Surface Area: Change in Dimensions

A lighthouse is located on an island 33 miles from the closest point on a straight shoreline. If the lighthouse light rotates cl

Zinaida [17]

Question:

A lighthouse is located on an island 3 miles from the closest point on a straight shoreline. If the lighthouse light rotates clockwise at a constant rate of 9 revolutions per minute, how fast does the beam of light move towards the point on the shore closest to the island when it is 52 miles from that point

Answer:

The beam of light moves at $16278\pi$ miles/min

Step-by-step explanation:

This question is illustrated with the attached image

Taking the instructions in the question, one at a time.

A revolution of 9 per minute implies that:

$\frac{d\theta}{dt} = \frac{9 * 2\pi\ rad}{1\ min}$

$\frac{d\theta}{dt} = 18\pi \frac{rad}{min}$

Take tan of the angle in the attachment:

$tan(\theta) =\frac{52}{3}$

Differentiate both sides with respect to time

$\frac{d\ tan(\theta)}{dt} =\frac{52}{3} * \frac{dx}{dt}$

Rewrite as:

$\frac{d\ tan(\theta)}{d\theta} * \frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}$

In calculus:

$sec^2(\theta) =\frac{d\ tan(\theta)}{d\theta}$ -- Chain rule

So:

$sec^2(\theta) *\frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}$

In trigonometry:

$sec^2(\theta) = tan^2(\theta) + 1$

So:

$(tan^2(\theta) + 1)\frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}$

Recall that:

$tan(\theta) =\frac{x}{3}$

$((\frac{52}{3})^2 + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}$

$(\frac{52^2}{9} + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}$

$(\frac{2704}{9} + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}$

$(\frac{2704+9}{9})\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}$

$(\frac{2713}{9})\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}$

Recall that: $\frac{d\theta}{dt} = 18\pi \frac{rad}{min}$

$(\frac{2713}{9}) * 18\pi =\frac{1}{3} * \frac{dx}{dt}$

$2713 * 2\pi =\frac{1}{3} * \frac{dx}{dt}$

Multiply both sides by 3

$3 * 2713 * 2\pi =\frac{1}{3} * \frac{dx}{dt} * 3$

$3 * 2713 * 2\pi =\frac{dx}{dt}$

$16278\pi =\frac{dx}{dt}$

$\frac{dx}{dt} = 16278\pi$ miles/min

Hence:

The beam of light moves at $16278\pi$ miles/min

6 0

3 years ago

Two sides of a parallelogram measure 60 centimeters and 40 centimeters. If one angle of the parallelogram measures 1320, find th

UkoKoshka [18]

Answer:

b. about 91.7 cm and 44.6 cm

Step-by-step explanation:

The lengths of the diagonals can be found using the Law of Cosines.

Consider the triangle(s) formed by a diagonal. The two given sides will form the other two sides of the triangle, and the corner angles of the parallelogram will be the measure of the angle between those sides (opposite the diagonal).

For diagonal "d" and sides "a" and "b" and corner angle D, we have ...

d² = a² +b² -2ab·cos(D)

The measure of angle D will either be the given 132°, or the supplement of that, 48°. We can use the fact that the cosines of an angle and its supplement are opposites. This means the diagonal measures will be ...

d² = 60² +40² -2·60·40·cos(D) ≈ 5200 ±4800(0.66913)

d² ≈ {1988.2, 8411.8}

d ≈ {44.6, 91.7} . . . . centimeters

The diagonals are about 91.7 cm and 44.6 cm.

4 0

3 years ago

A right triangle has a base of 14.3 cm, a slant height of 21.75 cm, and a height of 15 cm. What is the area of the triangle?

lina2011 [118]

Answer:

107.25cm²

Step-by-step explanation:

Area of triangle = 0.5 * base * height

Base = 14.3 cm

Height = 15 cm

Hence, Area of triangle :

Area = 0.5 * 14.3 * 15

Area = 107.25cm²

8 0

3 years ago

I need help with this probability question.

Ahat [919]

Answer:

20/total (u can calculate it)

Step-by-step explanation:

Number of dogs/Total number of pets

3 0

3 years ago

Given: <1 and <4 are supplementary; C || d.<br> Prove a || b

lys-0071 [83]

Answer:

4Ф64÷6→β44

Step-by-step explanation:

8 0

2 years ago