Final Assessment Test - Fall 2025-26: Question Set

General Instructions

• Use only Python list comprehensions for your computations. Do not use explicit for or while loops, and do not import NumPy, pandas, or any other external modules.
• Perform all calculations at full precision and round only in the final display using formatted string literals (e.g., f"{value:.2f}").
• Preserve the input order in your outputs — corresponding results must match the index positions of their inputs.
• Unless otherwise specified, use SI units and UK spelling (e.g., metres, centimetres, etc.).
• You may import only the standard Python math module (e.g., math.sin, math.sqrt, math.pi, math.tan).
• Each computation must be encapsulated within a clearly defined function that accepts the given input(s) and returns or prints the required rounded results.
• Display all outputs in list or tuple form, exactly as demonstrated in the examples provided for each question.

Questions Set

Each question involves the creation of a Python function, numerical computation, and rounding to the specified precision. No file reading or writing is permitted. Students must display computed results clearly.

On Inputs and Data Types

• Do not use input(). No string parsing or splitting is required.
• Your functions will be tested by calling them with native Python objects: lists (e.g., [20.0, 35.0]), tuples (e.g., (x, y)), or lists of tuples (e.g., [((2, 3), (5, 7)), ((1, 1), (4, 5))]).
• Assume valid types. Do not convert from strings; do not perform file I/O.
• Return the computed results (and also print them in the rounded format shown in each question).

Question 1: Degree–Minute–Second (DMS) to Decimal Degree Conversion (Theodolite Surveying)

Background: In theodolite surveying, angles are expressed as degrees (D), minutes (M), and seconds (S). They must be converted into decimal degrees for computation.

Formula:

\[ \text{Decimal Degrees} = D + \frac{M}{60} + \frac{S}{3600} \]

Task:

• Write a function dms_to_decimal(deg, min, sec) that accepts three lists of equal length.
• Return a list of decimal degree values rounded to 4 decimal places.
• Display the resulting list.

Expected Output:

[54.2583, 23.4583, 89.9997]

Question 2: Tangent Length for a Simple Circular Curve (Highway Alignment)

Background: The tangent length \(T\) for a simple circular curve is given by:

\[ T = R \tan\!\left(\frac{\Delta}{2}\right) \]

where \(R\) = radius (m), and \(\Delta\) = deflection angle (degrees).

Task:

• Write a function tangent_length(R, delta_deg) to compute the tangent length for each pair of values.
• Use the math module and round each result to 2 decimal places.

Expected Output:

Tangent lengths: [35.47, 56.78, 43.12]

Question 3: Time Period of a Compound Pendulum (Mechanics)

Background: For a compound pendulum of length \(L\) (in metres), the time period \(T\) is:

\[ T = 2\pi \sqrt{\frac{2L}{3g}} \]

where \(g = 9.81\,\mathrm{m\,s^{-2}}\).

Task:

• Write a function time_period(lengths) that returns a list of periods (seconds) rounded to 3 decimal places.

Expected Output:

Time Periods: ['2.197', '2.934', '3.208']

Question 4: Distance Between Two Points (Coordinate Geometry)

Background: The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Task:

• Write a function point_distance(points) where points is a list of pairs of coordinate tuples.
  Example input: [((2, 3), (5, 7)), ((1, 1), (4, 5))]
• The function should return a list of distances rounded to 2 decimal places.

Expected Output:

Distances: [5.00, 5.00]

Question 5: Centroid of a Triangle (Coordinate Geometry)

Background: For a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), the centroid \((x_c, y_c)\) is calculated as:

\[ x_c = \frac{x_1 + x_2 + x_3}{3}, \quad y_c = \frac{y_1 + y_2 + y_3}{3} \]

Task:

• Write a function centroid(coords) that accepts a list of three coordinate tuples.
• Return the centroid coordinates as a tuple rounded to 2 decimal places.

