1. Programming with Scripts.
Answer the following questions to check your understanding of scripts.
Exercises 💻 Coding Exercises
2. Programming with Functions.
Answer the following questions to check your understanding of functions.
(a) Temperature Conversion.
Write a function named
celsius_to_fahrenheit that converts a temperature from Celsius to Fahrenheit.
The conversion formula is: \(F = \frac{9}{5}C + 32\)
| Inputs: | celsius (1x1) double — temperature in degrees Celsius
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| Outputs: | fahrenheit (1x1) double — temperature in degrees Fahrenheit
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Test cases:
% Test 1: Freezing point of water
f1 = celsius_to_fahrenheit(0)
% Expected: f1 = 32
% Test 2: Boiling point of water
f2 = celsius_to_fahrenheit(100)
% Expected: f2 = 212
% Test 3: Room temperature
f3 = celsius_to_fahrenheit(20)
% Expected: f3 = 68
(b) Triangle Area.
Write a function named
triangle_area that computes the area of a triangle given its base and height.
The area formula is: \(A = \frac{1}{2} \times \text{base} \times \text{height}\)
| Inputs: | base (1x1) double — length of the triangle’s base
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height (1x1) double — height of the triangle
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| Outputs: | area (1x1) double — area of the triangle
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Test cases:
% Test 1: Simple triangle
a1 = triangle_area(6, 4)
% Expected: a1 = 12
% Test 2: Right triangle
a2 = triangle_area(5, 12)
% Expected: a2 = 30
% Test 3: Decimal values
a3 = triangle_area(7.5, 3.2)
% Expected: a3 = 12
(c) Sphere Properties.
Write a function named
sphere_properties that computes both the surface area and volume of a sphere given its radius.
| Inputs: | radius (1x1) double — radius of the sphere
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| Outputs: | surface_area (1x1) double — surface area of the sphere
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volume (1x1) double — volume of the sphere
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Test cases:
% Test 1: Unit sphere
[A1, V1] = sphere_properties(1)
% Expected: A1 ≈ 12.5664, V1 ≈ 4.1888
% Test 2: Radius of 2
[A2, V2] = sphere_properties(2)
% Expected: A2 ≈ 50.2655, V2 ≈ 33.5103
% Test 3: Radius of 5
[A3, V3] = sphere_properties(5)
% Expected: A3 ≈ 314.1593, V3 ≈ 523.5988
(d) Cylinder Properties.
Write a function named
cylinder_properties that computes the surface area, volume, and diagonal of a cylinder given its radius and height.
The formulas are:
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Surface area: \(A = 2\pi r^2 + 2\pi r h\)
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Volume: \(V = \pi r^2 h\)
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Diagonal: ?
Part of this problem is determining the correct formula for the diagonal of a cylinder. Hint: take out a piece of paper and sketch the cylinder, then draw the diagonal line from one edge of the base to the opposite edge of the top.
| Inputs: | radius (1x1) double — radius of the cylinder
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height (1x1) double — height of the cylinder
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| Outputs: | surface_area (1x1) double — total surface area of the cylinder
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volume (1x1) double — volume of the cylinder
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diagonal (1x1) double — diagonal length
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Test cases:
% Test 1: Unit cylinder (radius = 1, height = 1)
[A1, V1, d1] = cylinder_properties(1, 1)
% Expected: A1 ≈ 12.5664, V1 ≈ 3.1416, d1 ≈ 2.2361
% Test 2: Radius = 3, height = 5
[A2, V2, d2] = cylinder_properties(3, 5)
% Expected: A2 ≈ 150.7964, V2 ≈ 141.3717, d2 ≈ 7.8102
% Test 3: Radius = 2.5, height = 8
[A3, V3, d3] = cylinder_properties(2.5, 8)
% Expected: A3 ≈ 164.9336, V3 ≈ 157.0796, d3 ≈ 9.4340
(e) Compound Interest Calculator.
Write a function named
compound_interest that calculates the future value of an investment with compound interest, and also returns the total interest earned and the effective annual rate.
The compound interest formula is: \(A = P\left(1 + \frac{r}{n}\right)^{nt}\)
where:
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\(P\) = principal (initial investment)
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\(r\) = annual interest rate (as a decimal, e.g., 0.05 for 5%)
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\(n\) = number of times interest is compounded per year
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\(t\) = time in years
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\(A\) = final amount
The effective annual rate (EAR) is: \(\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1\)
| Inputs: | principal (1x1) double — initial investment amount
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rate (1x1) double — annual interest rate (as decimal)
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compounds_per_year (1x1) double — compounding frequency (e.g., 12 for monthly)
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years (1x1) double — time period in years
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| Outputs: | final_amount (1x1) double — total amount after compound interest
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interest_earned (1x1) double — total interest earned (final_amount - principal)
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effective_rate (1x1) double — effective annual rate
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Test cases:
% Test 1: $1000 at 5% annual rate, compounded monthly for 10 years
[final1, int1, effRate1] = compound_interest(1000, 0.05, 12, 10)
% Expected: final1 ≈ 1647.01, int1 ≈ 647.01, effRate1 ≈ 0.0512
% Test 2: $5000 at 3% annual rate, compounded quarterly for 5 years
[final2, int2, effRate2] = compound_interest(5000, 0.03, 4, 5)
% Expected: final2 ≈ 5805.92, int2 ≈ 805.92, effRate2 ≈ 0.03034
% Test 3: $10000 at 6% annual rate, compounded daily for 20 years
[final3, int3, effRate3] = compound_interest(10000, 0.06, 365, 20)
% Expected: final3 ≈ 33197.90, int3 ≈ 23197.90, effRate3 ≈ 0.0618
(f) Triangle Perimeter and Area in 3D Space.
Write a function named
triangle_3d that computes the perimeter and area of a triangle in 3D space given the coordinates of its three vertices. You may assume the three points do not lie on the same straight line. This ensures a valid triangle.
Hints:
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The distance formula between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is:\begin{equation*} d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \end{equation*}
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If the side lengths of the triangle are \(a\text{,}\) \(b\text{,}\) and \(c\text{,}\) then the area is given by Heron’s formula:\begin{equation*} A = \sqrt{s(s-a)(s-b)(s-c)} \end{equation*}where \(s = 0.5\left(a+b+c\right)\) (known as the semi-perimeter).
| Inputs: | x1, y1, z1 (1x1) double — coordinates of first vertex
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x2, y2, z2 (1x1) double — coordinates of second vertex
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x3, y3, z3 (1x1) double — coordinates of third vertex
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| Outputs: | perimeter (1x1) double — perimeter of the triangle
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area (1x1) double — area of the triangle
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Note: Typically you would pass coordinates as vectors or matrices, but we will save that for later exercises.
Test cases:
% Test 1: Triangle in xy-plane (vertices at (0,0,0), (4,0,0), (0,3,0))
[p1, a1] = triangle_3d(0, 0, 0, 4, 0, 0, 0, 3, 0)
% Expected: p1 = 12, a1 = 6 (3-4-5 right triangle)
% Test 2: Equilateral triangle in space
[p2, a2] = triangle_3d(0, 0, 0, 1, 0, 0, 0.5, sqrt(3)/2, 0)
% Expected: p2 = 3, a2 ≈ 0.4330
% Test 3: Triangle with vertices at (1,2,3), (4,5,6), (2,3,5)
[p3, a3] = triangle_3d(1, 2, 3, 4, 5, 6, 2, 3, 5)
% Expected: p3 ≈ 10.6456, a3 ≈ 2.1213
