Problem 15 - Basic Engineering Mathematics

Question

A boiler consists of a cylindrical section of length \(8 \mathrm{~m}\) and diameter \(6 \mathrm{~m}\), on one end of which is surmounted a hemispherical section of diameter \(6 \mathrm{~m}\), and on the other end a conical section of height \(4 \mathrm{~m}\) and base diameter \(6 \mathrm{~m}\). Calculate the volume of the boiler and the total surface area.

Step-by-Step Solution

Answer

Total volume is \( 102\pi \) m³, total surface area is \( 81\pi \) m².

Step 1: Understand the Problem

The boiler is made up of three different sections: a cylindrical section, a hemispherical section, and a conical section. The dimensions provided are necessary to calculate the volume and surface area of each section.

Step 2: Calculate Volume of the Cylinder

The volume of a cylinder is calculated using the formula \( V = \pi r^2 h \). Here, the radius \( r \) is half of the diameter, so \( r = 3 \) m, and the height \( h \) is 8 m. Thus, the volume is \( V = \pi \times 3^2 \times 8 = 72\pi \) cubic meters.

Step 3: Calculate Volume of the Hemisphere

The volume of a hemisphere is \( V = \frac{2}{3} \pi r^3 \). With a radius of 3 m, the volume is \( V = \frac{2}{3} \pi \times 3^3 = 18\pi \) cubic meters.

Step 4: Calculate Volume of the Cone

The volume of a cone is \( V = \frac{1}{3} \pi r^2 h \), where the radius \( r = 3 \) m and height \( h = 4 \) m. Therefore, the volume is \( V = \frac{1}{3} \pi \times 3^2 \times 4 = 12\pi \) cubic meters.

Step 5: Calculate Total Volume

Add the volumes of all three sections together: Cylinder (\(72\pi\)), Hemisphere (\(18\pi\)), and Cone (\(12\pi\)). The total volume is \( 72\pi + 18\pi + 12\pi = 102\pi \) cubic meters.

Step 6: Calculate Surface Area of the Cylinder

The lateral surface area of a cylinder is given by \( S = 2\pi rh \) and the top circular area is \( A = \pi r^2 \). However, this top is part of the hemisphere. Only the lateral area is considered: \( S = 2\pi \times 3 \times 8 = 48\pi \) square meters.

Step 7: Calculate Surface Area of the Hemisphere

The surface area of a hemisphere is \( S = 2\pi r^2 \). So, with a radius of 3 m, the area is \( S = 2\pi \times 3^2 = 18\pi \) square meters.

Step 8: Calculate Surface Area of the Cone

The surface area includes only the lateral surface, which is \( S = \pi r l \), where \( l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = 5 \) m, so \( S = \pi \times 3 \times 5 = 15\pi \) square meters.

Step 9: Calculate Total Surface Area

Add the areas of all sections: Cylinder lateral area (\(48\pi\)), Hemisphere area (\(18\pi\)), and Cone lateral area (\(15\pi\)). Total surface area is \( 48\pi + 18\pi + 15\pi = 81\pi \) square meters.

Key Concepts

Volume Calculation

Calculating the volume of a structure like a boiler that consists of multiple geometric shapes is crucial to understanding its capacity. Volume calculation involves finding out how much space is contained within an object. For the boiler in question, there are three sections to consider: a cylindrical section, a hemispherical section, and a conical section.

To find the total volume of the boiler, we calculate the volume for each shape separately using their respective formulas and then sum them up. For the cylinder, use the formula for volume:

• Volume: \( V = \pi r^2 h \)

The radius \( r \) is half the diameter, thereby \( 3 \text{ m} \), and the height \( h \) is \( 8 \text{ m} \). For a hemisphere, the formula is:

• Volume: \( V = \frac{2}{3} \pi r^3 \)

Here, the radius \( r \) remains \( 3 \text{ m} \). Finally, for the cone, calculate using:

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

The cone's radius is also \( 3 \text{ m} \) and height \( 4 \text{ m} \). Adding these separate volumes together gives the total volume of the boiler.

Surface Area Calculation

Surface area calculation helps determine how much material you would need to cover an object completely. When dealing with a composite shape like a boiler, we find the surface area for each section separately and add them up.

Like volume, surface areas have distinct formulas based on shape. For the cylinder, calculate only the lateral surface area since the top is covered by the hemisphere. Use:

• Lateral Surface Area: \( S = 2\pi rh \)

For the hemisphere, surface area includes only the outer curved surface:

• Surface Area: \( S = 2\pi r^2 \)

And for the cone, calculate:

• Lateral Surface Area: \( S = \pi rl \)

where \( l \) is the slant height, calculated using \( l = \sqrt{r^2 + h^2} \). Finally, by adding these individual surface areas, you determine the total surface area of the boiler.

Cylindrical Shapes

Cylinders are geometric shapes with straight parallel sides and a circular cross-section. They're prevalent in everyday objects such as cans and tubes. Cylindrical shapes have two main parameters: radius and height. Distinct formulas are used to calculate both their volume and surface area.

For the volume, use:

• \( V = \pi r^2 h \)

where \( r \) is the radius of the base and \( h \) is the height. Cylinders can also be defined by their lateral surface area, which does not include the top and bottom surfaces. It is calculated as:

• \( S = 2\pi rh \)

Understanding these formulas, you'll always be ready to assess or construct cylindrical objects.

Hemispherical Shapes

A hemisphere is half of a sphere. These shapes are significant in architecture and product design, offering aesthetic and functional value. To quantify a hemisphere, you need to know its radius.

The volume of a hemisphere differs from that of a full sphere. It's calculated using:

• Volume: \( V = \frac{2}{3} \pi r^3 \)

For the surface area, only the curved part of the hemisphere counts, assessed through:

• Surface Area: \( S = 2\pi r^2 \)

Grasping these concepts ensures you can calculate or estimate the characteristics of hemispherical objects accurately.

Conical Shapes

Cones are pointed shapes with a circular base and a single vertex. They're common in party hats, and ice cream cones, and even appear in industrial structures. In geometry, cones are characterized by the radius of their base and their height.

The volume formula for a cone is:

• \( V = \frac{1}{3} \pi r^2 h \)

where \( r \) is the radius and \( h \) is the height. The surface area primarily concerns the lateral surface, or the side "wall" of the cone. It's found by:

• Lateral Surface Area: \( S = \pi rl \)

Here, \( l \) is the slant height, calculated as \( l = \sqrt{r^2 + h^2} \). Understanding cones' mathematical properties enables you to measure or design these shapes efficiently and accurately.