Problem 1 - Basic Engineering Mathematics

Question

A water tank is the shape of a rectangular prism having length \(2 \mathrm{~m}\), breadth \(75 \mathrm{~cm}\) and height \(50 \mathrm{~cm}\). Determine the capacity of the tank in (a) \(\mathrm{m}^{3}\) (b) \(\mathrm{cm}^{3}\) (c) litres.

Step-by-Step Solution

Answer

(a) 0.75 \(\text{m}^3\), (b) 750,000 \(\text{cm}^3\), (c) 750 liters.

Step 1: Convert Measurements to Common Units

The length of the tank is given as \(2\, \text{m}\). The breadth and height are given in centimeters. Convert these to meters: \(75\, \text{cm} = 0.75\, \text{m}\), and \(50\, \text{cm} = 0.5\, \text{m}\). This will allow us to find the volume in cubic meters easily.

Step 2: Calculate Volume in Cubic Meters

The formula for the volume of a rectangular prism is \(V = \text{length} \times \text{breadth} \times \text{height}\). Use the measurements in meters: \(V = 2 \times 0.75 \times 0.5\). Calculate this to find the volume in cubic meters.

Step 3: Calculate Volume in Cubic Centimeters

Convert the calculated volume in cubic meters to cubic centimeters. Since \(1\, \text{m}^3 = 1,000,000\, \text{cm}^3\), multiply the volume in cubic meters by 1,000,000 to convert to cubic centimeters.

Step 4: Convert Volume to Liters

1 liter is equivalent to 1,000 cubic centimeters. Therefore, divide the volume in cubic centimeters by 1,000 to convert it to liters.

Key Concepts

Volume of a Rectangular Prism

The volume of a rectangular prism measures the space inside this 3D shape. It's calculated using the length, breadth, and height of the prism. The formula is simple and is given by:

• \( V = \text{length} \times \text{breadth} \times \text{height} \)

This formula helps us determine the capacity of objects shaped like a box. The units of measurement play an important role here. For the most accurate calculation, ensure that all measurements are in the same unit before using the formula. For instance, if working in meters, convert all dimensions to meters first. This ensures uniform calculation and correct volume determination. Once the volume is known in cubic meters, it can be adjusted to other units like liters or cubic centimeters as needed.

Unit Conversion

Unit conversion is crucial in calculations involving measurements expressed in different units. In the context of finding the volume of a rectangular prism, it ensures that measurements are compatible for mathematical operations.

• To convert from centimeters to meters, divide by 100 because 100 centimeters make a meter.
• Similarly, for converting back to centimeters, multiply by 100.

This conversion keeps calculations consistent. Failing to convert units can lead to inaccurate results. For instance, in the original problem, measurements were given in both meters and centimeters, requiring a conversion to a common unit to correctly use the volume formula.

Cubic Meters to Liters

Converting cubic meters to liters is a common step in volume-related exercises, especially in real-world applications like measuring liquid volumes. The conversion factor between these two units is straightforward:

• 1 cubic meter = 1,000 liters

This conversion is based on the definition of a liter as the volume occupied by a cube that is 10 centimeters (0.1 meter) on each side. So in a practical example like a water tank, calculating the capacity in cubic meters allows you to easily find out how many liters of water it can hold by simply multiplying the cubic meters by 1,000.

Geometry Problems

Geometry problems often involve understanding shapes and applying mathematical formulas to solve them. In the context of volumes, these problems test your ability to visualize 3D objects, perform unit conversions, and apply the correct formulas.

• Understanding how to break down complex shapes into simpler components for easier calculation.
• Ensuring units are consistent for accurate computation.
• Interpreting the physical meaning of calculated values, such as determining how much space is available in a tank.

By mastering these skills, students can more effectively tackle various kinds of geometry problems they may encounter in math courses or real-life scenarios, such as finding the storage capacity of containers or understanding architectural designs.