(a)A wire is in the shape of a square of a side 10 cm. If the wire is

1.	(a)A wire is in the shape of a square of a side 10 cm. If the wire is rebent into a rectangle oflength 12 cm, find its breadth.(b)The diameter of a wheel is 28 cm. How many revolutions will it make on covering 176m? (Take π = 22/7 )
Answer» Let's solve both the questions one by one! $\underline{\underline{ \mathfrak{ Answer \: a :-}}}$ The breadth of the rectangle is 8 cm. Given :- A wire is in the shape of a square of side 10 cm. The wire is rebent into a rectangle of length 12 cm. To find :- It's breadth. Step-by-step explanation :- Here, it has been given that a wire is in the shape of a square of side 10 cm. We are asked to find the breadth of the wire if it is rebent into a rectangle of length 12 cm. $\underline{\sf Calculations :-}$ To find the breadth of the rectangle made using the wire, we first have to find it's length. Length of the wire will be equal to the perimeter of the square, because it's the boundary of the figure. So, to find the length of the wire, we first have to find the perimeter of the square. We know that :- $\underline{\boxed{ \sf Perimeter \: of \: a \: square = 4 \times Side}}$ Here, Side = 10 cm. Hence, $\boxed{\tt Perimeter = 4 \times 10}$ $\overline{\boxed{\tt Perimeter = 40 \: cm}}$ Perimeter of the square = 40 cm. So, the length of the wire is 40 cm. ------------------- Now, since the wire is rebent into the shape of a rectangle, THEREFORE perimeter of the rectangle will be equal to the length of the wire. Length of the wire = 40 cm. Hence, the perimeter of the rectangle is 40 cm. Let's use the length and perimeter of the rectangle to find it's breadth. We know that :- $\underline{\boxed{\sf Perimeter \: of \: a \: rectangle = 2 \: (L + B)}}$ Here, Perimeter = 40 cm. Length = 12 cm. Let the breadth be b. $\underline{\underline{\mathfrak{Substituting \: these \: values \: in \: the \: formula,}}}$ $\rm \longmapsto40 = 2 \: (12 + b)$ Removing the brackets by multiplying, $\rm \longmapsto40 = <klux>24</klux> + 2b$ Transposing 24 from RHS to LHS, changing it's sign, $\rm \longmapsto40 - 24 = 2b$ Subtracting 24 from 40, $\rm\longmapsto16 = 2b$ Transposing 2 from RHS to LHS, changing it's sign, $\rm \longmapsto \dfrac{16}{2} = b$ Dividing 16 by 2, $\overline{\boxed{\rm\longmapsto8 \: cm = b.}}$ ------------------- Hence, the breadth of the rectangle is 8 cm. ----------------------------------------------------------- $\underline{\underline{\mathfrak{Answer \: b :-}}}$ The wheel will MAKE 200 revolutions for covering 176 m. Given :- The diameter of a wheel is 28 cm. To find :- The number of revolutions it will make for covering 176 m. Step-by-step explanation :- Here, the diameter of a wheel and the distance to be COVERED has been given to us. We have to find the number of revolutions the wheel will make for covering 176 m. $\underline{\sf Calculations :- }$ The distance covered by the wheel in one revolution will be equal to it's circumference. So, to find the number of revolutions the wheel will make for covering 176 m, we have to find the circumference of the wheel. ----------------- For finding the circumference of the wheel, it is essential for us to find it's radius. We know that :- $\underline{ \boxed{\sf Radius = \frac{Diameter}{2} }}$ Here, Diameter = 28 cm. Hence, $\boxed{\bf Radius = \dfrac{28}{2}}$ Dividing 28 by 2, $\overline{ \boxed{ \bf Radius = 14 \: cm}}$ ----------------- Now let's find it's circumference! We know that :- $\underline{\boxed{\sf Circumference \: of \: a \: circle = 2\pi r}}$ Here, Radius = 14 cm. pi = 22/7. Hence, $\tt Circumference = 2 \times \dfrac{22}{7} \times 14$ Reducing the numbers, $\tt Circumference = 2 \times \dfrac{22}{1} \times 2$ Now let's multiply the remaining numbers, since we can't reduce them anymore. $\tt Circumference = 2 \times 22 \times 2$ Multiplying the numbers, $\overline{ \boxed{\tt Circumference = 88 \: cm}}$ ----------------- We know the circumference of the wheel now. So let's find the number of revolutions it will make to cover 176 m. The wheel covers 88 cm in one revolution. So, the number of revolutions needed by the wheel is :- $\underline{ \boxed{\sf\dfrac{Distance \: to \: be \: covered}{Distance \: covered \: in \: one \: revolution}}}$ First let's convert the distance to be covered in centimetres, so the units become same. 1 m = 100 cm. Hence, $\boxed{\rm Distance\: to\: be\: covered = 176 \times 100}$ Multiplying 176 by 100, $\overline{\boxed{\rm Distance\: to\: be \:covered = 17600 \: cm}}$ Now, let's apply the given formula! Here, Distance to be covered = 17600 cm. Distance covered by the wheel in one revolution = 88 cm. So, $\boxed{\bf Number \: of \: revolutions = \dfrac{17600}{88}}$ Dividing 17600 by 88, $\overline{ \boxed{\bf Number \: of \: revolutions = 200}}$ ----------------- Hence, 200 revolutions will be needed by the wheel to cover 176 m. ----------------------------------------------------------- $\underline{\rm Abbreviations \: used}$ $\sf L = Length.$ $\sf B = Breadth.$ $\sf R = Radius.$

