We can solve this problem using trigonometry and geometry. Here's how:

Given Data:

• The triangular garden is bounded on two sides by streets intersecting at an angle of 31°15′ (or 31.25° in decimal).
• The sides along the streets are 5 m and 10 m.

Let's go through each part:

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Part (a): Find the perimeter of the garden

We can use the Law of Cosines to find the third side of the triangle, $c$, opposite to the angle $31°15′$.

The Law of Cosines formula is: $c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)$ where:

• $a = 5 \, \text{m}$
• $b = 10 \, \text{m}$
• $\theta = 31.25°$

Substitute the values: $c^2 = 5^2 + 10^2 - 2 \times 5 \times 10 \times \cos(31.25°)$ Now, we calculate the third side $c$ and then the perimeter.

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Part (b): Find the area of the garden

We can use the formula for the area of a triangle when two sides and the included angle are given: $\text{Area} = \frac{1}{2}ab \sin(\theta)$ Substitute the values: $\text{Area} = \frac{1}{2} \times 5 \times 10 \times \sin(31.25°)$

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Part (c): Total cost for seedlings

The cost of planting seedlings is Php 150 per square meter. The total cost is calculated as: $\text{Total Cost} = \text{Area} \times \text{Cost per square meter}$ Substitute the area found in part (b) and the cost per square meter: $\text{Total Cost} = \text{Area} \times 150$

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Now, let's calculate these values.### Solutions:

Part (a): The perimeter of the garden is approximately:

$\text{Perimeter} \approx 21.29 \, \text{meters}$

Part (b): The area of the garden is approximately:

$\text{Area} \approx 12.97 \, \text{square meters}$

Part (c): The total cost of planting seedlings is approximately:

$\text{Total Cost} \approx \text{Php} 1945.40$

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Do you have any questions or need more details on any part?

Here are 5 related questions:

1. How would the perimeter change if one side was increased by 2 meters?
2. Can we find the angles of the triangle if the third side is known?
3. What would be the cost if the price per square meter increased by 10%?
4. How would the area calculation differ if the angle was larger than 90°?
5. What other methods can we use to calculate the area of a triangle?

Tip: Knowing the Law of Cosines and Law of Sines helps solve triangles when angles and sides are involved!