Class 10th, Surface and Volume, 13.3 (Solution)

                    

                                 Mathematicsraj1.blogspot.com

Class 10th, Surface Areas and Volumes

 Exercise: 13.3 (Solution), Q1to Q9

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Q1. A metallie sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder.

Solution:

It is given that radius of the sphere (R) = 4.2 cm

Also, Radius of cylinder (r) = 6 cm

Now, let height of cylinder = h

It is given that the sphere is melted into a cylinder.

So, Volume of Sphere = Volume of Cylinder

∴ (4/3)× π× R3 = π× r2 × h.

= h = 2.74 cm

Q2. Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

Solution:

For Sphere 1:

Radius (r1) = 6 cm

∴ Volume (V1) = (4/3) × π× r13

For Sphere 2:

Radius (r2) = 8 cm

∴ Volume (V2) = (4/3) × π × r23

For Sphere 3:

Radius (r3) = 10 cm

∴ Volume (V3) = (4/3) × π× r33

Also, let the radius of the resulting sphere be “r”

Now,

Volume of resulting sphere = V1 + V2 + V3

(4/3)× π× r3 = (4/3)× π× r13 + (4/3)× π× r23 + (4/3) × π× r33

r3 = 63 + 83 + 103

r3 = 1728

r = 12 cm

Q3. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.

Solution:

It is given that the shape of the well is in the shape of a cylinder with a diameter of 7 m

So, radius = 7/2 m

Also, Depth (h) = 20 m

Volume of the earth dug out will be equal to the volume of the cylinder

∴ Volume of Cylinder = π × r2 × h

= 22 × 7 × 5 m3

Let the height of the platform = H

Volume of soil from well (cylinder) = Volume of soil used to make such platform

π × r2 × h = Area of platform × Height of the platform

We know that the dimension of the platform is = 22 × 14

So, Area of platform = 22 × 14 m2

∴ π × r2 × h = 22 × 14 × H

Or, H = 2.5 m

Q4. A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.

Solution:

The shape of the well will be cylindrical as given below.

Given, Depth (h1) of well = 14 m

Diameter of the circular end of the well =3 m

So, Radius (r1) = 3/2 m

Width of the embankment = 4 m

From the figure, it can be said that the embankment will be a cylinder having an outer radius (r2)

as (4 + 3/2) = 11/2 m and inner radius (r1) as 3/2m

Now, let the height of embankment be h2

∴ Volume of soil dug from well = Volume of earth used to form embankment

=> π × r12 × h = π × (r22 – r12) × h2

Solving this, we get,

The height of the embankment (h2) as 1.125 m.

Q5. A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

Solution:

Number of cones will be = Volume of cylinder / Volume of ice cream cone

For the cylinder part,

Radius = 12/2 = 6 cm

Height = 15 cm

∴ Volume of cylinder = π × r2 × h = 540π

For the ice cone part,

Radius of conical part = 6/2 = 3 cm

Height = 12 cm

Radius of hemispherical part = 6/2 = 3 cm

Now,

Volume of ice cream cone = Volume of conical part + Volume of hemispherical part

= (⅓) × π × r2 × h + (⅔) × π × r3

= 36π + 18π

= 54π

∴ Number of cones = (540π / 54π)

= 10

Q6. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form a cuboid of dimensions 5.5 cn × 10 cm × 3.5 cm?

Solution:

It is known that the coins are cylindrical in shape.

So, height (h1) of the cylinder = 2 mm = 0.2 cm

Radius (r) of circular end of coins =1.75/2 = 0.875 cm

Now, the number of coins to be melted to form the required cuboids be “n”

So, Volume of n coins = Volume of cuboids

n × π × r2 × h1 = l × b × h

n × π × (0.875)2 × 0.2 = 5.5 × 10 × 3.5

Or, n = 400

Q7. A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.

Solution:

The diagram will be as-

Given,

Height (h1) of cylindrical part of the bucket = 32 cm

Radius (r1) of circular end of the bucket = 18 cm

Height of the conical heap ((h2) = 24 cm

Now, let “r2” be the radius of the circular end of the conical heap.

We know that volume of the sand in the cylindrical bucket will be equal to the volume of sand in the conical heap.

∴ Volume of sand in the cylindrical bucket = Volume of sand in conical heap

π × r12 × h1 = (⅓) × π × r22 × h2

π × 182 × 32 = (⅓) × π × r22 × 24

Or, r2 = 36 cm

And,

Slant height (l) = √(362 + 242) = 12√13 cm.

Q8. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed?

