Math / Trigonometry

Question6. When the Sun is at an angle of elevation $x$ , a tree with height 12 m casts a shadow of length $s$ , in metres. a) Show that the length of the shadow can be modelled by the relation $s=12 \cot x$ . b) Sketch a graph of $s=12 \cot x$ on the interval $x \in\left[0, \frac{\pi}{2}\right]$ . c) Interpret the meaning of the graph as $x$ approaches 0 and as $x$ approaches $\frac{\pi}{2}$ .

Studdy Solution

STEP 1

1. The height of the tree is constant at 12 meters.
2. The angle of elevation $x$ is measured from the horizontal ground to the line of sight to the top of the tree.
3. The shadow length $s$ is directly related to the angle of elevation $x$ .

STEP 2

1. Derive the relation for the length of the shadow.
2. Sketch the graph of the derived relation.
3. Interpret the behavior of the graph as $x$ approaches 0 and $\frac{\pi}{2}$ .

STEP 3

The angle of elevation $x$ forms a right triangle with the tree height as the opposite side and the shadow length $s$ as the adjacent side.

STEP 4

Using trigonometry, the tangent of the angle $x$ is given by: $\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{s}$

STEP 5

Rearrange the equation to solve for $s$ : $s = \frac{12}{\tan x} = 12 \cot x$

STEP 6

To sketch the graph of $s = 12 \cot x$ , note that the cotangent function $\cot x$ is undefined at $x = 0$ and has a vertical asymptote there. It decreases to 0 as $x$ approaches $\frac{\pi}{2}$ .

STEP 7

Plot the graph of $s = 12 \cot x$ over the interval $x \in \left(0, \frac{\pi}{2}\right)$ . The graph starts from infinity at $x = 0$ and approaches 0 as $x$ approaches $\frac{\pi}{2}$ .

STEP 8

As $x$ approaches 0, the angle of elevation is very small, causing the shadow length $s$ to become very large (approaches infinity).

STEP 9

As $x$ approaches $\frac{\pi}{2}$ , the angle of elevation increases, causing the shadow length $s$ to decrease and approach 0.
The graph and its interpretation show how the shadow length changes with the angle of elevation:
- As $x \to 0$ , $s \to \infty$ . - As $x \to \frac{\pi}{2}$ , $s \to 0$ .

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Studdy Solution

STEP 1

1. The height of the tree is constant at 12 meters.
2. The angle of elevation $x$ is measured from the horizontal ground to the line of sight to the top of the tree.
3. The shadow length $s$ is directly related to the angle of elevation $x$ .

STEP 2

1. Derive the relation for the length of the shadow.
2. Sketch the graph of the derived relation.
3. Interpret the behavior of the graph as $x$ approaches 0 and $\frac{\pi}{2}$ .

STEP 3

The angle of elevation $x$ forms a right triangle with the tree height as the opposite side and the shadow length $s$ as the adjacent side.

STEP 4

Using trigonometry, the tangent of the angle $x$ is given by: $\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{s}$

STEP 5

Rearrange the equation to solve for $s$ : $s = \frac{12}{\tan x} = 12 \cot x$

STEP 6

To sketch the graph of $s = 12 \cot x$ , note that the cotangent function $\cot x$ is undefined at $x = 0$ and has a vertical asymptote there. It decreases to 0 as $x$ approaches $\frac{\pi}{2}$ .

STEP 7

Plot the graph of $s = 12 \cot x$ over the interval $x \in \left(0, \frac{\pi}{2}\right)$ . The graph starts from infinity at $x = 0$ and approaches 0 as $x$ approaches $\frac{\pi}{2}$ .

STEP 8

As $x$ approaches 0, the angle of elevation is very small, causing the shadow length $s$ to become very large (approaches infinity).

STEP 9

As $x$ approaches $\frac{\pi}{2}$ , the angle of elevation increases, causing the shadow length $s$ to decrease and approach 0.
The graph and its interpretation show how the shadow length changes with the angle of elevation:
- As $x \to 0$ , $s \to \infty$ . - As $x \to \frac{\pi}{2}$ , $s \to 0$ .

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Studdy Solution

STEP 1

1. The height of the tree is constant at 12 meters.2. The angle of elevation x x x is measured from the horizontal ground to the line of sight to the top of the tree.3. The shadow length s s s is directly related to the angle of elevation x x x.

STEP 2

1. Derive the relation for the length of the shadow.2. Sketch the graph of the derived relation.3. Interpret the behavior of the graph as x x x approaches 0 and π2 \frac{\pi}{2} 2π​.

STEP 3

The angle of elevation x x x forms a right triangle with the tree height as the opposite side and the shadow length s s s as the adjacent side.

STEP 4

Using trigonometry, the tangent of the angle x x x is given by: tan⁡x=oppositeadjacent=12s \tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{s} tanx=adjacentopposite​=s12​

STEP 5

Rearrange the equation to solve for s s s: s=12tan⁡x=12cot⁡x s = \frac{12}{\tan x} = 12 \cot x s=tanx12​=12cotx

STEP 6

To sketch the graph of s=12cot⁡x s = 12 \cot x s=12cotx, note that the cotangent function cot⁡x \cot x cotx is undefined at x=0 x = 0 x=0 and has a vertical asymptote there. It decreases to 0 as x x x approaches π2 \frac{\pi}{2} 2π​.

STEP 7

Plot the graph of s=12cot⁡x s = 12 \cot x s=12cotx over the interval x∈(0,π2) x \in \left(0, \frac{\pi}{2}\right) x∈(0,2π​). The graph starts from infinity at x=0 x = 0 x=0 and approaches 0 as x x x approaches π2 \frac{\pi}{2} 2π​.

STEP 8

As x x x approaches 0, the angle of elevation is very small, causing the shadow length s s s to become very large (approaches infinity).

STEP 9

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1. The height of the tree is constant at 12 meters.
2. The angle of elevation $x$ is measured from the horizontal ground to the line of sight to the top of the tree.
3. The shadow length $s$ is directly related to the angle of elevation $x$ .

1. Derive the relation for the length of the shadow.
2. Sketch the graph of the derived relation.
3. Interpret the behavior of the graph as $x$ approaches 0 and $\frac{\pi}{2}$ .

The angle of elevation $x$ forms a right triangle with the tree height as the opposite side and the shadow length $s$ as the adjacent side.

Using trigonometry, the tangent of the angle $x$ is given by: $\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{s}$

Rearrange the equation to solve for $s$ : $s = \frac{12}{\tan x} = 12 \cot x$

To sketch the graph of $s = 12 \cot x$ , note that the cotangent function $\cot x$ is undefined at $x = 0$ and has a vertical asymptote there. It decreases to 0 as $x$ approaches $\frac{\pi}{2}$ .

Plot the graph of $s = 12 \cot x$ over the interval $x \in \left(0, \frac{\pi}{2}\right)$ . The graph starts from infinity at $x = 0$ and approaches 0 as $x$ approaches $\frac{\pi}{2}$ .

As $x$ approaches 0, the angle of elevation is very small, causing the shadow length $s$ to become very large (approaches infinity).