Light Houses Galore

building Small Lighthouses
Math help please. triangles, pythagorean theorm?

my math class is confusing . i need help on learning how to do this on my practice test.
The captain of a ship wants to determine his distance from shore. Seeking a familiar landmark, he finds a 90-ft-high lighthouse on top of a cliff. He sights both the top and bottom of the lighthouse. The measures of the two angles of elevation are 46 and 39. How far, to two significant digits, is he from the base of the cliff? A)325 B) 425 C)300 D)400
Opposite corners of a small rectangular park are joined by diagonal paths, each 360 m long. What are the dimensions of the park if the paths intersect at a 65° angle?
A)166.6 m by 56.4 m B)151.8 m by 96.7 m C)193.4 m by 303.6 m D)172.34 m by 45.6 m
How far from the base of a building is the bottom of a 30 foot ladder that makes an angle of 75° with the ground?
A)111.96 ft B)28.98 ft C)7.7646 ft D)94.56 ft

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Problem 1
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There are two angles of elevation.
They would look like:

/_

with one angled lower than the other.

The wider angle (46­°) is from sea level to the _top_ of the light house, and the smaller angle (39°) is from sea to the _bottom_ of the light house. So if we take the far side of the bigger triangle and subtract it from the same side of the smaller triangle, it would equal the height of the lighthouse.

Since we have the far side and are interested in the distance from shore (the bottom side):
tan(x) = opposite/adjacent

(big tan) - (small tan) = (big far side - small far side)/a
tan (46°) - tan (39°) = 90/a

Multiply by x (or cross-multiply):
a [ tan (46°) - tan (39°) ] = 90

Divide by the tangents:
a = 90 / [ tan (46°) - tan (39°) ]
a ≈ 398.677665

Two significant digits would be 3 and 9, so round to the tenth:

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Answer:
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D) 400 ft from shore.

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Problem 2
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/ 65° angle of two paths
/

Split it in the middle:
/_ two 32.5° angles
/

This makes a right angle with the side of the park:
.../|
_/_| <- right triangle
../..|

The hypotenuse is half of the path diagonal across the park, since it only goes to the middle.
360 / 2 = 180

This gives us:
The angle at the center of the triangle = 32.5°
The hypotenuse = 180m
We want:
The edge of the park (opposite), which is half the length
AND the near edge (adjacent), which is half the width

So we will use both sin and cos

Angle: 32.5°
Hypotenuse: 180
opposite: ½ L
sin(x) = opposite/hypotenuse
sin(32.5°) = (½ L)/180
180·sin(32.5°) = (½ L)
360·sin(32.5°) = L
L ≈ 193.427859

Angle & Hypotenuse are the same
opposite: ½ W
cos(x) = adjacent/hypotenuse
cos(32.5°) = (½ W)/180
180·cos(32.5°) = (½ W)
360·cos(32.5°) = W
W ≈ 303.62092

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Answer:
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C) 193.4 m by 303.6 m

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Problem 3
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A ladder set like:
../|
./.l
/_|
30 feet is the length of the ladder, and it is at an angle.
Hypotenuse: 30
Angle at bottom: 75°
We want: the far side from the angle, the distance up the wall

......./|
30ft./.l <- distance up wall
75°./_|

Oppsite side and hypotenuse:

sin(x) = opposite/hypotenuse
sin(75°) = w/30
30­·sin(75°) = w
w ≈ 28.9777748

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Answer:
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B) 28.98 ft