Surface Area & Volumes- 2 - Free MCQ Practice Test with solutions,

Surface area of a cube = 96 cm²
Surface area of a cube = 6 (Side)² = 96 ⇒  (Side)² = 16
⇒ (Side) = 4 cm
[taking positive square root because side is always a positive quantity]
Volume of cube = (Side)³ = (4)3 = 64 cm³
Hence, the volume of the cube is 64 cm³.

To calculate the curved surface area of a pipe, use the formula:

• Curved Surface Area = 2 × π × r × h

Where:

• r is the radius of the pipe
• h is the height or length of the pipe

Given:

• Outer diameter = 1 m, thus radius (r) = 0.5 m
• Length of the pipe (h) = 21 m

Now, substituting the values into the formula:

• Curved Surface Area = 2 × π × 0.5 m × 21 m
• ≈ 66 m² (using π ≈ 3.14)

The curved surface area of the pipe is therefore approximately 66 m².

Let ABCD be cube which contain sphere touching its sides.The diameter of sphere is the sides of the cube.

Now,

The volume of the gap in between cube and sphere

= volume of cube - volume of sphere


\

= 30.48 cm³

To calculate the cost of digging the pit, follow these steps:

• Calculate the volume of the pit:
  • Dimensions: 4.5m × 2.5m × 2.5m
  • Volume = 4.5 × 2.5 × 2.5 = 28.125 cubic metres
• Determine the cost per cubic metre:
  • Rate = Rs 20 per cubic metre
• Calculate total cost:
  • Total cost = Volume × Rate
  • Total cost = 28.125 × 20 = Rs 562.50

The total cost of digging the pit is Rs 562.50.

The volume of a sphere is given as 4851 cm³.

To find the surface area, we can use the following formulas:

• The volume of a sphere is calculated using the formula: V = (4/3)πr³, where r is the radius.
• The surface area of a sphere is given by: A = 4πr².

First, we need to find the radius:

• Rearranging the volume formula to find r: r = ((3V)/(4π))^1/3.
• Substituting the volume: r = ((3 × 4851)/(4π))^1/3.

Next, we calculate the surface area:

• Using the radius in the surface area formula: A = 4πr².
• After calculations, the surface area is found to be: 1386 cm².

Thus, the surface area of the sphere is 1386 cm².

Depth of well(h) = 14m
radius of well(r) = 2m

Inner surface area of well(like a cylinder) = 2πrh 

surface area= 2 x 22/7 x 2 x 14 = 176 m²

cost of cementing = ₹2 per m²

Total Cost = 176 x 2 = ₹352

The ratio of the radii of two spheres, whose volumes are in the ratio 64 : 27, can be determined as follows:

• The volume of a sphere is calculated using the formula: V = (4/3)πr³, where r is the radius.
• If the volumes are in the ratio 64 : 27, we can express this as:
• V1/V2 = 64/27
• Substituting the volume formula gives us:
• (4/3)πr1³ / (4/3)πr2³ = 64/27
• This simplifies to:
• r1³/r2³ = 64/27
• Taking the cube root of both sides, we find:
• r1/r2 = (64/27)^(1/3)
• This results in:
• r1/r2 = 4/3

Thus, the ratio of the radii of the two spheres is 4 : 3.

The volume of the cuboid formed by joining two cubes with a side of 6 cm is calculated as follows:

• Each cube has a volume of 216 cm³ (calculated as 6 cm x 6 cm x 6 cm).
• When two cubes are joined end to end, the new dimensions of the cuboid are:
• Length: 12 cm (6 cm + 6 cm)
• Width: 6 cm
• Height: 6 cm
• The volume of the cuboid is calculated as:
• Volume = Length x Width x Height
• Volume = 12 cm x 6 cm x 6 cm = 432 cm³

To find the outer curved surface area of a hemispherical bowl, follow these steps:

• Identify the inner radius: The inner radius is given as 3.25 cm.
• Calculate the outer radius: Add the thickness of the steel (0.25 cm) to the inner radius:
  • Outer radius = 3.25 cm + 0.25 cm = 3.5 cm.
• Use the formula for curved surface area: The formula for the curved surface area of a hemisphere is:
  • Curved Surface Area = 2 * π * r².
• Calculate the outer curved surface area: Substitute the outer radius into the formula:
  • Curved Surface Area = 2 * π * (3.5 cm)².
  • Curved Surface Area = 2 * π * 12.25 cm².
  • Curved Surface Area ≈ 2 * 3.14 * 12.25 cm².
  • Curved Surface Area ≈ 76.9 cm².

