Question

It is required to construct a big rectangular hall that can accommodate 400 people with \(25 \mathrm{~m}^{3}\) space for each person. The height of the wall has been fixed at \(10 \mathrm{~m}\) and the total inner surface area of the walls must be \(1300 \mathrm{~m}^{2}\). What is the length and breadth of the hall (in metres)? (a) 30,20 (b) 45,20 (c) 40,25 (d) 35,30

Short answer

Answer: The length (L) of the rectangular hall is 40 meters, and the breadth (B) is 25 meters.

Step-by-step solution

Define the variables and equations

Let the length of the hall be L meters and the breadth be B meters. The volume of the hall (V), can be calculated using the formula: V = L * B * H Let the total surface area of the walls (S) be calculated as: S = 2 * (L * H + B * H) Now, given the information, we can construct the following two equations: 1. 400 * 25 = L * B * 10 (400 people, 25 cubic meters per person, and 10m height) 2. 1300 = 2 * (L * 10 + B * 10) (total inner surface area of the walls is 1300m^2)

Solve the first equation to get an expression for L in terms of B

From equation 1: 10000 = L * B * 10 Divide both sides by 10: 1000 = L * B Therefore: L = 1000 / B

Substitute the expression for L into the second equation

Substituting the expression for L from Step 2 into equation 2: 1300 = 2 * ((1000 / B) * 10 + B * 10)

Solve for B

Simplify the equation by dividing both sides by 20: 65 = (1000 / B) + B Multiply both sides by B: 65B = 1000 + B^2 Rearrange the equation as a quadratic equation: B^2 - 65B + 1000 = 0

Solve the quadratic equation for B

Solving the quadratic equation, we get: B = 25 m

Find the value of L

We already have an expression for L from Step 2: L = 1000 / B Now, plug in the value of B we found: L = 1000 / 25 Therefore: L = 40 m So, the length and breadth of the hall are L = 40 meters and B = 25 meters, which corresponds with answer choice (c).

Key concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree, generally represented in the form of \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients and \( x \) represents an unknown variable. They are fundamental in algebra and are characterized by their distinctive 'U' shaped curve when graphed, which is known as a parabola.

In the problem mentioned, we encountered a quadratic equation while trying to determine the breadth (B) of the hall. To solve for \( B \), we manipulated the equation to achieve the standard form and then proceeded to find the roots, or solutions, for \( B \). Quadratic equations can be solved by various methods such as factoring, completing the square, or using the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). Understanding how to form and solve these equations is crucial for solving complex real-world problems that can be modeled by quadratic relationships.

An important note for students: always ensure that your quadratic equation is simplified to its standard form before attempting to solve it.

Volume Calculation

Volume calculation is a fundamental aspect of geometry involving finding the amount of space enclosed within a three-dimensional figure. For a rectangular hall, the volume \(V\) is calculated by multiplying the length \(L\), breadth \(B\), and height \(H\) of the hall: \( V = L \times B \times H \).

In our exercise, the problem stated that the hall needs to accommodate 400 people with each person requiring \(25 \text{ m}^3\) of space. This specification translates directly to a volume calculation, since volume measures the capacity of the hall.

Tips for Accurate Volume Calculation

• Ensure all dimensions are in the same unit of measurement before multiplying.
• Understand the shape of the object to apply the correct formula for volume.
• Check the values used for each dimension to avoid computational errors.

These tips will help prevent common mistakes and ensure precise results when dealing with volume-related problems in geometry.

Surface Area Calculation

Surface area calculation deals with finding the total area that the surface of a three-dimensional object occupies. For our rectangular hall, the surface area \(S\) includes all the interior wall surfaces and is calculated by summing the areas of these walls.

Specifically, the formula to calculate the total inner surface area for our hall, excluding the floor and ceiling, is given by \( S = 2 \times (L \times H + B \times H) \), where \(L\) is the length, \(B\) is the breadth, and \(H\) is the height. This equation allows us to understand the amount of material needed for the wall surfaces, which can be critical for construction budgeting and planning.

Surface area is quite essential in various fields, especially in architecture and engineering, where it pertains directly to the cost of materials.

Key Points in Surface Area Calculation

• Identify which surfaces are to be included in the calculation.
• Use consistent units for all measurements.
• Account for each distinct area separately before summing them up for the total.

By mastering surface area calculations, students can better understand the design and cost implications of three-dimensional spaces.