Math  /  Trigonometry

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1. Larissa stands at point K , the top of a 218 m vertical cliff. The base of the cliff is located at point B . A ship is located at point S,1200 m\mathrm{S}, 1200 \mathrm{~m} from Larissa. (a) Find the angle of elevatien from the ship to Larissa. (b) Find the horizontal distance from the base of the cliff to the ship. [2] [2] [2]

Studdy Solution

STEP 1

What is this asking? We need to find the angle looking up from a ship to a person on a cliff, and also the distance between the ship and the bottom of the cliff. Watch out! Make sure your calculator is in degree mode, and don't mix up the sides of the triangle!

STEP 2

1. Draw a diagram
2. Find the angle of elevation
3. Find the horizontal distance

STEP 3

Let's **imagine** Larissa at the top of the cliff, waving down at a ship out at sea.
We can draw a right triangle with Larissa at the top, the base of the cliff at the bottom, and the ship out at sea.

STEP 4

We'll label the top of the cliff KK, the base of the cliff BB, and the ship SS.
The distance from KK to SS is \(1200 m\).
The height of the cliff, KBKB, is \(218 m\).
We want to find the angle of elevation from SS to KK, which we'll call θ\theta, and the horizontal distance BSBS, which we'll call xx.

STEP 5

We now have a right triangle KBSKBS with hypotenuse KSKS, height KBKB, and base BSBS.

STEP 6

We know the hypotenuse (KS=1200KS = 1200 m) and the opposite side (KB=218KB = 218 m) relative to the angle θ\theta.
So, we'll use the **sine function**: sin(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

STEP 7

sin(θ)=2181200\sin(\theta) = \frac{218}{1200}

STEP 8

To find θ\theta, we take the **inverse sine** (arcsin) of both sides: θ=arcsin(2181200)\theta = \arcsin\left(\frac{218}{1200}\right) θ10.5\theta \approx 10.5^\circ

STEP 9

We can use the **Pythagorean theorem** since we have a right triangle: (hypotenuse)2=(opposite)2+(adjacent)2(\text{hypotenuse})^2 = (\text{opposite})^2 + (\text{adjacent})^2 KS2=KB2+BS2KS^2 = KB^2 + BS^2

STEP 10

(1200)2=(218)2+x2(1200)^2 = (218)^2 + x^2

STEP 11

x2=(1200)2(218)2x^2 = (1200)^2 - (218)^2 x2=144000047524x^2 = 1440000 - 47524 x2=1392476x^2 = 1392476x=1392476x = \sqrt{1392476}x1180x \approx 1180

STEP 12

(a) The angle of elevation from the ship to Larissa is approximately **10.510.5^\circ**. (b) The horizontal distance from the base of the cliff to the ship is approximately **11801180 m**.

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