Question

A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is 17°44'. When the boat stops, the angle of depression is 48°13' . The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.

Answer 1

Answer:

Distance covered by the boat was 446.27 feet

Step-by-step explanation:

A boat which is at the point C approaching the light house.

At point C angle of depression to the boat was 17°44' and when boat stops at point B angle of depression was 48°13'.

Length of the tower is 200 feet then distance DC traveled by the boat.

Since at point B angle of depression was 17°44' so we will convert the angle in degrees.

Angle of depression = ∠ACB = 17°44' = $17+\frac{44}{60}=17+0.73=17.73$

From ΔABC

tan 17.73° = $\frac{200}{BC}$

$0.320=\frac{200}{BC}$

$BC=\frac{200}{0.32}= 625 feet$

Now in ΔABD

∠ADB = 48°13' = 48° + $\frac{13}{60}=48 + 0.217$ = 48.217°

Now $tan(48.217)=\frac{200}{BD}$

$BD=\frac{200}{1.119}=178.73 feet$

Therefore distance DC = BC - BD

DC = 625 - 178.73 = 446.27 feet

So the answer is 446.27 feet was distance covered by the boat.

Answer 2

The distance covered by the boat is $\boxed{446.26{\text{ feet}}}.$

Further explanation:

The Pythagorean formula can be expressed as,

$\boxed{{H^2} = {P^2} + {B^2}}.$

Here, H represents the hypotenuse, P represents the perpendicular and B represents the base.

The formula for sin of angle a can be expressed as

$\boxed{\sin a = \frac{P}{H}}$

The formula for cos of angle a can be expressed as

$\boxed{\cos a = \frac{B}{H}}$

The formula for tan of angle a can be expressed as

$\boxed{\tan a = \frac{P}{B}}$

Explanation:

The perpendicular AB. The length of AB is $200{\text{ feet}}.$

The angle of depression is $\angle ACB = {17^ \circ }44'.$

One degree has 60 minutes.

${1^ \circ } = 60'$

$\begin{aligned}\angle ACB &= 17 + \frac{{44}}{{60}}\\&= 17 + 0.73\\&= 17.73\\\end{aligned}$

The angle ABD is

$\begin{aligned}\angle ADB &= {48^ \circ }13' \\ &= 48 + \frac{{13}}{{60}}\\& = 48 + 0.217\\&= {48.217^ \circ }\\\end{aligned}$

In triangle ABC.

$\begin{aligned}\tan \left( {{{17.73}^ \circ }} \right) &= \frac{{200}}{{BC}}\\0.320&= \frac{{200}}{{BC}}\\BC&= \frac{{200}}{{0.320}}\\BC&= 625{\text{ feet}}\\\end{aligned}$

In triangle ABD.

$\begin{aligned}\tan \left( {48.217} \right) &= \frac{{200}}{{BD}}\\1.12&= \frac{{200}}{{BD}}\\BD&= \frac{{200}}{{1.12}}\\BD &= 178.74{\text{ feet}}\\\end{aligned}$

The distance boat can travel can be obtained as follows,

$\begin{aligned}DC &= BD - BC\\&= 625 - 178.74\\&= 446.26{\text{ feet}}\\\end{aligned}$

The distance covered by the boat is $\boxed{446.26{\text{ feet}}}.$

Kindly refer to the image attached.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Trigonometry

Keywords: perpendicular, person watching boat, top, lighthouse, angle of depression, angle of elevation, 200 feet tall, travel, sides, right angle triangle, triangle, altitudes, hypotenuse, on the triangle, hypotenuse, trigonometric functions.