Let's break down the problem.

Part (a) - Finding angles $x$ and $y$

We are given two triangular roof sections, with the dimensions of the triangle for each house:

• House A: A triangle with the base of $7 \, \text{m}$ and the slant height of $5.2 \, \text{m}$, and the vertical height from the ground of $5 \, \text{m}$. The angle $x$ is the angle formed between the slant height and the vertical side.

• House B: A triangle with the base of $7.6 \, \text{m}$, slant height of $5.6 \, \text{m}$, and a vertical height from the ground of $5 \, \text{m}$. The angle $y$ is the angle formed similarly.

Using Trigonometry:

For a right-angled triangle, we can use the tangent formula:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

where $\theta$ is the angle in question, the opposite side is the vertical height, and the adjacent side is the base (half of the total roof width, as it's a symmetric triangle).

Angle $x$ for House A:

The base for this triangle will be half of the total roof width of $7 \, \text{m}$, which is $3.5 \, \text{m}$.

$\tan(x) = \frac{5 \, \text{m}}{3.5 \, \text{m}}$ $x = \tan^{-1}\left( \frac{5}{3.5} \right)$

Angle $y$ for House B:

Similarly, the base for this triangle will be half of the total roof width of $7.6 \, \text{m}$, which is $3.8 \, \text{m}$.

$\tan(y) = \frac{5 \, \text{m}}{3.8 \, \text{m}}$ $y = \tan^{-1}\left( \frac{5}{3.8} \right)$

Let me calculate these values for you.The calculated angles to one decimal place are:

• Angle $x$ for House A: $55.0^\circ$
• Angle $y$ for House B: $52.8^\circ$

Part (b) - Snowfall Suitability

A house in a country with a lot of snowfall is more likely to have a steeper roof to allow snow to slide off easily. In this case, House A has a steeper roof with a greater angle ($55.0^\circ$) compared to House B ($52.8^\circ$). Therefore, House A is more likely to be in a country that experiences heavy snowfall.

Would you like any further details or clarifications on this? Here are 5 related questions to explore this further:

1. How do different roof angles affect snow load on buildings?
2. What other factors should be considered when designing roofs for snowy climates?
3. Can we calculate the area of the triangular sections of these roofs?
4. How would the length of the slant height change if the roof angle was altered?
5. What is the significance of roof angle in architectural design beyond snow considerations?

Tip: When designing for heavy snowfall, angles steeper than 50 degrees are often preferred to reduce snow accumulation.