(a)A wire is in the shape of a square of a side 10 cm. If the wire is rebent into a rectangle oflength 12 cm, find its breadth.(b)The diameter of a wheel is 28 cm. How many revolutions will it make on covering 176m? (Take π = 22/7 )

Answer»

Let's solve both the questions one by one!

$\underline{\underline{ \mathfrak{ Answer \: a :-}}}$

The breadth of the rectangle is 8 cm.

Given :-

A wire is in the shape of a square of side 10 cm.
The wire is rebent into a rectangle of length 12 cm.

To find :-

It's breadth.

Step-by-step explanation :-

Here, it has been given that a wire is in the shape of a square of side 10 cm. We are asked to find the breadth of the wire if it is rebent into a rectangle of length 12 cm.

$\underline{\sf Calculations :-}$

To find the breadth of the rectangle made using the wire, we first have to find it's length.

Length of the wire will be equal to the perimeter of the square, because it's the boundary of the figure. So, to find the length of the wire, we first have to find the perimeter of the square.

We know that :-

$\underline{\boxed{ \sf Perimeter \: of \: a \: square = 4 \times Side}}$

Here,

Side = 10 cm.

Hence,

$\boxed{\tt Perimeter = 4 \times 10}$

$\overline{\boxed{\tt Perimeter = 40 \: cm}}$

Perimeter of the square = 40 cm.
So, the length of the wire is 40 cm.

-------------------

Now, since the wire is rebent into the shape of a rectangle, THEREFORE perimeter of the rectangle will be equal to the length of the wire.

Length of the wire = 40 cm.

Hence, the perimeter of the rectangle is 40 cm.

Let's use the length and perimeter of the rectangle to find it's breadth.

We know that :-

$\underline{\boxed{\sf Perimeter \: of \: a \: rectangle = 2 \: (L + B)}}$

Here,

Perimeter = 40 cm.
Length = 12 cm.

Let the breadth be b.

$\underline{\underline{\mathfrak{Substituting \: these \: values \: in \: the \: formula,}}}$

$\rm \longmapsto40 = 2 \: (12 + b)$

Removing the brackets by multiplying,

$\rm \longmapsto40 = <klux>24</klux> + 2b$

Transposing 24 from RHS to LHS, changing it's sign,

$\rm \longmapsto40 - 24 = 2b$

Subtracting 24 from 40,

$\rm\longmapsto16 = 2b$

Transposing 2 from RHS to LHS, changing it's sign,

$\rm \longmapsto \dfrac{16}{2} = b$

Dividing 16 by 2,

$\overline{\boxed{\rm\longmapsto8 \: cm = b.}}$

-------------------

Hence, the breadth of the rectangle is 8 cm.

-----------------------------------------------------------

$\underline{\underline{\mathfrak{Answer \: b :-}}}$

The wheel will MAKE 200 revolutions for covering 176 m.

Given :-

The diameter of a wheel is 28 cm.

To find :-

The number of revolutions it will make for covering 176 m.

Step-by-step explanation :-

Here, the diameter of a wheel and the distance to be COVERED has been given to us. We have to find the number of revolutions the wheel will make for covering 176 m.

$\underline{\sf Calculations :- }$

The distance covered by the wheel in one revolution will be equal to it's circumference.

So, to find the number of revolutions the wheel will make for covering 176 m, we have to find the circumference of the wheel.

-----------------

For finding the circumference of the wheel, it is essential for us to find it's radius.

We know that :-

$\underline{ \boxed{\sf Radius = \frac{Diameter}{2} }}$

Here,

Diameter = 28 cm.

Hence,

$\boxed{\bf Radius = \dfrac{28}{2}}$

Dividing 28 by 2,

$\overline{ \boxed{ \bf Radius = 14 \: cm}}$

-----------------

Now let's find it's circumference!

We know that :-

$\underline{\boxed{\sf Circumference \: of \: a \: circle = 2\pi r}}$

Here,

Radius = 14 cm.

pi = 22/7.

Hence,

$\tt Circumference = 2 \times \dfrac{22}{7} \times 14$

Reducing the numbers,

$\tt Circumference = 2 \times \dfrac{22}{1} \times 2$

Now let's multiply the remaining numbers, since we can't reduce them anymore.

$\tt Circumference = 2 \times 22 \times 2$

Multiplying the numbers,

$\overline{ \boxed{\tt Circumference = 88 \: cm}}$

-----------------

We know the circumference of the wheel now. So let's find the number of revolutions it will make to cover 176 m.