Question

The volume of a cuboidal block of silver is 10368 cm³. If its dimensions are in the ratio 3 : 2 : 1, find
(i) The dimensions of the block.
(ii) The cost of gold polishing its entire surface at ₹0.50 per cm²​

Answer 1

Given :-

The volume of a cuboidal block = 10368 cm³

Ratio of it's dimensions = 3 : 2 : 1

Cost per centimeter = Rs. 0.50

To Find :-

The length of the cuboid.

The breadth of the cuboid.

The height of the cuboid.

The cost of gold polishing its entire surface.

Analysis :-

Take the dimensions as variables and make an equation.

Find the value of the variable and substitute them in the dimensions.

Then using the formula of TSA, find the area to be polished.

Multiply the TSA by the cost per centimeter.

Solution :-

We know that,

• l = Length
• b = Breadth
• h = Height

Let the length, breadth and height be 3x, 2x and x respectively.

By the formula,

$\underline{\boxed{\sf Volume \ of \ cuboid = Length \times Breadth \times Height}}$

Given that,

Length (l) = 3x

Breadth (b) = 2x

Height (h) = x

Volume = 10368 cm³

Substituting their values,

$\sf 10368=3x \times 2x \times x$

$\sf 10368=6 x^{3}$

$\sf x^{3}=\dfrac{10368}{6}=1728$

$\sf x=12$

Therefore, the value of x is 12.

Finding dimensions,

Height = 12 cm

Length (l) = 3x = 3 × 12

Length = 36 cm

Breadth (b) = 2x = 2 × 12

Breadth = 24 cm

Finding the TSA,

$\underline{\boxed{\sf TSA \ of \ cuboid=2(lb+bh+hl) }}$

Substituting them,

= 2 (864 + 288 + 432)

= 2 × 1584

= 3168 cm²

Finding the cost,

Cost of gold polishing per cm = Rs. 0.50

Area of gold polishing = Cost per cm × TSA of the cuboid

Substituting them,

Area of gold polishing = 0.50 × 3168

= Rs. 1584

Therefore, it takes Rs. 1584 for gold polishing.

Answer 2

Answer :-

i) Dimensions of Cuboid :-

• Length, l = 36 cm

• Breadth, b = 24 cm

• Height, h = 12 cm

ii) Cost of Polishing entire surface

= ₹ 1584

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★ Concept :-

Here the concept of Volume and Total surface area of Cuboid are used.

=> Volume = l × b × h

=> Total Surface Area = 2(lb+bh+lh)

_____________________

★ Solution :-

Given,

» The dimensions of cuboid are in ratio of 3:2:1

» Volume of cuboid = 10368 cm³

Then,

▶Let the length (l) of the cuboid be 3x

▶Let the breadth (b) of the cuboid be 2x

▶Let the height (h) of the cuboid be 1x

where x is the constant by which all teh dimensions are multiplied.

________________________________

By applying the length, breadth and height in the formula of Volume, we get,

✒ Volume = length × breadth × height

✒ 3x × 2x × 1x = 10368

✒ 6x³ = 10368

✒ x³ = 10368/6 = 1728

✒ x = ³√1728 = 12

Hence, x = 12 .

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By applying the value of x, in length, breadth and height, we get,

▶ Length, l = 3x = 3(12) = 36 cm

▶Breadth, b = 2x = 2(12) = 24 cm

▶Height, h = 1x = 1(12) = 12 cm

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In order to find the total cost of polishing we must multiply the total area to be painted with the rate of painting an area.

☞ Total Cost = Area to be painted × Rate

Let us find the area to be painted. In order to find the area to be painted, we must take total surface area of solid.

→ T.S.A. = 2(lb + bh + lh)

→ T.S.A. = 2(36×24 + 24×12 + 36×12)

→ T.S.A. = 2(864 + 288 + 432)

→ T.S.A. = 2 × 1584

→ T.S.A. = 3168 cm²

Hence, we get total surface to be painted = 3168 cm².

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• Rate of Painting per area = ₹ 0.50 per cm²

Then,

✏ Total Cost of Painting = 3168 × 0.5

✏ Total Cost of Painting = ₹ 1584

Hence, we get the total cost of painting the cuboid = ₹ 1584.

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★ More to know :-

• Volume of Cube = (Side)³

• Volume of cylinder = πr²h

• Volume of Sphere = 4/3 (πr³)

• Volume of cone = ⅓(πr²h)