Two lighthouses are located 75 miles from one another on a north-south line. If a boat is spotted S 40o E from the northern lighthouse and N 21o E from the southern lighthouse, how much closer is the closest lighthouse to the boat than the lighthouse furthest away? The northern lighthouse is 51.3 miles closer than the southern lighthouse. The northern lighthouse is 24.4 miles closer than the southern lighthouse. The northern lighthouse is 79.9 miles closer than the southern lighthouse. The northern lighthouse is 17.0 miles closer than the southern lighthouse.

Answer:

The northern lighthouse is approximately [tex]24.4\; \rm mi[/tex] closer to the boat than the southern lighthouse.

Step-by-step explanation:

Refer to the diagram attached. Denote the northern lighthouse as [tex]\rm N[/tex], the southern lighthouse as [tex]\rm S[/tex], and the boat as [tex]\rm B[/tex]. These three points would form a triangle.

It is given that two of the angles of this triangle measure [tex]40^{\circ}[/tex] (northern lighthouse, [tex]\angle {\rm N}[/tex]) and [tex]21^{\circ}[/tex] (southern lighthouse [tex]\angle {\rm S}[/tex]), respectively. The three angles of any triangle add up to [tex]180^{\circ}[/tex]. Therefore, the third angle of this triangle would measure [tex]180^{\circ} - (40^{\circ} + 21^{\circ}) = 119^{\circ}[/tex] (boat [tex]\angle {\rm B}[/tex].)

It is also given that the length between the two lighthouses (length of [tex]\rm NS[/tex]) is [tex]75\; \rm mi[/tex].

By the law of sine, the length of a side in a given triangle would be proportional to the angle opposite to that side. For example, in the triangle in this question, [tex]\angle {\rm B}[/tex] is opposite to side [tex]\rm NS[/tex], whereas [tex]\angle {\rm S}[/tex] is opposite to side [tex]{\rm NB}[/tex]. Therefore:

[tex]\begin{aligned} \frac{\text{length of NS}}{\sin(\angle {\rm B})} = \frac{\text{length of NB}}{\sin(\angle {\rm S})} \end{aligned}[/tex].

Substitute in the known measurements:

[tex]\begin{aligned} \frac{75\; \rm mi}{\sin(119^{\circ})} = \frac{\text{length of NB}}{\sin(21^{\circ})} \end{aligned}[/tex].

Rearrange and solve for the length of [tex]\rm NB[/tex]:

[tex]\begin{aligned} & \text{length of NB} \\ =\; & (75\; \rm mi) \times \frac{\sin(21^{\circ})}{\sin(119^{\circ})} \\ \approx\; & 30.73\; \rm mi\end{aligned}[/tex].

(Round to at least one more decimal places than the values in the choices.)

Likewise, with [tex]\angle {\rm N}[/tex] is opposite to side [tex]{\rm SB}[/tex], the following would also hold:

[tex]\begin{aligned} \frac{\text{length of NS}}{\sin(\angle {\rm B})} = \frac{\text{length of SB}}{\sin(\angle {\rm N})} \end{aligned}[/tex].

[tex]\begin{aligned} \frac{75\; \rm mi}{\sin(119^{\circ})} = \frac{\text{length of SB}}{\sin(40^{\circ})} \end{aligned}[/tex].

[tex]\begin{aligned} & \text{length of SB} \\ =\; & (75\; \rm mi) \times \frac{\sin(40^{\circ})}{\sin(119^{\circ})} \\ \approx\; & 55.12\; \rm mi\end{aligned}[/tex].

In other words, the distance between the northern lighthouse and the boat is approximately [tex]30.73\; \rm mi[/tex], whereas the distance between the southern lighthouse and the boat is approximately [tex]55.12\; \rm mi[/tex]. Hence the conclusion.

6 + 1 2/3 in fraction

PLEASE ANWSER CORRECTLY!! I WILL MARK BRAINIEST!!!

PLEASE HELP ILL GIVE BRIANALIST!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Decrease 12kg by 25%

FRACTIONAL INDICIES
PLEASE HELP!!!

A forest covers 54,000 acres. A survey finds that 0.4​% of the forest is​ old-growth trees. How many acres of​ old-growth trees are​ there?

WILL MARK BEST ANSWER BRAINLIEST


Miranda compared the function f (x) = 1/4x + 5 to the linear function, g(x), shown in the table.

table:

X G(X)

-9 -10
-6 -8
-3 -6
0 -4

What is the slope of the function with the larger slope?

A: 5/1

B: 1/4

C: 3/2

D: 2/3

I need help with my delta math

write an inequality for the following statement: a is greater than or equal to 9

6x – 2 + 2x = 2(4x – 3) + 4.

Find the perimeter P of ▱JKLM with vertices J(0,1), K(3,4), L(8,4), and M(5,1). Round your answer to the nearest tenth, if necessary. Please help, got to get this done.

Need help ASAP 50 points ?!!!

Kay and Karen went on a fishing trip.Kay caught 20 fish and Karen caught 12 fish. What is the value of the ratio

(I will give brainiest for BEST answer) Why is a negative integer time a negative integer positive?

13 A wheel covers la distance of 88m after making 100 revolutions What is the radiu of the wheel?​

how much is 32 yards and 24 feet then it it divided by 4

For the figure shown on the right, find the value of the variable and the measures of the angles.

What is the slope-intercept equation
for the following line?

Calculate the slope (50,10) (77,25)

In the diagram on the side, BCDE and ACF are straight lines. Calculate the values ​​of x and y. ​

Am confused
Pls help

Vanessa has 7 pieces of pizza and she wants to split them evenly between herself and 4 friends. How much will each person receive?​

Help needed ASAP!!!!!!!!

5/6x10 tyyyyyyyyyyyyyyyyyyyyyy

Lines a, b, c, d, and e are distinct lines in
the same plane. Decide if lines a and e
are parallel, perpendicular, or if there is not
enough information given.
alb, b || c, cid, die
A. parallel
B. perpendicular
C. not enough information

Find Square Root of the Following
49

Find the negative Square root of 16​

Simplify the following expression:
3[(x^3– 7 x + 1) – (x+ 4)]

plz fill this out. in trying to pass. will give best answer​

List the multiples of 6 through 100

slove of the quotient of 952 and 4