DAV Class 8 Maths Chapter 14 Worksheet 5
Mensuration Worksheet 5
1. Two cubes each of side 6 cm are joined together. Name the new solid formed and, also find its volume.
Solution
\[ \begin{align*} \text{When two cubes } & \text{are joined, new solid (cuboid) is formed} \\ Length \ (AC) &= 12 \, \text{cm} \\ Breadth \ (CD) &= 6 \, \text{cm} \\ Height \ (DE) &= 6 \, \text{cm} \\ \\ \color{green} \text{Volume of cuboid} & = \color{green} l \times b \times h \\ &= 12 \times 6 \times 6 \\ &= 432 \ cm^3 \\ \end{align*} \]
Answer Volume of the new cuboid \( = \color{red} 432 \ cm^3 \)
2. Find the height of a cuboid whose volume is 275 cm3 and the base area is 25 cm2.
Solution
\[ \begin{aligned} \text{Base area} &= 25 \ \text{cm}^2 \\ \implies l \times b &= 25 \ \text{cm}^2 \\[6pt] \text{Volume of cuboid} &= 275 \ \text{cm}^3 \\[6pt] l \times b \times h &= 275 \\[6pt] 25 \times h &= 275 \\[6pt] h &= \frac{275}{25} \\[6pt] & = 11 \ \text{cm} \end{aligned} \]
Answer The height of the cuboid \(= \color{red}{11 \ \text{cm}}\)
3. The dimensions of a box are 60 cm × 54 cm × 30 cm. How many small cubes of side 6 cm can be placed in the box?
Solution
\[ \begin{aligned} \text{Volume of box} & = 60 \times 54 \times 30 \\ \text{Volume of cube} & = 6 \times 6 \times 6 \\[6pt] \text{No. of cubes} &= \frac{\text{Volume of box}}{\text{Volume of small cube}} \\ \\ &= \frac{\cancel{60}^{10} \times \cancel{54}^{9} \times \cancel{30}^{5}}{\cancel6_1 \times \cancel6_1 \times \cancel6_1} \\ \\ &= 10 \times 9 \times 5 \\[6pt] &= 450 \end{aligned} \]
Answer The number of small cubes that can be placed in the box \(=\ \color{red}{450}\)
4. If each edge of a cube is doubled,
(i) How many times will its surface area increase?
Solution
\[ \begin{align*} \color{magenta} \text{Small Cube} \\ l = x \\ \text{TSA of small cube} &= 6 l^2\\ TSA & = 6x^2 \\ \\ \color{magenta} \text{Big Cube} \\ l = 2x \\ \text{TSA of small cube} &= 6 l^2 \\ TSA & = 6 (2x)^2 \\ & = 6 \times 4x^2 \\ TSA & = 24x^2 \\\\ \text{Increase in surface area} & = \frac{24x^2}{6x^2} \\[6pt] \implies & 4 \ times \end{align*} \]
Answer If each edge of a cube is doubled surface area of cube is increased by \( = \color{red} 4 \ times \)
(ii) how many times will its volume increase?
Solution
\[ \begin{align*} \color{magenta} \text{Small Cube} \\ side = x \\ \text{Volume of small cube} &= (side)^3\\ Volume & = x^3 \\ \\ \color{magenta} \text{Big Cube} \\ side = 2x \\ \text{Volume of small cube} &= (side)^3 \\ & = (2x)^3 \\ Volume & = 8x^3 \\\\ \text{Increase in Volume} & = \frac{8x^3}{x^3} \\[6pt] \implies & 8 \ times \end{align*} \]
Answer If each edge of a cube is doubled Volume of cube is increased by \( = \color{red} 8 \ times \)
5. A match box measures 4 cm × 2.5 cm × 1.5 cm. What will be the volume of a packet containing 12 such boxes?
Solution
\[ \begin{aligned} \color{green} \text{Volume of one box} &=\color{green} l \times b \times h \\ &= 4 \times 2.5 \times 1.5 \\ &= 10 \times 1.5 \\ &= 15\ \text{cm}^3 \\[10pt] \text{Number of boxes} &= 12 \\[6pt] \text{Volume of 12 boxes} &= 12 \times 15 \\ &= 180\ \text{cm}^3 \end{aligned} \]
Answer The volume of the packet containing 12 boxes \(= \color{red}{180 \ \text{cm}^3}\)
