QuestionApplication
1. The angle of elevation of the sun is $65^{\circ}$ when a tree casts a shadow of 18.3 m long. How tall is the tree? (3 marks)
2. The angle of elevation of the top of a mountain peak, $P$, as observed from a point $A$ on a level plane below is $22^{\circ}$. The angle of elevation from a second point B, 1000 m closer to the base of the mountain is $32^{\circ}$. Find the height of the peak. ( 6 marks)
3. Football goal posts are measured and found to be 6.7 m apart. A player is to attempt a field goal from a point where the ball is 44 m and 42 m from the ends of the goal posts. Within what angle must the he kick the ball? ( 4 marks)

STEP 1

1. For problem 1, the tree, its shadow, and the line of sight to the sun form a right triangle.
2. For problem 2, the observer's line of sight to the mountain peak forms right triangles with the horizontal plane.
3. For problem 3, the player, the two goal posts, and the ball form a triangle.
_HIGH_LEVEL_APPROACH_ for Problem 1:
1. Identify the right triangle and the known values.
2. Use trigonometric ratios to find the height of the tree.
_HIGH_LEVEL_APPROACH_ for Problem 2:
1. Set up two right triangles for points A and B.
2. Use trigonometric ratios to find expressions for the height of the mountain.
3. Solve the system of equations to find the height of the peak.
_HIGH_LEVEL_APPROACH_ for Problem 3:
1. Use the law of cosines to find the angle at which the player must kick the ball.
**Problem 1:**

STEP 2

STEP 3

Identify the right triangle formed by the tree, its shadow, and the line of sight to the sun. The angle of elevation is $65^\circ$, and the shadow length is $18.3$ m.

STEP 4

Use the tangent function, which relates the angle of elevation to the opposite side (height of the tree) and the adjacent side (length of the shadow):
$$\tan(65^\circ) = \frac{\text{Height of the tree}}{18.3}$$
Solve for the height of the tree:
$$\text{Height of the tree} = 18.3 \times \tan(65^\circ)$$
**Problem 2:**
STEP_1: Set up right triangles for points A and B. Let $h$ be the height of the peak, and $d$ be the distance from point A to the base of the mountain.
STEP_2: For point A, use the tangent function:
$$\tan(22^\circ) = \frac{h}{d}$$
For point B, which is 1000 m closer:
$$\tan(32^\circ) = \frac{h}{d - 1000}$$

STEP 5

Solve the system of equations:
1. $h = d \times \tan(22^\circ)$
2. $h = (d - 1000) \times \tan(32^\circ)$
Equate the two expressions for $h$ and solve for $d$, then substitute back to find $h$.
**Problem 3:**
STEP_1: Use the law of cosines in the triangle formed by the player and the two goal posts. Let $a = 44$ m, $b = 42$ m, and $c = 6.7$ m (distance between the goal posts).
The law of cosines states:
$$c^2 = a^2 + b^2 - 2ab \cos(\theta)$$
Solve for $\theta$:
$$\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}$$
Calculate $\theta$ using the inverse cosine function.
**Solutions:**
1. Height of the tree: $\boxed{18.3 \times \tan(65^\circ)}$
2. Height of the peak: Solve the system of equations to find $h$.
3. Angle for the field goal: Calculate $\theta$ using the law of cosines.