Frustum of Cone MCQ Quiz - Objective Question with Answer for Frustum of Cone - Download Free PDF

Given:

Volume = 3696 cubic units

Height (h) = 18 units

\(\pi = \frac{22}{7}\)

Formula used:

Volume of cone = \(\frac{1}{3} \pi r^2 h\)

Calculation:

\(3696 = \frac{1}{3} \times \frac{22}{7} \times r^2 \times 18\)

⇒ \(3696 = \frac{22 \times r^2 \times 18}{21}\)

⇒ \(3696 \times 21 = 22 \times r^2 \times 18\)

⇒ \(77616 = 396 \times r^2\)

⇒ \(r^2 = \frac{77616}{396} = 196\)

⇒ \(r = 14\)

⇒ Diameter = 2 × r = 2 × 14 = 28 units

∴ The diameter of the cone is 28 units.

Given:

Height (h) = 7 cm

Capacity (Volume) = 6.6 litres = 6600 cm³ (1 litre = 1000 cm³)

Formula used:

Volume of cone = \(\frac{1}{3} \pi r^2 h\)

Where r = radius of the base

Diameter = 2 × r

Calculations:

Volume = \(\frac{1}{3} \pi r^2 h\)

⇒ 6600 = \(\frac{1}{3} \pi r^2 \times 7\)

⇒ 6600 = \(\frac{22}{7} \times \frac{1}{3} \times 7 \times r^2\)

⇒ 6600 = \(\frac{22 \times r^2}{3}\)

⇒ \(\frac{6600 \times 3}{22} = r^2\)

⇒ r² = 900

⇒ r = √900 = 30 cm

Diameter = 2 × r = 2 × 30 = 60 cm

∴ The correct answer is option (3).

Given:

Diameter (d) = 42 cm

Height (h) = 20 cm

Radius (r) = d / 2 = 42 / 2 = 21 cm

Formula used:

Total surface area of a cone = πr(r + l)

Where, l = slant height = √(r² + h²)

Calculation:

l = √(21² + 20²)

⇒ l = √(441 + 400)

⇒ l = √841

⇒ l = 29 cm

Total surface area = π × 21 × (21 + 29)

⇒ Total surface area = π × 21 × 50

⇒ Total surface area = 22/7 × 21 × 50

⇒ Total surface area = 3300 cm²

∴ The correct answer is option 4.

Given the height ratio of two cones \( C_1 \) and \( C_2 \) as \( h_1 : h_2 = 2 : 5 \) and the ratio of their diameters as \( d_1 : d_2 = 6 : 7 \), the radii ratio is \( r_1 : r_2 = 6 : 7 \). The volume of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

The ratio of their volumes is:

\[ \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \]

Substituting the given ratios:

\[ \frac{V_1}{V_2} = \left(\frac{6}{7}\right)^2 \times \frac{2}{5} = \frac{36}{49} \times \frac{2}{5} = \frac{72}{245} \]

Thus, the ratio of their volumes is:

\[ \boxed{\dfrac{72}{245}} \]

Given:

R = radius of the top circular face = 35 cm

r = radius of the bottom circular face = 28 cm

h = height of the bucket

Formula used:

Volume of frustum = \(\dfrac{1}{3} \pi h (R^2 + r^2 + Rr)\)

Volume of the frustum = 187.88 litres = 187.88 × 1000 cm3 = 187880 cm3

Calculations:

Volume of the frustum = 187880 cm3

\(\dfrac{1}{3} \pi h (R^2 + r^2 + Rr) = 187880\)

\(\dfrac{1}{3} \times \dfrac{22}{7} \times h \times (35^2 + 28^2 + 35 \times 28) = 187880\)

\(\dfrac{22}{21} \times h \times (1225 + 784 + 980) = 187880\)

\(\dfrac{22}{21} \times h \times 2989 = 187880\)

⇒ \(h = \dfrac{187880 \times 21}{22 \times 2989}\)

⇒ \(h = \dfrac{3945480}{65758}\)

⇒ h = 60 cm

∴ The correct answer is option 3.

∵ Base area of cone = π(radius)² & Volume of cone = (1/3)π × (radius)² × height

⇒ The volume of cone = (1/3) × base area × height

Now, Volume of small cone = 12π cm³

⇒ 12π = 1/3 × 4π × Height

⇒ Height of small cone = 9 cm

Similarly, Volume of large cone = 96π cm³

⇒ 96π = 1/3 × 16π × Height

⇒ Height of large cone = 18 cm

Given:

Height of cone = 3√ 23 cm

Slant height of cone = 16 cm

Formula:

l² = h² + r²

Area of the curved surface of a cone = πrl

Where r is the radius, h = height, and l is the slant height of the cone.

