Test: Cylinders - 1 - GMAT MCQ

Given:

The diameter of a sphere = 8 cm

The diameter of wire = 3 mm = 0.3 cm

Concept used:

Volume of sphere = 4/3 πr³

Volume of cylinder = πr²h

1 mm = 0.1 cm

Calculation:

The radius of a sphere = 8/2 = 4 cm

The radius of wire = 0.3/2 = 0.15 cm

Now, according to the above concept

⇒ Volume of sphere = Volume of wire

⇒ (4/3 × π × 4 × 4 × 4) = (π × 0.15 × 0.15 × h)

⇒ 256/3 = 0.225h

⇒ h = 256/0.0675

⇒ h = 3792.59 cm

⇒ h = 37.9 m   [1 m = 100 cm]

∴ The length of the wire is 37.9 m.

Given:

Inner radius, r = 15 cm

Outer radius, R = 16 cm

Height, h = 63 cm

Concept:

Volume of a hollow cylinder = π(R² - r²)h

Calculation:

We know that,

Volume of a hollow cylinder = π(R2 - r2)h

⇒ (22/7) × [16² - 15²] × 63

⇒ (22/7) × 31 × 63

⇒ 6138 cm³ 

Hence, the iron required to construct a hollow circular cylinder is 6138 cm³.

Given:

The total surface area of the right circular cylinder is 50% more than the curved surface area of that cylinder.

Volume of right circular cylinder = 5488π m³

Formula used:

(1.) Total surface area of cylinder = 2πr(r + h)

(2.) Curved surface area of cylinder = 2πrh

(3.) Volume of cylinder = πr²h

Where, 

r = radius 

h = height

Calculation:

According to the question,

⇒ 2πr(r + h) = (100% + 50%) of 2πrh

⇒ r + h = 150% of h

⇒ r + h = 3/2 of h

⇒ 2r + 2h = 3h

⇒ h = 2r

Now,

Volume of right circular cylinder = 5488π m³

⇒ πr²h = 5488π

⇒ r²(2r) = 5488

⇒ 2r³ = 5488

⇒ r³ = 2744

⇒ r = 14 m

Therefore, 

⇒ h = 2r

⇒ h = 28 m

Again according to the question, 

Curved surface area of cylinder = 2πrh = 784π m²

Therefore, '784π m²' is the required answer.

Given:

Width of canal  6 m 

Depth of canal = 1.5 m 

Speed of water in the canal = 10 km/hr

Time of irrigation is 30 min = 1/2 hr

8 cm of standing water is needed 

Concept Used:

The volume of a Cuboid = (Length × Breadth × Height) cubic units.

Water flow through canal = water required to irrigate

Calculation:

According to the question 

Length of water flow in 1/2 hr = l = 10 × (1/2) km

⇒ 5 km = 5000 m

⇒ Volume of water flown in 30 min = 6 × 1.5 × 5000

⇒ 45000 m³.

Now, According to the concept used

The volume of irrigated land = Area × Height

⇒ 45000 = Area × (8/100)

∴ The area of land of irrigation = 562500 m².

Given:

Height of the cylinder= 6 cm

Volume of the cylinder = 231 cm³

Formula used:

Volume of cylinder = πr²h

where,

r = radius of cylinder

h = height of cylinder

Calculations:

Volume of cylinder = πr2h

⇒ 231 = (22/7) × r² × 6

⇒ r² = 231 × (7/22) × (1/6)

⇒ r2 = 12.25

⇒ r = √12.25

⇒ r = 3.5 cm

∴ The radius of the cylinder is 3.5 cm.

Calculation:

Let the radius and height of the cylinder be r and h respectively,

Total surface area = 231 cm²,

⇒ 2πr(r + h) = 231      ----(1)

Curved surface area is 2/3 of the total surface area,

⇒ 2πrh = 2/3 × 2πr(r + h)

⇒ h = 2r      ----(2)

From equation 1 and 2,

⇒ h = 7 cm and r = 7/2 cm

Volume of Cylinder = πr²h

⇒ Volume of cylinder = 539/2 cm³

∴ The volume of cylinder is 539/2 cm³

Given:

Height of cylinder (h) = 15 cm

Radius of cylinder (r) = 14 cm

Formula:

Total surface area of cylinder = 2πr(r + h)

Calculation:

Total surface area of cylinder = 2πr(r + h)

⇒ 2 × (22/7) × 14 × (14 + 15)

⇒ 2 × 22 × 2 × 29

⇒ 2552 cm²

Given, height of the cylinder (H) = Base radius (R) of the cylinder

Outer radius of the sphere = Base radius of (R) of the cylinder

Inner radius (r) of the sphere = (outer radius of the sphere)/2

As we know, volume of the cylinder = πR²H

Given, R = H

Volume of the cylinder = π R² × R = π R³

Volume of the hollow sphere = (4/3) × π (R³ – r³)

Given, r = R/2

Volume of the hollow sphere = (4/3) × π [R³ – (R/2)³] = (4/3) × π × (R³ – R³/8) = (4/3) × π × (7R³/8)

Required ratio of the volumes of the cylinder to the hollow sphere = π R³ : (4/3) × π × (7R³/8)

⇒ 6 : 7

Given:

Circumference of the base of cylinder = 66 cm

Height of cylinder = 40 cm

Formula Used:

Circumference of a circle = 2πr

Where r = Radius of circle

Volume of cylinder = πr²h

Where h = height of the cylinder

Calculation:

Let the radius of the base of the cylinder be r

2πr = 66 cm

⇒ 2 × 22/7 × r = 66 cm

⇒ r = (3 × 7)/2 cm

⇒ r = 21/2 cm

Volume of cylinder = π(21/2 cm)² × 40 cm

⇒ 22/7 × 441/4 × 40 cm³

⇒ 220 × 63 cm³

⇒ 13860 cm³

∴ The Volume of the cylinder is 13860 cm³

Formula Used:

Volume of right circular cylinder = πr²h

Calculation:

r = 1 cm and h = 2 cm (Given)

∴ Volume = (22/7) × 1 × 2 = 44/7 cubic cm.