Let's solve both problems step by step.

First Problem: Cost of the Fence

Given Data:

• The field is a right-angled triangle.
• One side (adjacent to 52°) = 151 m.
• Fence cost = £1.42 per metre.

Step 1: Finding the Other Two Sides

We use trigonometry to find the missing sides.

1. Finding the opposite side (height of the triangle)
   $\text{opposite} = \text{adjacent} \times \tan(52^\circ)$ $= 151 \times \tan(52^\circ)$ $\approx 151 \times 1.279$ $\approx 193.13 \text{ m}$

2. Finding the hypotenuse
   $\text{hypotenuse} = \frac{\text{opposite}}{\sin(52^\circ)}$ $= \frac{193.13}{\sin(52^\circ)}$ $\approx \frac{193.13}{0.788}$ $\approx 245.1 \text{ m}$

Step 2: Finding the Total Perimeter

The total perimeter = sum of all three sides $151 + 193.13 + 245.1 = 589.23 \text{ m}$

Step 3: Finding the Total Cost

$\text{Total Cost} = 589.23 \times 1.42$ $\approx 836.71$

Final Answer:

The total cost of the fence is £837 (rounded to the nearest pound).

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Second Problem: Smallest Possible Angle of Take-off

Given Data:

• Height = 9 km.
• Hypotenuse (flight distance) = 49 km (smallest possible value).

Step 1: Using Sine Rule

Since we have the opposite side (height) and the hypotenuse (flight path), we use:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

$\sin \theta = \frac{9}{49}$

$\sin \theta \approx 0.1837$

Step 2: Find the Angle

$\theta = \sin^{-1} (0.1837)$

$\theta \approx 10.6^\circ$

Final Answer:

The smallest possible angle is 10.6° (rounded to 1 decimal place).

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Do you need more details or explanations?

Here are 5 related questions:

1. How would the cost change if the price per meter increased to £1.60?
2. What if the field was a different type of triangle—how would we calculate the perimeter?
3. How would the take-off angle change if the plane traveled 53 km instead of 49 km?
4. How can we use the Pythagorean theorem to verify the hypotenuse in the first problem?
5. What are some real-life applications of trigonometry in aviation and construction?

Tip:

In trigonometry, always check whether to use sine, cosine, or tangent based on the sides given.