6. Volume of a cube is 1000 cm3. Find its total surface area.
Solution
\[ \begin{aligned} \color{green} Volume &= \color{green} (side)^3 \\[6pt] Volume&= 1000 \ \text{cm}^3 \\[6pt] (side)^3 &= 1000 \ \text{cm}^3 \\[6pt] side & = \sqrt[3]{1000} \\[6pt] & = \sqrt[3]{10^3} \\[6pt] side & = 10 \ cm \\ \\ \color{green}\text{T.S.A} &= \color{green}6l^2 \\[6pt] &= 6 \times 10^2 \\[4pt] &= 6 \times 100 \\[4pt] &= 600 \ \text{cm}^2 \end{aligned} \]
Answer Total surface area \(= \color{red}{600 \ \text{cm}^2}\)
7. How many planks each of which is 2 m long, 2.5 cm broad and 4 cm thick can be cut off from a wooden block 6 m long, 15 cm broad and 40 cm thick?
Solution
\[ \begin{aligned} & \textbf{Convert to } (cm)\\ & \color{magenta} \text{Wodden block} \\ l & = 6 \ m \implies 600 \ cm \\ b & = 15 \ cm \\ h & = 40 \ cm \\ \\ & \color{magenta} \text{Plank} \\ l & = 2 \ m \implies 200 \ cm \\ b & = 2.5 \ cm \\ h & = 4 \ cm \\ \\ \color{green} \text{No. of planks} &= \color{green} \frac{\text{Volume of Wodden block}}{\text{Volume of Plank}} \\ \\ &= \frac{\cancel{600}^{\ 3} \times 15 \times 40}{\cancel{200}_1 \times 2.5 \times 4} \\ \\ &= \frac{45 \times \cancel{40}^4}{\cancel{10}_1} \\ \\ &= 45 \times 4 \\ & = 180 \end{aligned} \]
Answer The number of planks that can be cut \(=\ \color{red}{180}\)
8. Three solid metal cubes with edges 6 cm, 8 cm and 10 cm respectively are melted together and formed into a single cube. Find the volume of the new cube.
Solution
\[ \begin{aligned} \text{Volume of cube 1 }(V_1) &= 6^3 \implies 216\ \text{cm}^3\\ \text{Volume of cube 2 }(V_2) &= 8^3 \implies 512\ \text{cm}^3\\ \text{Volume of cube 3 }(V_3) &= 10^3 \implies 1000\ \text{cm}^3\\[6pt] \text{Volume of new cube} &= V_1 + V_2 + V_3 \\[4pt] &= 216 + 512 + 1000 \\[4pt] &= 1728\ \text{cm}^3 \\[8pt] \end{aligned} \]
Answer The volume of the new cube \(= \color{red}{1728\ \text{cm}^3}\)
9. Find the volume of a cube whose surface area is 150 m2.
Solution
\[ \begin{aligned} \color{green} \text{TSA of cube} & = \color{green} 6l^2 \\ TSA &= 150\ \text{m}^2 \\[6pt] 6l^2 &= 150 \\[4pt] l^2 &= \frac{150}{6} \\ l^2 & = 25 \\[6pt] l &= \sqrt{25} \\[6pt] l& = 5\ \text{m} \\[10pt] \color{green}\text{Volume of the cube} &= \color{green}(side)^3 \\ & = 5^3 \\ & = 125\ \text{m}^3 \end{aligned} \]
Answer Volume of the cube \(= \color{red}{125\ \text{m}^3}\)
10. Find the volume of a cube, one face of which has an area of 81 m2.
Solution
\[ \begin{aligned} \text{Area of one face} &= (side)^2 \\[6pt] \text{Area of one face} &= 81\ \text{m}^2 \\[6pt] (side)^2 &= 81 \\[6pt] side &= \sqrt{81} \\[6pt] side & = 9\ \text{m} \\[10pt] \color{green}\text{Volume} &= \color{green}(side)^3 \\ &= 9^3 \\ &= 729\ \text{m}^3 \end{aligned} \]
Answer Volume of the cube \(= \color{red}{729 \ \text{m}^3}\)