Surface Areas and Volumes | Exercise 12.2

Question 6

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 cm31\text{ cm}^3 of iron has approximately 8g mass. (Use π=3.14\pi = 3.14)

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Solution

Let's find the total volume of the iron pole first, then calculate its mass.

Step 1 — Identify Dimensions of Larger Cylinder

We are given the dimensions of the larger cylinder. Its base diameter is 24 cm. So, we find its radius. r1=242 cmr_1 = \frac{24}{2} \text{ cm}

r1=12 cm\boxed{r_1 = 12 \text{ cm}} Its height is given as 220 cm. h1=220 cmh_1 = 220 \text{ cm}

Step 2 — Identify Dimensions of Smaller Cylinder

Next, we identify the dimensions of the smaller cylinder. Its radius is given as 8 cm. r2=8 cmr_2 = 8 \text{ cm} Its height is given as 60 cm. h2=60 cmh_2 = 60 \text{ cm}

Diagram 1

Step 3 — Calculate the total volume of the pole

The pole is made of two cylinders — a larger one at the bottom and a smaller one on top. Its total volume is the sum of the two cylinder volumes.

The volume of a cylinder is:

V=πr2hV = \pi r^2 h

Larger cylinder (r=12 cmr = 12\text{ cm}, h=220 cmh = 220\text{ cm}):

V1=π×(12)2×220V_1 = \pi \times (12)^2 \times 220

V1=π×144×220V_1 = \pi \times 144 \times 220

V1=31680 π cm3V_1 = 31680\,\pi \text{ cm}^3

Smaller cylinder (r=8 cmr = 8\text{ cm}, h=60 cmh = 60\text{ cm}):

V2=π×(8)2×60V_2 = \pi \times (8)^2 \times 60

V2=π×64×60V_2 = \pi \times 64 \times 60

V2=3840 π cm3V_2 = 3840\,\pi \text{ cm}^3

Total volume:

Vtotal=V1+V2=(31680+3840) π=35520 π cm3V_{\text{total}} = V_1 + V_2 = (31680 + 3840)\,\pi = 35520\,\pi \text{ cm}^3

Now using π=3.14\pi = 3.14:

Vtotal=35520×3.14V_{\text{total}} = 35520 \times 3.14

Vtotal=111532.8 cm3\boxed{V_{\text{total}} = 111532.8 \text{ cm}^3}

Step 4 — Calculate the mass of the pole

We are given that 1 cm31\text{ cm}^3 of iron has a mass of 8 g. So to find the total mass, we multiply the total volume by this value:

Mass=Vtotal×8\text{Mass} = V_{\text{total}} \times 8

Mass=111532.8×8\text{Mass} = 111532.8 \times 8

Mass=892262.4 g\text{Mass} = 892262.4 \text{ g}

Finally, we convert grams to kilograms. Since 1 kg=1000 g1\text{ kg} = 1000\text{ g}, we divide by 1000:

Mass=892262.41000 kg\text{Mass} = \frac{892262.4}{1000} \text{ kg}

Mass=892.262 kg\boxed{\text{Mass} = 892.262 \text{ kg}}

Answer

The mass of the pole is 892.262 kg.

More questions in Exercise 12.2

Q1

Unless stated otherwise, take π=227\pi = \frac{22}{7}.

  1. 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\pi.
Q2
  1. 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.)
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 Fig. 12.15).

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. 12.16).

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-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

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 cm31\text{ cm}^3 of iron has approximately 8g mass. (Use π=3.14\pi = 3.14)

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.

Q8

A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm3345\text{ cm}^3. Check whether she is correct, taking the above as the inside measurements, and π=3.14\pi = 3.14.

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