Math / Trigonometry

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From the observation deck of a skyscraper, Ian measures a $48^{\circ}$ angle of depression to a ship in the harbor below. If the observation deck is 871 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 1 out of 2 feet Submit Answer

Studdy Solution

STEP 1

What is this asking? If we're 871 feet up in a skyscraper, and we look down at a $48^\circ$ angle to see a ship, how far away is the ship horizontally? Watch out! Don't mix up angle of depression with angle of elevation!
Angle of depression is from the horizontal downwards, while angle of elevation is from the horizontal upwards.

STEP 2

1. Draw a diagram
2. Find the horizontal distance

STEP 3

Let's imagine ourselves up on that observation deck!
We're looking down at the ship.

STEP 4

We can draw a right triangle where: * The vertical leg is the height of the skyscraper (871 feet). * The horizontal leg is the distance we want to find.
Let's call it $x$ . * The angle of depression is the angle from the horizontal down to the ship, which is $48^\circ$ .

STEP 5

Since the angle of depression is measured from the horizontal downwards, the angle inside our triangle is actually $90^\circ - 48^\circ = 42^\circ$ .
This angle is between the horizontal leg and the hypotenuse.

STEP 6

We have the opposite side ( $871$ feet) and we want the adjacent side ( $x$ ).
Which trig function relates opposite and adjacent?
It's the tangent function!

STEP 7

Remember, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ .
In our case, $\tan(42^\circ) = \frac{871}{x}$ .

STEP 8

To get $x$ by itself, we can multiply both sides by $x$ and then divide both sides by $\tan(42^\circ)$ : $x \cdot \tan(42^\circ) = 871$ $x = \frac{871}{\tan(42^\circ)}$

STEP 9

Using a calculator, we find that $\tan(42^\circ) \approx 0.9004$ . $x = \frac{871}{0.9004}$ $x \approx 967.34$

STEP 10

The horizontal distance from the base of the skyscraper to the ship is approximately 967.34 feet.

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Studdy Solution

STEP 1

STEP 2

1. Draw a diagram
2. Find the horizontal distance

STEP 3

Let's imagine ourselves up on that observation deck!
We're looking down at the ship.

STEP 4

We can draw a right triangle where: * The vertical leg is the height of the skyscraper (871 feet). * The horizontal leg is the distance we want to find.
Let's call it $x$ . * The angle of depression is the angle from the horizontal down to the ship, which is $48^\circ$ .

STEP 5

Since the angle of depression is measured from the horizontal downwards, the angle inside our triangle is actually $90^\circ - 48^\circ = 42^\circ$ .
This angle is between the horizontal leg and the hypotenuse.

STEP 6

We have the opposite side ( $871$ feet) and we want the adjacent side ( $x$ ).
Which trig function relates opposite and adjacent?
It's the tangent function!

STEP 7

Remember, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ .
In our case, $\tan(42^\circ) = \frac{871}{x}$ .

STEP 8

To get $x$ by itself, we can multiply both sides by $x$ and then divide both sides by $\tan(42^\circ)$ : $x \cdot \tan(42^\circ) = 871$ $x = \frac{871}{\tan(42^\circ)}$

STEP 9

Using a calculator, we find that $\tan(42^\circ) \approx 0.9004$ . $x = \frac{871}{0.9004}$ $x \approx 967.34$

STEP 10

The horizontal distance from the base of the skyscraper to the ship is approximately 967.34 feet.

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Studdy Solution

STEP 1

STEP 2

1. Draw a diagram2. Find the horizontal distance

STEP 3

Let's **imagine** ourselves up on that observation deck!We're looking *down* at the ship.

STEP 4

We can draw a **right triangle** where: * The **vertical leg** is the height of the skyscraper (**871 feet**). * The **horizontal leg** is the distance we want to find.Let's call it xxx. * The **angle of depression** is the angle from the horizontal *down* to the ship, which is 48∘48^\circ48∘.

STEP 5

Since the angle of depression is measured from the horizontal downwards, the angle *inside* our triangle is actually 90∘−48∘=42∘90^\circ - 48^\circ = 42^\circ90∘−48∘=42∘.This angle is between the horizontal leg and the hypotenuse.

STEP 6

We have the **opposite side** (871871871 feet) and we want the **adjacent side** (xxx).Which trig function relates opposite and adjacent?It's the **tangent** function!

STEP 7

Remember, tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}tan(θ)=adjacentopposite​.In our case, tan⁡(42∘)=871x\tan(42^\circ) = \frac{871}{x}tan(42∘)=x871​.

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∘)=871 x \cdot \tan(42^\circ) = 871 x⋅tan(42∘)=871 x=871tan⁡(42∘) x = \frac{871}{\tan(42^\circ)} x=tan(42∘)871​

STEP 9

Using a calculator, we find that tan⁡(42∘)≈0.9004\tan(42^\circ) \approx 0.9004tan(42∘)≈0.9004. x=8710.9004 x = \frac{871}{0.9004} x=0.9004871​ x≈967.34 x \approx 967.34 x≈967.34

STEP 10

The horizontal distance from the base of the skyscraper to the ship is approximately **967.34 feet**.

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1. Draw a diagram
2. Find the horizontal distance

Let's imagine ourselves up on that observation deck!
We're looking down at the ship.

We can draw a right triangle where: * The vertical leg is the height of the skyscraper (871 feet). * The horizontal leg is the distance we want to find.
Let's call it $x$ . * The angle of depression is the angle from the horizontal down to the ship, which is $48^\circ$ .

Since the angle of depression is measured from the horizontal downwards, the angle inside our triangle is actually $90^\circ - 48^\circ = 42^\circ$ .
This angle is between the horizontal leg and the hypotenuse.

We have the opposite side ( $871$ feet) and we want the adjacent side ( $x$ ).
Which trig function relates opposite and adjacent?
It's the tangent function!

Remember, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ .
In our case, $\tan(42^\circ) = \frac{871}{x}$ .

To get $x$ by itself, we can multiply both sides by $x$ and then divide both sides by $\tan(42^\circ)$ : $x \cdot \tan(42^\circ) = 871$ $x = \frac{871}{\tan(42^\circ)}$

Using a calculator, we find that $\tan(42^\circ) \approx 0.9004$ . $x = \frac{871}{0.9004}$ $x \approx 967.34$

The horizontal distance from the base of the skyscraper to the ship is approximately 967.34 feet.