Diameter of the cart wheel is 4.3 m[Given.]

Circumference of the wheel = 4.3

[Circumference = .]

Distance covered for 10 revolutions = 10 × 4.3 = 43 m.

[Distance covered = circumference × no. of revolutions.]

Correct answer : (3)

#### Solution:

Radius of the wheel is 50 cm[Given.]

Diameter = 100 cm = 1 m

Circumference of the wheel = 1

[Circumference = .]

No. of revolutions made in traveling a distance of 270 m = = 90

[.]

Correct answer : (3)

#### Solution:

Base of the right triangle = 13.5 cm.Height of the right triangle = 18 cm.

Radius of the circle = = 11.25 cm.

[The radius of the circumcircle of a right triangle is half of the hypotenuse.]

Circumference of the circle = 2r = 2 × × 11.25 = 67.5 cm.

Correct answer : (1)

#### Solution:

Perimeter of the figure = Length of the line segment AB + Length of arc BC + Length of line segment CD + length of arc DE + length of line segment EF + length of line segment FA

AB = 2 cm

[Given.]

Length of arc BC = × × 2 = 1.57 cm

[Formula for the length of the arc = × × radius.]

CD = 5 cm

[Given.]

Length of arc DE = × = 6.28 cm

Δ AEF is equilateral.

[AF = EF, AFE = 60.]

AF = EF = AE = 1 + 5 + 4 = 10 cm

[AE = AC + CD + DE.]

Perimeter of the figure = 1 + 1.57 + 5 + 6.28 + 10 + 10 = 33.85 cm

[Step 1.]

Correct answer : (3)

#### Solution:

When the circle is inscribed in a square (figure A), perimeter of the square = 4 × 2 = 8

[Side of the square = 2.]

Circumference of the circle = 2

2 < 8

So, in figure(A), the perimeter of the circle is always less than the perimeter of the square.

When the square is inscribed in the circle, side of the square = ×

[Diagonal of the square = 2, side = = × .]

Perimeter of the square = 4 ×

Perimeter of the circle = 2

4 < 2

So, in figure(B), the Perimeter of the square is less than perimeter of the circle.

Statements I and III are correct.

Correct answer : (4)

#### Solution:

Length of arc 1 = × 2 × 6 =[Formula for the length of arc = × 2 × .]

Length of arc 2 = × 2 × 3 = ×

Length of arc 3 = × 2 = ×

The ascending order is arc 3, arc 1, arc 2.

Correct answer : (3)

#### Solution:

The radius of the first semicircle is 17.Radius of the second semi circle =

Radius of the third semicircle =

Circumference of the first semicircle = × 17

Circumference of the second semicircle = ×

Circumference of the third semicircle = ×

Sum of the circumferences = × 17 + × + × + ------------------

= (17 + + + ---------------- )

= 2(17 + + + ---------------- ) = 7 m

[Given, the semicircles are drawn on a line of length 7 m.]

So, the sum of the arclengths of all the semi circular portions = × m = m.

[Given, line length = 7 m.]

Correct answer : (4)

#### Solution:

Perimeter of the outer curve = × 2 × 5 = 14.39 cmPerimeter of the inner curve = × 2 × 6 = 8.06 cm

Perimeter = 14.39 - 8.06 = 6.33 cm

Correct answer : (4)

#### Solution:

The distance covered by the cyclists are the circumference of the semi circles.Distance covered by the cyclist R = Circumference of the semi circle with AE as diameter.

=

[Circumference of semi circle with diameter is .]

= = 4

[Substitute AE = 8.]

Distance covered by the cyclist Q = Circumference of semicircle with diameter AC + circumference of semi circle with diameter CE.

=

=

Distance covered by the cyclist P = circumference of the semi circle with diameter AB + circumference of the semi circle with diameter BC + circumference of the semi circle with diameter CD + circumference of the semicircle with diameter DE.

= × × AB + × × BC + × × CD + × × DE

= × × 2 + × × 2 + × × 2 + × × 2 = 4

The distances covered by the cyclists P, Q and R are same.

Therefore, three cyclists P, Q and R reach the point E in same time.

Correct answer : (1)

#### Solution:

Radius of the cake = = 15 cm.[Radius = × diameter.]

Length of the decorative band is 100 cm.

[Given.]

Perimeter of the cake = 2.

[Perimeter of the circle = 2 .]

2 × 3 × 15 = 90 cm

[Substitute the values and simplify.]

The required length of the band is 90 cm, therefore the given band is sufficient to decorate the cake.

Correct answer : (1)

1.

The diameter of a cart wheel is 4.3 m. How much distance will it cover if it makes 10 revolutions?

2.

The radius of a bicycle wheel is 50 cm. How many revolutions does the wheel make when the bicycle travels 270 m?[Take = 3.]

3.

4.

5.

Select the correct statement / statements.

I. When a square is inscribed in a circle, the perimeter of the square is always less than the perimeter of the circle.

II. When a circle is inscribed in a square, the perimeter of the circle will be less than the perimeter of the square if the radius of the circle is greater than 2.

III. When a circle is inscribed in a square, the perimeter of the circle will be always less than the perimeter of the square.

6.

7.

8.

[Take = 77

^{o}, = 165

^{o}, = 6 cm, = 5 cm.]

9.

10.

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