Math / Trigonometry

QuestionTrig Word Problems (Level 1) Score: $2 / 10$ Penalty: 1 off
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From the observation deck of a skyscraper, Lavaughn measures a $42^{\circ}$ angle of depression to a ship in the harbor below. If the observation deck is 872 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest hundredth of a foot if necessary.
Answer Attempt 2 out of 2 968.62 $\square$ feet Submit Answer

Studdy Solution

STEP 1

What is this asking? If Lavaughn is 872 feet up in a skyscraper and sees a ship at a 42° angle downwards, how far is the ship horizontally from the skyscraper? Watch out! Don't mix up angle of depression with angle of elevation!
Also, make sure your calculator is in degree mode, not radians!

STEP 2

1. Draw a diagram
2. Identify the relevant trigonometric function
3. Solve for the horizontal distance

STEP 3

Let's sketch this out!
We've got our skyscraper, right?
And way up high, there's Lavaughn on the observation deck.
Then, boom, there's a ship out on the water.

STEP 4

Draw a vertical line to represent the skyscraper.
Label the top of the line "Lavaughn" and the bottom "Base of Skyscraper." Draw a horizontal line from the base of the skyscraper to represent the sea level.
Draw another line from Lavaughn to the ship, representing Lavaughn's line of sight.
Label the point where the line of sight meets the sea level as "Ship." Finally, draw a horizontal line from Lavaughn to a point directly below Lavaughn on the sea level.
This creates a right triangle.
Label the angle between Lavaughn's horizontal line and the line of sight as 42°.
Label the vertical side of the triangle 872 feet.

STEP 5

We know the angle of depression is 42°.
This is the angle downwards from the horizontal.
The angle inside our triangle is the same as this angle of depression!

STEP 6

We know the opposite side ( $872$ feet) and we want the adjacent side (the horizontal distance).
Which trig function uses opposite and adjacent?
It's the tangent function!

STEP 7

We can write the equation: $\tan(42^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{872}{\text{horizontal distance}}$ Let's use $x$ to represent the horizontal distance. $\tan(42^{\circ}) = \frac{872}{x}$

STEP 8

To get $x$ by itself, we can multiply both sides by $x$ and then divide both sides by $\tan(42^{\circ})$ : $x \cdot \tan(42^{\circ}) = 872$ $x = \frac{872}{\tan(42^{\circ})}$

STEP 9

Now, grab your calculator and make sure it's in degree mode! $x = \frac{872}{\tan(42^{\circ})} \approx \frac{872}{0.9004} \approx 968.45$

STEP 10

The horizontal distance from the base of the skyscraper to the ship is approximately 968.45 feet.

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Studdy Solution

STEP 1

What is this asking? If Lavaughn is 872 feet up in a skyscraper and sees a ship at a 42° angle downwards, how far is the ship horizontally from the skyscraper? Watch out! Don't mix up angle of depression with angle of elevation!
Also, make sure your calculator is in degree mode, not radians!

STEP 2

1. Draw a diagram
2. Identify the relevant trigonometric function
3. Solve for the horizontal distance

STEP 3

Let's sketch this out!
We've got our skyscraper, right?
And way up high, there's Lavaughn on the observation deck.
Then, boom, there's a ship out on the water.

STEP 4

STEP 5

We know the angle of depression is 42°.
This is the angle downwards from the horizontal.
The angle inside our triangle is the same as this angle of depression!

STEP 6

We know the opposite side ( $872$ feet) and we want the adjacent side (the horizontal distance).
Which trig function uses opposite and adjacent?
It's the tangent function!

STEP 7

We can write the equation: $\tan(42^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{872}{\text{horizontal distance}}$ Let's use $x$ to represent the horizontal distance. $\tan(42^{\circ}) = \frac{872}{x}$

STEP 8

To get $x$ by itself, we can multiply both sides by $x$ and then divide both sides by $\tan(42^{\circ})$ : $x \cdot \tan(42^{\circ}) = 872$ $x = \frac{872}{\tan(42^{\circ})}$

STEP 9

Now, grab your calculator and make sure it's in degree mode! $x = \frac{872}{\tan(42^{\circ})} \approx \frac{872}{0.9004} \approx 968.45$

STEP 10

The horizontal distance from the base of the skyscraper to the ship is approximately 968.45 feet.

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Studdy Solution

STEP 1

What is this asking? If Lavaughn is 872 feet up in a skyscraper and sees a ship at a 42° angle downwards, how far is the ship horizontally from the skyscraper? Watch out! Don't mix up *angle of depression* with *angle of elevation*!Also, make sure your calculator is in *degree* mode, not radians!

STEP 2

1. Draw a diagram2. Identify the relevant trigonometric function3. Solve for the horizontal distance

STEP 3

Let's **sketch** this out!We've got our skyscraper, right?And way up high, there's Lavaughn on the observation deck.Then, boom, there's a ship out on the water.

STEP 4

STEP 5

We know the angle of depression is **42°**.This is the angle *downwards* from the horizontal.The angle inside our triangle is the same as this angle of depression!

STEP 6

We know the **opposite side** (872872872 feet) and we want the **adjacent side** (the horizontal distance).Which trig function uses opposite and adjacent?It's the **tangent** function!

STEP 7

STEP 8

To get xxx by itself, we can multiply both sides by xxx and then divide both sides by tan⁡(42∘)\tan(42^{\circ})tan(42∘): x⋅tan⁡(42∘)=872 x \cdot \tan(42^{\circ}) = 872 x⋅tan(42∘)=872 x=872tan⁡(42∘) x = \frac{872}{\tan(42^{\circ})} x=tan(42∘)872​

STEP 9

Now, grab your calculator and make sure it's in *degree* mode! x=872tan⁡(42∘)≈8720.9004≈968.45 x = \frac{872}{\tan(42^{\circ})} \approx \frac{872}{0.9004} \approx 968.45 x=tan(42∘)872​≈0.9004872​≈968.45

STEP 10

The horizontal distance from the base of the skyscraper to the ship is approximately **968.45 feet**.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? If Lavaughn is 872 feet up in a skyscraper and sees a ship at a 42° angle downwards, how far is the ship horizontally from the skyscraper? Watch out! Don't mix up angle of depression with angle of elevation!
Also, make sure your calculator is in degree mode, not radians!

1. Draw a diagram
2. Identify the relevant trigonometric function
3. Solve for the horizontal distance

Let's sketch this out!
We've got our skyscraper, right?
And way up high, there's Lavaughn on the observation deck.
Then, boom, there's a ship out on the water.

We know the angle of depression is 42°.
This is the angle downwards from the horizontal.
The angle inside our triangle is the same as this angle of depression!

We know the opposite side ( $872$ feet) and we want the adjacent side (the horizontal distance).
Which trig function uses opposite and adjacent?
It's the tangent function!

To get $x$ by itself, we can multiply both sides by $x$ and then divide both sides by $\tan(42^{\circ})$ : $x \cdot \tan(42^{\circ}) = 872$ $x = \frac{872}{\tan(42^{\circ})}$

Now, grab your calculator and make sure it's in degree mode! $x = \frac{872}{\tan(42^{\circ})} \approx \frac{872}{0.9004} \approx 968.45$

The horizontal distance from the base of the skyscraper to the ship is approximately 968.45 feet.