Question

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【A̸̸】An open box is made of wood 3 cm thick. Its external Length, breadth and height are 1.46 m, 1.16 m and 8.3 dm respectively. What is the cost of painting the inner surface of the box at 50p per 100 cm².

【B̸】66 cm³ of silver is drawn into a wire 1 mm in diameter. What is the length of wire in metres ?

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Answer 1

A
Given thickness of wood = 3 cm
Length of open box = 1.48 m = 148 cm 
Breadth of open box = 1.16 m = 116 cm 
Height of open box = 8.3 dm = 83 cm 
Inner length, l = outer length – twice width = 148 – 6 = 142 cm 
Outer breath = 116 – 6 = 110 cm 
Outer height = 83 – 3 = 80 cm
[since it is an open box]Area of inner surface = 2(lb + bh + hl) – lb= 2[142 x 110 + 110 x 80 + 80 x 142] – [142 x 110] = 55940 sq cm 
Cost of painting inner surface = Rs (50/100) x (55940/100) = Rs 279.90

B
As we know that volume of wire is volume of cylinder.

And in given question silver wire is drawn from 66 cubic centimetres of silver.

So we can equivalent the volume of cylinder with 66 cubic centimetres.

we know that volume of cylinder is pi*r^2*l.  [where l = length and r = radius]

Given, r = 1/2mm.

i.e. r = 0.5mm.

i.e. r =0.05cm.

According to question pi*r^2*l = 66

i.e. 22/7 * (0.05)^2* l = 66

by solving this question we get l = 8400cm.

So, length of wire in metres is 84m.

Answer 2

$\underline{\underline{\Huge{\textbf{Solution\: for\: Question-1:}}}}$

Given,
Thickness of the wooden box $= 3\: m$
Length of the wooden box $= 1.46\: m$
Breadth of the wooden box $= 1.16\: m$
Height of the wooden box $= 8.3\: dm$

Cost of painting the $\textit{inner surface}$ of the box at $50\: ps$ per $100\: cm^{2}$= ?

$\underline{\Large{\textsf{Step-1:}}}$
Express all the dimensions in cm

We have,
$\bigstar\: 1\: m= 100\: cm$
$\bigstar\: 1\: dm = 10\: cm$

Length of the wooden box $= 146\: cm$
Breadth of the wooden box $= 116\: cm$
Height of the wooden box $=83\: cm$

Note that all these dimensions are Outer dimensions of the wooden box.

$\underline{\Large{\textsf{Step-2:}}}$
Find the inner dimensions of the wooden box

Inner Length(and breadth) of the box is twice its width removed from its Outer length(and breadth).

Inner height of the box is equal to width of the box removed from Outer height.
($\textsf{As the box is Ope}\textsf{n}$, only the thickness of base has to be removed from Outer length)

$\therefore$

$\textsf{Inner length = Outer length -2(Width)}$

$\implies$ Inner length $= 146-2(3)$

$\implies$ Inner length $l = 140\: cm$

$\textsf{Inner breadth = Outer breadth -2(Width)}$

$\implies$ Inner breadth $= 116-2(3)$

$\implies$ Inner breadth $b = 110\: cm$

$\textsf{Inner height = Outer height - width}$

$\implies$ Inner height $= 83-3$

$\implies$ Inner height $h = 80\: cm$

$\underline{\Large{\textsf{Step-3:}}}$
Find the inner area of the wooden box

Removing the Area of the Base from the Outer Area gives Inner Area

$\textsf{Inner Area = 2(lb+bh+lh)-lb}$

Substituting Values

$\implies$ Inner Area = $2\left[(140\times110)+(110\times80)+(80\times140)\right]-\left(140\times110\right)$

$\implies$ Inner Area = $2\left[(140\times110)+(110\times80)+(80\times140)\right]-\left(140\times110\right)$

$\implies$ Inner Area = $2\left[(15400)+(8800)+(11200)\right]-\left(15400\right)$

$\implies$Inner Area = $2\left[35400\right]-\left(15400\right)$

$\implies$Inner Area = 70800-15400

$\implies$Inner Area $= 55400\: cm^{2}$

$\underline{\Large{\textsf{Step-4:}}}$
Find the cost of painting the inner surface

$\textsf{Cost of Painting = Rate of painting \times Area}$

Cost of Painting $= \dfrac{50}{100}\times 55400$

Cost of Painting $= 27700\: ps$

$\underline{\Large{\textsf{Step-5:}}}$
Express the cost of Painting in Rs

We have,
$\bigstar\: \mathbf{1\: Re = 100\: ps}$
$\implies 1\: ps = \dfrac{1}{100}\: Re$

Cost of Painting $= 27700\: \times \dfrac{1}{100}\: Rs$

Cost of Painting $= 277\: Rs$



$\underline{\underline{\Huge{\textbf{Solution\: for\: Question-2:}}}}$

Given,
Volume of silver = $66\: cm^{3}$
Diameter of wire = $1\: mm$

$\underline{\Large{\textsf{Step-1:}}}$
Find the radius of the wire

Since the wire is Cylindrical in shape,
It's base is a circle

For a circle
We have,
$\bigstar\: \textbf{Diameter = 2 \times Radius}$
$\implies Radius = \dfrac{Diameter}{2}$

Substitute value

$\implies Radius = \dfrac{1\: mm}{2}$

$\implies Radius = 0.5\: mm$

$\underline{\Large{\textsf{Step-2:}}}$
Express the volume of silver in $mm^{3}$

We have,
$\bigstar\: \mathbf{1\: cm^{3}= 1000\: mm^{3}}$

$\implies$ Volume of silver = $66\times 1000\: mm^{3}$

$\implies$ Volume of silver = $66000\: mm^{3}$

$\underline{\Large{\textsf{Step-3:}}}$
Assume a variable for the height of wire

Let height of the wire be 'h' mm

Since the Wire is Cylindrical in shape, it's height is equal to it's length.

$\underline{\Large{\textsf{Step-4:}}}$
Find the height of the wire

Since the shape of the wire is Cylindrical
We have,
$\bigstar\: \mathbf{Volume= \pi r^{2} h}$

Substitute Values

$\implies 66000 = \left(\dfrac{22}{7}\right) \times \left(\dfrac{1}{2}\right)^{2} \times h$

$\implies 66000= \left(\dfrac{11}{14}\right) \times h$

$\implies 6000= \left(\dfrac{1}{14}\right) \times h$

$\implies h = 6000\times 14$

$\implies h = 84000\: mm$

$\underline{\Large{\textsf{Step-5:}}}$
Express the height of the wire in mts

$\bigstar\: \mathbf{1\: m= 1000\: mm}$
$\implies 1\: mm = \dfrac{1}{1000}\: m$

$\implies h = 84000\times \dfrac{1}{1000}\: m$

$\implies h = 84\: m$