Surface Areas and Volumes | Exercise 12.1

Question 1

Unless stated otherwise, take π=227\pi = \dfrac{22}{7}.

2 cubes each of volume 64 cm364\text{ cm}^3 are joined end to end. Find the surface area of the resulting cuboid.

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Solution
Understand the Question
  • Given two cubes, each of volume 64 cm364\text{ cm}^3, joined end to end.
  • Using the formula for the volume of a cube, V=a3V = a^3, we first find the side length aa.
  • When two cubes of side aa are placed end to end, the resulting shape is a cuboid where:
    • Length l=a+a=2al = a + a = 2a
    • Breadth b=ab = a
    • Height h=ah = a
  • The total surface area of the resulting cuboid is then calculated using the formula Surface Area=2(lb+bh+hl)\text{Surface Area} = 2(lb + bh + hl).

Step 1 · Find the Side Length of One Cube

Let the edge length of each cube be aa.Diagram 1

Given Volume of cube=a3=64 cm3\text{Volume of cube} = a^3 = 64\text{ cm}^3

a=643=4 cm\begin{aligned} a &= \sqrt[3]{64} \\ &= 4\text{ cm} \end{aligned}

Step 2 · Find the Dimensions of the Resulting Cuboid

When two cubes of edge 4 cm4\text{ cm} are joined end to end, only the length changes while the breadth and height remain the same.Diagram 2

Dimensions of the resulting cuboid:

l=4+4=8 cmb=4 cmh=4 cm\begin{aligned} l &= 4 + 4 = 8\text{ cm} \\ b &= 4\text{ cm} \\ h &= 4\text{ cm} \end{aligned}

Step 3 · Calculate the Surface Area of the Cuboid

Using the surface area formula of a cuboid

Surface Area=2(lb+bh+hl)=2((8×4)+(4×4)+(4×8))=2(32+16+32)=2(80)=160 cm2\begin{aligned} \text{Surface Area} &= 2(lb + bh + hl) \\[0.6em] &= 2((8 \times 4) + (4 \times 4) + (4 \times 8)) \\[0.6em] &= 2(32 + 16 + 32) \\[0.6em] &= 2(80) \\[0.6em] &= 160\text{ cm}^2 \end{aligned}
Answer

160 cm2160\text{ cm}^2

Common Mistakes
  • Simply Adding Surface Areas: Calculating the surface area of two separate cubes as 2×6a2=2×96=192 cm22 \times 6a^2 = 2 \times 96 = 192\text{ cm}^2. When joined, two square faces meet inside, reducing the exposed surface area by 2a2=32 cm22a^2 = 32\text{ cm}^2.
  • Modifying All Dimensions: Doubling all three dimensions (l,b,hl, b, h) instead of only the length along which the cubes are joined.

More questions in Exercise 12.1

Q1

Unless stated otherwise, take π=227\pi = \dfrac{22}{7}.

2 cubes each of volume 64 cm364\text{ cm}^3 are joined end to end. Find the surface area of the resulting cuboid.

Q2

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm14\text{ cm} and the total height of the vessel is 13 cm13\text{ cm}. Find the inner surface area of the vessel.

Q3

A toy is in the form of a cone of radius 3.5 cm3.5\text{ cm} mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm15.5\text{ cm}. Find the total surface area of the toy.

Q4

A cubical block of side 7 cm7 \text{ cm} is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

Q5

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter ll of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

Q6

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is 14 mm14\text{ mm} and the diameter of the capsule is 5 mm5\text{ mm}. Find its surface area.

Q7

A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m2.1\text{ m} and 4 m4\text{ m} respectively, and the slant height of the top is 2.8 m2.8\text{ m}, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ₹500500 per m2\text{m}^2. (Note that the base of the tent will not be covered with canvas.)

Q8

From a solid cylinder whose height is 2.4 cm2.4 \text{ cm} and diameter 1.4 cm1.4 \text{ cm}, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm2\text{cm}^2.

Q9

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is 10 cm10\text{ cm}, and its base is of radius 3.5 cm3.5\text{ cm}, find the total surface area of the article.

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