Solution:

It is given that the canal is the shape of a cuboid with dimensions as:

Breadth (b) = 6 m and Height (h) = 1.5 m

It is also given that

The speed of canal = 10 km/hr

Length of canal covered in 1 hour = 10 km

Length of canal covered in 60 minutes = 10 km

Length of canal covered in 1 min = 1/60 x 10 km

Length of canal covered in 30 min (l) = 30/60 x 10 = 5km = 5000 m

We know that the the canal is cuboidal in shape. So,

Volume of canal = l x b x h

= 5000 x 6 x 1.5 m3

= 45000 m3

Now,

Volume of water in canal = Volume of area irrigated

= Area irrigated x Height

So, Area irrigated = 56.25 hectares

∴ Volume of canal = l x b x h

45000 = Area irrigated x 8 cm

45000 = Area irrigated x (8/100)m

Or, Area irrigated = 562500 m2 = 56.25 hectares.

Q9. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled?

Solution:

This is Diagram-

Volume of water that flows in t minutes from pipe = t × 0.5π m3

Volume of water that flows in t minutes from pipe = t × 0.5π m3

Radius (r2) of circular end of cylindrical tank =10/2 = 5 m

Depth (h2) of cylindrical tank = 2 m

Let the tank be filled completely in t minutes.

Volume of water filled in tank in t minutes is equal to the volume of water flowed in t minutes from the pipe.

Volume of water that flows in t minutes from pipe = Volume of water in tank

t × 0.5π = π × r22 × h2

 t = 100 minutes

See next blog for more Solution...........Thank you.

Class 10th, Surface and Volume, 13.2 (Solution)

                           Mathematicsraj1.blogspot.com

Class 10th, Surface Areas and Volumes

 Exercise: 13.2 (Solution), Q1to Q8

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Q1. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of π.Sol. Here, r = 1 cm and h = 1 cm.

Solution:

The diagram.

Now, Volume of solid = Volume of conical part + Volume of hemispherical part

We know the volume of cone = ⅓ πr2h

And,

The volume of hemisphere = ⅔ π r3

So, volume of solid will be

                                                           = π cm3

Q.2. Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)

Solution:

Given,

Height of cylinder = 12 – 4 = 8 cm

Radius = 1.5 cm

Height of cone = 2 cm

Now, the total volume of the air contained will be = Volume of cylinder + 2 × (Volume of cone)

∴ Total volume = πr2h + [2 × (⅓ πr2h )]

= 18 π + 2(1.5 π)

= 66 cm3.

Q3. A gulab jamun contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see figure).

Solution:

It is known that the gulab jamuns are similar to a cylinder with two hemispherical ends.

So, the total height of a gulab jamun = 5 cm.

Diameter = 2.8 cm

So, radius = 1.4 cm

∴ The height of the cylindrical part = 5 cm – (1.4 + 1.4) cm

=2.2 cm

Now, total volume of One Gulab Jamun = Volume of Cylinder + Volume of two hemispheres

= πr2h + 4/3 πr3

= 4.312 π + (10.976/3) π

                                        = 25.05 cm3

We know that the volume of sugar syrup = 30% of total volume

So, volume of sugar syrup in 45 gulab jamuns = 45 x 30% (25.05 cm3)

= 45 x 7.515 = 338.184 cm3

Q4. A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand (see Fig.).

The Diagram.

Solution:

Volume of cuboid = length x width x height

We know the cuboid’s dimensions as 15 cm x 10 cm x 3.5 cm

So, the volume of the cuboid = 15 x 10 x 3.5 = 525 cm3

Here, depressions are like cones and we know,

Volume of cone = ⅓ πr2h

Given, radius (r) = 0.5 cm and depth (h) = 1.4 cm

∴ Volume of 4 cones = 4x ⅓ πr2h

= 1.46 cm2

Now, volume of wood = Volume of cuboid – 4 x volume of cone

= 525 – 1.46 = 523.54 cm2

Q5. A vessel is in the form of an inverted cone. its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-forth of the water flows out. Find the number of lead shots dropped in the vessel.

Solution:

For the cone,

Radius = 5 cm,

Height = 8 cm

Also,

Radius of sphere = 0.5 cm

The diagram will be like

It is known that,

Volume of cone = volume of water in the cone

= ⅓ πr2h = (200/3)π cm3

Now,

Total volume of water overflown= (¼) × (200/3) π =(50/3) π

Volume of lead shot

= 4/3 πr3

= (1/6) π

Now,

The number of lead shots = Total Volume of Water overflown/ Volume of Lead shot

= (50/3) π /(⅙) π

= 50/36 × 6 = 100

Q6. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has approximately 8 g mass.

Solution:

Given, the height of the big cylinder (H) = 220 cm

Radius of the base (R) = 24/12 = 12 cm

So, the volume of the big cylinder = πR2H

= π(12)2 × 220 cm3

= =99565.8 cm3

Now, the height of smaller cylinder (h) = 60 cm

Radius of the base (r) = 8 cm

So, the volume of the smaller cylinder = πr2h

= π(8)2 × 60 cm3

= 12068.5 cm3

∴ Volume of iron = Volume of the big cylinder + Volume of the small cylinder

99565.8 + 12068.5

=111634.5 cm3

We know,

Mass = Density x volume

So, mass of the pole = 8×111634.5

= 893 Kg (approx.)

