Let's break down and solve the problem in three parts based on the given triangular garden.

Given:

• Two sides of the triangular garden: $a = 5 \, \text{m}$, $b = 10 \, \text{m}$
• Angle between these sides: $\theta = 31^\circ 15'$ or $31.25^\circ$
• Cost of seedlings per square meter: Php 150

Step 1: Find the perimeter of the garden

To find the perimeter, we first need the third side of the triangle, which we can calculate using the Cosine Rule: $c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)$ Substituting the values: $c^2 = 5^2 + 10^2 - 2 \cdot 5 \cdot 10 \cdot \cos(31.25^\circ)$ $c^2 = 25 + 100 - 100 \cdot \cos(31.25^\circ)$

After finding the third side, the perimeter $P$ will be: $P = a + b + c$

Step 2: Find the area of the garden

The area of a triangle can be calculated using the formula for two sides and the included angle: $\text{Area} = \frac{1}{2}ab \sin(\theta)$ Substituting the values: $\text{Area} = \frac{1}{2} \cdot 5 \cdot 10 \cdot \sin(31.25^\circ)$

Step 3: Calculate the total cost of seedlings

Once we have the area of the garden, the total cost of planting seedlings at Php 150 per square meter will be: $\text{Total Cost} = \text{Area} \times 150$

Let me calculate these values for you.Here are the results for the problem:

1. Perimeter of the garden: Approximately 21.29 meters.
2. Area of the garden: Approximately 12.97 square meters.
3. Total cost for seedlings: Approximately Php 1,945.40.

Would you like more details or further explanation on any step?

Here are five related questions you might explore:

1. What if the angle between the sides was different, say $45^\circ$?
2. How does the area change if both sides were increased proportionally?
3. Can you use the Sine Rule to find the other angles of the triangle?
4. How would the perimeter change if one side was increased by 2 meters?
5. What would be the effect of changing the cost per square meter to Php 200?

Tip: Always convert angles to radians when using trigonometric functions in calculations.