A farmer decided to fence a triangular vegetable garden from his field bounded on two sides by streets intersecting at 31°15′. The sides of the garden along the streets are 5 m and 10 m.

a.Find the perimeter of the garden.

b.Find the area of the garden.

c.Seedlings are to be planted in the garden at a cost of Php150 per square meter. What is the total cost?

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:

---

### 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.

---

### 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:
\[
\text{Area} = \frac{1}{2} \times 5 \times 10 \times \sin(31.25°)
\]

---

### 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:
\[
\text{Total Cost} = \text{Area} \times 150
\]

---

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:
\[
\text{Total Cost} \approx \text{Php} 1945.40
\]

---

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!

A farmer decided to fence a triangular vegetable garden from his field bounded on two sides by streets intersecting at 31°15′. The sides of the garden along the streets are 5 m and 10 m.

(a) Find the perimeter of the garden.
(b) Find the area of the garden.
(c) Seedlings are to be planted in the garden at a cost of Php150 per square meter. What is the total cost?

Learn how to calculate the perimeter, area, and planting cost of a triangular garden. This problem involves trigonometry concepts such as the Law of Cosines and the sine rule for the area of a triangle. Perfect for students in Grades 10-12.

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