1. Cuboid
- Volume of the cuboid = l × b × h
- Lateral surface Area = Perimeter of Base × Height Base = 2(l + b) × h
- Total surface area = Lateral surface Area + 2 × Area of base = 2 (lh + bh + lb)
- Diagonal= √(l²+b²+h² )
- V = √(A1×A2×A3)
A2 ⟹ Area of one side face = bh
A3 ⟹ Area of another side face = hl
- To find the total surface area of a cuboid if the sum of all three sides and diagonals are given.
- Total surface area= (sum of all three side)² – (Diagonal)²
- For painting the surface area of a box or to know how much tin sheet is required, we will use, Total surface area.
- To find the length of the longest pole to be placed is a room, we will calculate diagonal i.e. √(l²+b²+h² )
Q1. If the sum of three dimensions and the total surface area of the rectangular box are 12 cm and 94 cm² respectively, then the maximum length of the stick that can be placed in the box is?
(a) 5√2 cm
(b) 5 cm
(c) 6 cm
(d) 2√5 cm
S1. Ans.(a)
Sol.
Maximum length of the stick that can be placed inside box=Diagonal of box
Total surface area = (sum of L + b + h)² – (Diagonal)²
94 = (12)² – (diagonal)²
(Diagonal)² = 144 – 94 = 50
Q2. In a swimming pool measuring 90m by 40m, if the displacement of water by one man is 800 cm³, what will be the rise in water level if 150 men entered the swimming pool?
(a) 0.66 m
(b) 0.44 m
(c) 0.33 m
(d) 0.25 m
S2. Ans.(c)
Sol. Volume of water displaced by 150 men = Volume of water came out
Let height raised in water = h cm
800cm³ × 150 = 9000 cm × 4000 cm × h
120 = 9 × 4000 × h
h=12/(9×400) cm
h=1/300 cm
h=1/3 m=0.33m
2. Cube(घन)
- Volume = (side)³ = a³
- Lateral surface area = 4a²
- Total surface area = 6a²
- Diagonal of the cube= √3a
- Face diagonal of the cube = √2 a
- Volume of the cube = (√(total surface area)/6)³
- In Radius of cube= a/2
- Circumradius of cube = √3/2 a
Q1. Three cubes of volume 1 cm³, 216 cm³ and 512 cm³ are melted to form a new cube. Find the diagonal of the New cube?
(a) 15.6 cm
(b) 16.6 cm
(c) 17.6 cm
(d) 18.6 cm
S1. Ans.(a)
Sol.
Volume of New cube
= 1 cm² + 216 cm² + 512 cm³
= 729 cm³
a³ = 729 cm³
a = 9 cm
Diagonal of New cube = √3a
= √3×a
=9√3
= 9 × 1.7320
≅ 15.56 cm
Q2. The cost of painting the whole surface area of a cube at the rate of B paise per sq. cm is Rs. 343.98. Then the volume of the cube is?
(a) 9261cm³
(b) 9264cm³
(c) 10248cm³
(d) 13310cm³
S2. Ans.(a)
Sol.
Cost of painting the whole surface = Rs. 343.98 = 34398 paise
Total surface area=34398/13=2646 cm²
Total surface area = cube = 6a²
6a² = 2646
a² = 441, a = 21 cm
Volume of cube = a³
= (21)³ = 9261 cm³
3. Right circular cone
- Slant height , l = √(r²+h² )
- Volume = 1/3×area of base×height = 1/3 πr² h
- Curved surface area= 1/2 (Perimeter of base) × slant height = 1/2 × 2πr×l=πrl=πr√(r²+h² )
- Total surface area = C.S.A + Area of base = πrl+πr²=πr(l+r)
- If cone is formed by sector of a circle then.(a) Slant height = radius of circle , (b) circumference of base of cone = length of arc of sector
- Radius of maximum size sphere in a cone = (h × r)/(l + r) [ r → radius of the cone , l →slant height of the cone , h→height of the cone)]
- If cone is cut parallel to its base and ratio of heights, radius or slant height of both parts is given as →x∶y. ,Then Ratio of their volume = x³ ∶y³
Q1. 5 persons live in a tent. If each person requires 16 m² of floor area and 100 m³ space for air then the height of the cone of smallest size to accommodate these persons would be?
(a) 16 m
(b) 18.75 m
(c) 10.25 m
(d) 20 m
S1. Ans.(b)
Sol. Let the height of cone h meter
⇒ Total area of ground will be required
= 5 × 16 m²= 80 m²
⇒ Total volume of air is needed
= 100 × 5 m³ = 500 m³
⇒ volume of cone = 500 m³
⇒ 1/3 area of ground = 500
⇒ 1/3 × πr² × h = 500
=1/3×80×h = 500
⇒ height = (500 × 3)/80
⇒ height of cone = 18.75 metres
Q2. In a right circular cone, the radius of its base is 7 cm and its height 24 cm. A cross-section is made through the midpoint of the height parallel to the base. The volume of the upper portion is?
(a) 169 cm³
(b) 154 cm³
(c) 1078 cm³
(d) 800 cm³
More Questions
Q1. The lateral surface area of frustum of a right circular cone , if the area of its base is 16π cm² and the diameter of circular upper surface is 4 cm and slant height 6 cm, will be?
(a) 30π cm²
(b) 48π cm²
(c) 36π cm²
(d) 60π cm²
Q2. An inverted conical shaped vessel is filled with water to its brim. The height of the vessel is 8 cm and radius of the open end is 5 cm. When a few solid spherical metallic balls each of radius 1/2 cm are dropped in the vessel, 25% water is overflowed. The number of balls is:
(a) 100
(b) 400
(c) 200
(d) 150
Q3. If the height and the radius of the base of a cone are each increased by 100%, then the volume of the cone becomes
(a) double that of the original
(b) three times that of the original
(c) six times that of the original
(d) eight times that of the original
Q4. A plane divides a right circular cone into two parts of equal volume. If the plane is parallel to the base, then the ratio, in which the height of the cone is divided, is?
(a) 1 : √2
(b) 1 : ∛2
(c) 1 : ∛2-1
(d) 1 : ∛2+1
Q5. A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of the sphere. What is the ratio of the height and radius of the cone so formed?
(a) 4 : 3
(b) 2 : 3
(c) 3 : 4
(d) 4 : 1
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