Rounding to the nearest whole number, the outer curved surface area of the bowl is approximately 77 cm².

To find the difference between the total surface area and the lateral surface area of a cube with a side length of 4 cm, follow these steps:

• The formula for the total surface area of a cube is: 6 × (side length)².
• For a side length of 4 cm, the total surface area is: 6 × (4 cm)² = 6 × 16 cm² = 96 cm².
• The formula for the lateral surface area of a cube is: 4 × (side length)².
• For a side length of 4 cm, the lateral surface area is: 4 × (4 cm)² = 4 × 16 cm² = 64 cm².
• Now, calculate the difference:
• Difference = Total Surface Area - Lateral Surface Area = 96 cm² - 64 cm² = 32 cm².

To find the diameter of the base of the cylinder:

• The formula for the curved surface area (CSA) of a cylinder is given by:
• CSA = 2πrh, where r is the radius and h is the height.
• Given the CSA is 88 cm² and the height is 14 cm, we can set up the equation:
• 88 = 2πr × 14
• This simplifies to:
• 88 = 28πr
• Next, solve for r:
• r = 88 / (28π)
• Calculating this gives:
• r ≈ 1 cm
• Since the diameter (d) is twice the radius:
• d = 2r
• This results in:
• d ≈ 2 cm

To find the diameter of a spherical shot put given its surface area:

• The formula for the surface area (A) of a sphere is: A = 4πr², where r is the radius.
• Given that the surface area is 616 cm², we can set up the equation: 4πr² = 616.
• To isolate r², divide both sides by 4π: r² = 616 / (4π).
• Calculating this gives: r² ≈ 49.
• Taking the square root of both sides, we find: r ≈ 7 cm.
• The diameter (d) is twice the radius: d = 2r, which results in: d ≈ 14 cm.

The diameter of the shot put is 14 cm.

Volume = Length × Width × Height

• Substituting the dimensions:
• Volume = 6m × 5m × 4.5m
• Calculating the volume:
• Volume = 30m² × 4.5m = 135m³

To convert cubic metres to litres, use the conversion factor:

• 1m³ = 1000 litres
• Therefore, 135m³ = 135 × 1000 litres = 135000 litres

The tank can hold 135000 litres of water.

To find the total surface area of a solid hemisphere given the surface area of a sphere:

• The formula for the surface area of a sphere is 4πr².
• Given that the surface area is 1386 cm², we can set up the equation:
• 4πr² = 1386
• To find the radius (r), we first solve for r²:
• r² = 1386 / (4π)
• Next, we calculate the surface area of a hemisphere:
• The total surface area of a hemisphere is given by the formula:
• 2πr² + πr² = 3πr²
• Now, substitute the value of r²:
• Total surface area = 3 * (1386 / 4) = 1039.5 cm²

The total surface area of the solid hemisphere is therefore 1039.5 cm².

To find the volume of a cube when the diagonal is given:

• The formula for the diagonal D of a cube in terms of its side length s is:
• D = s√3
• Given that the diagonal is 8√3 cm, we can set up the equation:
• 8√3 = s√3
• Dividing both sides by √3 gives:
• s = 8 cm
• To find the volume V of the cube, use:
• V = s³
• Calculating the volume:
• V = 8³ = 512 cm³

The volume of the cube is 512 cm³.

To calculate the total surface area of a right circular cylinder:

• Use the formula: Total Surface Area = 2πr(h + r)
• Where:
  • r is the radius of the base
  • h is the height of the cylinder
• Given:
  • Height (h) = 4 cm
  • Radius (r) = 3 cm
• Substituting the values into the formula:
  • Total Surface Area = 2π(3)(4 + 3)
  • Total Surface Area = 2π(3)(7)
  • Total Surface Area = 42π cm²
• Approximating π as 3.14:
  • Total Surface Area ≈ 42 × 3.14 ≈ 131.88 cm²

The total surface area of the cylinder is approximately 132 cm².

When a spherical balloon expands to twice its radius, the volume ratio between the inflated and original balloons can be calculated as follows:

• The formula for the volume of a sphere is V = (4/3)πr³.
• If the original radius is r, the original volume is V₁ = (4/3)πr³.
• When the radius is doubled, the new radius becomes 2r.
• The volume of the inflated balloon is V₂ = (4/3)π(2r)³.
• Calculating this gives V₂ = (4/3)π(8r³) = (32/3)πr³.
• The ratio of the volumes is V₂ / V₁ = ((32/3)πr³) / ((4/3)πr³).
• Simplifying the ratio results in V₂ / V₁ = 8.