Q7. A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

Solution:

Here, the volume of water left will be = Volume of cylinder – Volume of solid

Given,

Radius of cone = 60 cm,

Height of cone = 120 cm

Radius of cylinder = 60 cm

Height of cylinder = 180 cm

Radius of hemisphere = 60 cm

Now,

Total volume of solid = Volume of Cone + Volume of hemisphere

Volume of cone = π × 122 × 103cm3 = 144 × 103 π cm3

So, Total volume of solid = 144 × 103 π cm3 – (⅔) × π × 103 cm3

Volume of hemisphere = (⅔) × π × 103 cm3

Volume of cylinder = π × 602 × 180 = 648000 = 648 × 103 π cm3

Now, volume of water left will be = Volume of cylinder – Volume of solid

= ( 648-288) × 103 × π = 1.131 m3

Q8. A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter o the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm3. Check whether she is correct, taking the above as the inside measurements, and π = 3.14.

Solution:

Given,

For the cylinder part, Height (h) = 8 cm and Radius (R) = (2/2) cm = 1 cm

For the spherical part, Radius (r) = (8.5/2) = 4.25 cm

 The Diagram

Now, volume of this vessel = Volume of cylinder + Volume of sphere

= π × (1)2 × 8 + (4/3) π (1)3

= 346.51 cm3

See next blog for more solution........ Thank you.

Class 1oth, Surface and Volume, Exe-13.1 (Solution 2)

                  Mathematicsraj1.blogspot.com

Class 10th, Surface Areas and Volumes

 Exercise: 13.1 (Solution), Q5 to Q8

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Q5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

Answer:

The diagram :-

Now, the diameter of hemisphere = Edge of the cube = l

So, the radius of hemisphere = l/2

∴ The total surface area of solid = surface area of cube + CSA of hemisphere – Area of base of hemisphere

=> TSA of remaining solid = 6 (edge)2 + 2πr2 – πr2

= 6l2 + πr2

= 6l2 + π(l/2)2

= 6l2 + πl2/4

= l2/4 (24 + π) sq. units

Q6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

Answer:

Two hemisphere and one cylinder are shown in the figure given below.

Here, the diameter of the capsule = 5 mm

∴ Radius = 5/2 = 2.5 mm

Now, the length of the capsule = 14 mm

So, the length of the cylinder = 14 – (2.5 + 2.5) = 9 mm

∴ The surface area of a hemisphere = 2πr2 = 2 × 22/7 × 2.5 × 2.5

= 275/7 mm2

Now, the surface area of the cylinder = 2πrh

= 2 × 22/7 × 2.5 x 9

=> 22/7 × 45 = 990/7 mm2

Thus, the required surface area of medicine capsule will be

= 2 × surface area of hemisphere + surface area of the cylinder

= (2 × 275/7) × 990/7

=> 550/7 + 990/7 = 1540/7 = 220 mm2

Q7. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of Rs 500 per m2. (Note that the base of the tent will not be covered with canvas.)

Answer:

It is known that a tent is a combination of cylinder and a cone.

From the question we know that

Diameter = 4 m

Slant height of the cone (l) = 2.8 m

Radius of the cone (r) = Radius of cylinder = 4/2 = 2 m

Height of the cylinder (h) = 2.1 m

So, the required surface area of tent = surface area of cone + surface area of cylinder

= πrl + 2πrh

= πr (l+2h)

= 22/7 × 2 (2.8 + 2×2.1)

= 44/7 (2.8 + 4.2)

= 44/7 × 7 = 44 m2

∴ The cost of the canvas of the tent at the rate of ₹500 per m2 will be

= Surface area × cost per m2

=> 44 × 500 = ₹22000

So, Rs. 22000 will be the total cost of the canvas.

Q8. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the

same height and same diameter is hollowed out. Find the total surface area of the

remaining solid to the nearest cm2.

Answer:

The diagram for the question is as follows:

From the question we know the following:

The diameter of the cylinder = diameter of conical cavity = 1.4 cm

So, the radius of the cylinder = radius of the conical cavity = 1.4/2 = 0.7

Also, the height of the cylinder = height of the conical cavity = 2.4 cm

Now, the TSA of remaining solid = surface area of conical cavity + TSA of the cylinder

= πrl + (2πrh + πr2)

= πr (l + 2h + r)

= 22/7 × 0.7 (2.5 + 4.8 + 0.7)

= 2.2 × 8 = 17.6 cm2

So, the total surface area of the remaining solid is 17.6 cm2

See next blog for more Solution........ Thank you.