Expected Output:

Centroid: (4.67, 2.33)

Question 6: Moment of Inertia for a Rectangular Beam Section (Mechanics of Materials)

Background: For a rectangular cross-section, the moment of inertia \(I\) about its base is given by:

\[ I = \frac{b h^3}{12} \]

where \(b\) = breadth (m) and \(h\) = depth (m).

Task:

• Write a function moment_inertia(b, h) that accepts lists of breadths and depths.
• Return moments of inertia rounded to 5 decimal places.

Expected Output:

Moments of Inertia: [0.00123, 0.00289, 0.00500]

Common Rational Rubric (Applicable to All Questions)

Demonstration: Calling the Functions in Python

The following snippet shows one possible way in which each function can be called from a Python session or script. Inputs are supplied directly as lists and tuples, matching the constraints given above.

# Example calls for testing:

# 1. DMS to Decimal Degrees
dms_to_decimal(
    [54, 23, 89],
    [15, 27, 59],
    [30, 30, 59]
)

# 2. Tangent Lengths
tangent_length(
    [200, 350], 
    [45, 60]
)

# 3. Time Periods
time_period([0.8, 1.2, 1.5])

# 4. Point Distances
point_distance([((2, 3), (5, 7)), ((10, 4), (13, 9))])

# 5. Centroid
centroid([(2, 3), (6, 1), (6, 3)])

# 6. Moment of Inertia
moment_inertia([0.2, 0.3], [0.4, 0.5])

Worked Solutions: Narrative and Code

The sections below present one possible implementation for each question. The code respects the given constraints: use of functions, list comprehensions, full-precision calculations, and rounding only at the display stage. Each explanation is placed in a note box for quick reference.

1. DMS to Decimal Degrees

Angle observations recorded in D–M–S format are converted to decimal degrees using a direct numeric translation of the surveying formula.

def dms_to_decimal(deg, mins, secs):
    values = [d + m/60 + s/3600 for d, m, s in zip(deg, mins, secs)]
    rounded = [round(v, 4) for v in values]
    print(rounded)
    return rounded

2. Tangent Length of a Simple Circular Curve

The tangent length depends on both the curve radius and the deflection angle, and is obtained by applying the standard geometric relation for circular curves.

import math

def tangent_length(R, delta_deg):
    values = [r * math.tan(math.radians(d/2)) for r, d in zip(R, delta_deg)]
    rounded = [round(v, 2) for v in values]
    print(rounded)
    return rounded

3. Time Period of a Compound Pendulum

The time period depends on the pendulum length and gravitational acceleration, with the formula involving a square root and the constant 2π.

import math

def time_period(lengths):
    g = 9.81
    values = [2 * math.pi * math.sqrt((2*l)/(3*g)) for l in lengths]
    rounded = [f"{v:.3f}" for v in values]
    print(rounded)
    return rounded

4. Distance Between Two Points

Planar distances between pairs of points are obtained by applying the Euclidean distance formula to each tuple of coordinates.

import math

def point_distance(points):
    values = [math.sqrt((p2[0]-p1[0])**2 + (p2[1]-p1[1])**2) 
              for p1, p2 in points]
    rounded = [round(v, 2) for v in values]
    print(rounded)
    return rounded

5. Centroid of a Triangle

The centroid is found by averaging the coordinates of the three vertices, giving the balance point of the triangle.

def centroid(coords):
    x_vals = [c[0] for c in coords]
    y_vals = [c[1] for c in coords]
    xc = round(sum(x_vals)/3, 2)
    yc = round(sum(y_vals)/3, 2)
    result = (xc, yc)
    print(result)
    return result

6. Moment of Inertia of a Rectangular Section

The moment of inertia about the base is computed from the section breadth and depth using the standard expression for rectangular sections.

def moment_inertia(b, h):
    values = [bi * (hi**3) / 12 for bi, hi in zip(b, h)]
    rounded = [round(v, 5) for v in values]
    print(rounded)
    return rounded