Calculation:

According to the given data,

⇒ r = √ [16² – (3√23)²]

⇒ √[256 - 207]

⇒ √49 = 7 cm

So, Curved surface area = (22/7) × 7 × 16 = 352 cm²

∴ The curved surface area of the cone is 352 cm2.

Given that the radius of the cone = the height of the cone

Let the radius of cone be r and the radius of the hemisphere be R.

Given that the volume of the cone is equal to the volume of the hemisphere

⇒ 1/3πr².r = 2/3πR³

⇒ r³ = 2R³

⇒ r³ = 2R³

LaGiven:

Base Area = 36π cm²

The radius of the circular upper surface, r = 3 cm.

The slant height, l = 5 cm

Formula used:

The lateral surface area of frustum = (πRL - πrl)

Calculation:

Base, Area = 36π 

⇒ πR² = 36π 

⇒ R = √ 36 = 6 cm

Given r = 3 cm

∵ ΔABC ≅ ΔADE

\(BC\over DE\) = \(AC\over AE\)

⇒ \(6\over12\) = \(5\over AE\)

⇒​ AE = 10 cm = L

The lateral surface area of the frustum

⇒ πRL - πrl

⇒ π × 6 × 10 - π × 3 × 5

⇒ 60π - 15π = 45π  

Shortcut Trick Formula Used:

Lateral Surface Area of a Frustum = π (R + r) × l

where,

R & r are the radii of a cone 

L is the slant height

Calculation:

Using the formula, 

⇒ π (6 + 3) × 5

⇒ π (9) × 5

⇒ 45 π 

∴ The lateral surface area of a frustum is 45π.

Volume of cone = (1/3)πr²h

⇒ r = h/3

⇒ h = 3r

⇒ Volume of cone = (1/3) × π × r² × 3r

⇒ Volume of cone = πr³  

Volume of sphere = (4/3)πR³

According to the question

⇒ Volume of cone = volume of sphere

⇒ πr³ = (4/3)πR³

⇒ r3 = (4/3)R3

∴ Radius of cone/ radius of sphere = ∛4 : ∛3

Given:

Radius of the base (R) = 5 cm 

Radius of the top (r) = 3 cm

Height (H) of frustum = 6 cm

Formula used:

Volume of Frustum (V) = 1/3 πH (R² + Rr + r²)

Calculations:

According to the question,

Volume of Frustum = 1/3 π × 6 × [(5)2 + (5 × 3) + (3)2]

⇒ V = π × 2 × [25 + 15 + 9]

⇒ V = π × 2 × [49] = 98π cm³ 

∴ The volume of frustum is 98π cm3. 

r = 1 cm, R = 2.5 cm, h = 7 cm

height of frustum = 7 - 1 = 6 cm

⇒ l = √[h² + (R – r)]²

⇒ l = √[(6)² + (2.5 – 1)²]

⇒ l = √(36 + 2.25)

⇒ Slant height, l = 6.2 cm

External Surface area = (curved surface area of frustum) + (curved surface area of hemisphere)

⇒ External Surface area = 22/7 × (r + R) × l + 2 × (22/7) × r²

⇒ External Surface area = 22/7 × [(1 + 2.5)6.2 + 2 × (1)²]

⇒ External Surface area = 22/7 × (3.5 × 6.2 + 2)

⇒ External Surface area = 22/7 × (21.7 + 2) = 74.29 cm²

Given:

Height =  21 cm

R = 15 cm

r = 7 cm

Formula:

Volume of Frustum = \(\frac{\pi H}{3}\)(R² + Rr + r²) 

Calculation:

Volume of the Tumbler = \(\frac{22}{7}×\frac{21}{3}\)(152 + 15 × 7 + 72) 

⇒ 8338 cm³ 

∴ The volume of a tumbler 8338 cm3 

Given:

Slant height = 15cm

Height of frustum = 12cm

Formula:

Slant height = √[(R - r)² + h²]

Calculation:

⇒ 15² = [(R - r)² + 12²]

⇒ (R - r)² = 225 - 144 = 81

⇒ (R - r) must be positive.

⇒ (R - r) = 9cm

∴ The difference between the radius of two circular bases is 9cm.

Formula used:

Volume of frustum of cone =  \(\frac{1}{3}\) × π × h ×(r12 + r22 + r1r2)

Where r₁ and r₂ are radius of the frustum of cone

Given:

Radius (r1) of the upper base = 6/2 = 3 cm

Radius (r2) of lower the base = 4/2 = 2 cm

Height = 14 cm

Calculation:

Now, Capacity of glass = Volume of a frustum of a cone

We know that,

Capacity of glass container =  \(\frac{1}{3}\) × π × h ×(r12 + r22 + r1r2)

=  \(\frac{1}{3}\) × π × 14 × (32 + 22 + 3 × 2)

= \(\frac{1}{3}\) × 266 × π cm³

∴ The capacity of the glass = 88.67π cm3