Presume that a ladder is put upright against a wall. Let variables length and angle store the length of the la

Exercises · Q21

Q. Presume that a ladder is put upright against a wall. Let variables length and angle store the length of the ladder and the angle that it forms with the ground as it leans against the wall. Write a Python program to compute the height reached by the ladder on the wall for the following values of length and angle:

a) 16 feet and 75 degrees

b) 20 feet and 0 degrees

c) 24 feet and 45 degrees

d) 24 feet and 80 degrees

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Concept understanding — Simple Interest Calculation

Simple Interest: The Cost of Borrowing (or the Reward for Saving)

Imagine you lend a friend ₹100. A year later, they return the ₹100. You're even — but you also missed the chance to use that ₹100 yourself for a whole year. That inconvenience has a price. In finance, that price is called interest.

Simple interest is the most straightforward way to calculate that price. It says: the interest you earn (or pay) each year is a fixed percentage of the original amount you started with — no matter how many years pass.

The Core Intuition

Think of it like renting money. If you rent a car for a day, you pay a fixed daily rate. If you rent it for 5 days, you pay 5 times that daily rate. Simple interest works the same way:

The Principal (P) is the amount you lend or borrow — the "car."
The Rate (R) is the daily rental charge, expressed as a percentage of the car's value — the "rent per day."
The Time (T) is how many days you keep the car — the number of years.

The total rent (interest) is just: daily rent × number of days.

The Precise Statement

\text{Simple Interest (SI)} = \frac{P \times R \times T}{100}

Where:

P = Principal (the original sum of money)
R = Rate of interest per year (in percent)
T = Time period (in years)

The Total Amount (A) you get back (or owe) at the end is:

A = P + \text{SI}

Why Divide by 100?

The rate $R$ is given as a percentage — say 10%. "10% per year" means you get ₹10 for every ₹100 you lend, each year. So for a principal $P$ , the interest for one year is:

\frac{10}{100} \times P = 0.10 \times P

That's the same as $\frac{P \times R}{100}$ . For $T$ years, you multiply by $T$ , giving $\frac{P \times R \times T}{100}$ .

A Worked Example

You deposit ₹2000 in a bank that offers 8% simple interest per year. You leave it for 3 years.

P = ₹2000
R = 8
T = 3

\text{SI} = \frac{2000 \times 8 \times 3}{100} = \frac{48000}{100} = ₹480

So after 3 years, you earn ₹480 in interest. Your total amount becomes:

A = 2000 + 480 = ₹2480

Tip

Notice that each year you earn exactly the same interest: $\frac{2000 \times 8}{100} = ₹160$ . Year 1: ₹160, Year 2: another ₹160, Year 3: another ₹160. The interest never grows because it's always calculated on the original ₹2000, not on the growing balance.

The Key Property (and its Limitation)

Simple interest is linear — the interest grows at a constant rate each year. This makes it easy to calculate and understand. …

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The ladder is the hypotenuse, so height on the wall = length × sin(angle). Python's math.sin needs radians, hence math.radians():

import math
for length, angle in [(16, 75), (20, 0), (24, 45), (24, 80)]:
    height = length * math.sin(math.radians(angle))
    print(length, "feet at", angle, "degrees -> height", round(height, 2), "feet")

Expected output:

16 feet at 75 degrees -> height 15.45 feet
20 feet at 0 degrees -> height 0.0 feet …

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height = length × sin(angle), converting degrees to radians first — giving ≈ 15.45 ft, 0.0 ft, ≈ 16.97 ft and ≈ 23.64 ft for the four cases.

The geometry: the ladder, the wall and the ground form a right triangle in which the ladder is the hypotenuse and the angle is at the ground. The height reached on the wall is the side opposite the angle, so height = length × sin(angle). The programming concept is that math.sin() expects radians, so the degree inputs must go through math.radians() first.

# Height reached on the wall by a leaning ladder
import math

for length, angle in [(16, 75), (20, 0), (24, 45), (24, 80)]:
    height = length * math.sin(math.radians(angle))
    print(length, "feet at", angle, "degrees -> height", round(height, 2), "feet")

Key lines:

math.radians(angle) converts degrees to radians (75° → 1.3089…). Feeding degrees straight into math.sin is the classic bug — sin(75 radians) ≈ −0.39, a nonsense height.
The loop evaluates all four required cases in one run.

Working for each case:

Case	length	angle	sin(angle)	height = length × sin
a	16 ft	75°	0.9659	15.45 ft
b	20 ft	0°	0.0	0.0 ft
c	24 ft	45°	0.7071	16.97 ft
d	24 ft	80°	0.9848	23.64 ft

Expected output:

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