The final ratio of the volume of the inflated balloon to the original balloon is 8:1.

The surface area of a cuboid can be calculated using the formula:

• Surface Area = 2(lw + lh + wh)

Where:

• l = length
• w = breadth
• h = height

For the given cuboid:

• Length = 15 cm
• Breadth = 10 cm
• Height = 20 cm

Now, substituting the values into the formula:

• Surface Area = 2(15 cm × 10 cm + 15 cm × 20 cm + 10 cm × 20 cm)
• = 2(150 cm² + 300 cm² + 200 cm²)
• = 2(650 cm²)
• = 1300 cm²

Thus, the surface area of the cuboid is 1300 cm².

Step 1: Convert all dimensions into meters

• Diameter of each bullet = 5 dm = 0.5 m
• Radius of each bullet = 0.25 m

Step 2: Volume of the rectangular block

Length = 11 m
Breadth = 10 m
Height = 5 m

Volume = 11 × 10 × 5 = 550 m³

Step 3: Volume of one spherical bullet

Volume = (4/3) × π × r³
= (4/3) × π × (0.25)³
= π / 48 m³

Step 4: Number of bullets

Number = Total volume ÷ Volume of one bullet
= 550 ÷ (π / 48)
= 26400 / π

Using π = 22/7,
Number = (26400 × 7) / 22 = 8400

8400 spherical bullets

To determine the number of planks that can fit in the pit, follow these steps:

• Calculate the volume of the pit:
  • Length: 20 m
  • Width: 6 m
  • Depth: 80 cm (0.8 m)
  • Volume = Length × Width × Depth = 20 m × 6 m × 0.8 m = 96 m³
• Calculate the volume of one plank:
  • Dimensions: 5 m × 25 cm (0.25 m) × 10 cm (0.1 m)
  • Volume = 5 m × 0.25 m × 0.1 m = 0.125 m³
• Determine the number of planks:
  • Number of planks = Volume of pit / Volume of one plank
  • Number of planks = 96 m³ / 0.125 m³ = 768

The pit can accommodate 768 planks.

To find the base diameter of the cylinder:

We start with the formula for the lateral surface area of a cylinder:

• Lateral Surface Area = 2 × π × r × h

Where:

• r = radius of the base
• h = height of the cylinder

Given:

• Lateral Surface Area = 132 cm²
• Height (h) = 7 cm

Substituting the values into the formula:

• 132 = 2 × π × r × 7

To isolate r:

• 132 = 14πr
• r = 132 / (14π)

Now, calculate the radius:

• Using π ≈ 3.14, r ≈ 132 / 43.96 ≈ 3 cm

To find the diameter:

• Diameter = 2 × r = 2 × 3 cm = 6 cm

Therefore, the base diameter of the cylinder is 6 cm.

To find the volume of a cylinder:

• First, we need to determine the radius of the base using the circumference.
• The formula for circumference (C) is: C = 2πr, where r is the radius.
• Given the circumference is 132 cm, we can rearrange the formula to find r:
• r = C / (2π)
• Substituting the values: r = 132 / (2π) ≈ 21 cm.
• Next, we calculate the volume (V) of the cylinder using the formula:
• V = πr²h, where h is the height.
• We know that the height (h) is 25 cm. Now, substitute the values:
• V ≈ π(21)²(25).
• Calculating further:
• V ≈ π(441)(25) ≈ π(11025) ≈ 34650 cm³.

The volume of the cylinder is approximately 34650 cm³.

To determine how many solid spheres can be moulded from a solid cylinder, follow these steps:

• Calculate the volume of the cylinder:
  • Diameter of the cylinder = 4 cm, so radius = 2 cm.
  • Height of the cylinder = 45 cm.
  • Volume = πr²h = π(2)²(45) = 180π cm³.
• Calculate the volume of one sphere:
  • Diameter of the sphere = 6 cm, so radius = 3 cm.
  • Volume = (4/3)πr³ = (4/3)π(3)³ = 36π cm³.
• Determine the number of spheres:
  • Number of spheres = Volume of cylinder / Volume of sphere.
  • Number of spheres = 180π / 36π = 5.

The number of solid spheres that can be moulded from the cylinder is 5.