1. A man stands 12 feet from a statue. The angle of elevation from the eye level to the top of the statue is 30 degrees, and the angle of depression to the base of the statue is 15. How tall in the statue?
2. Two boats lie on a straight line with the base of a lighthouse. From the top of the lighthouse, 21 meters above water level, it is observed that the angle of depression of the nearest boat is 53 degrees and the angle of depression of the farthers boat is 27 degrees. How far aprt are the boats?
ok,
draw a triangle ABE with AB a vertical line, that is your statue.
E is to left of the line AB
Join EA and EB
From E draw a line perpendicular to AB to meet AB at P, P is between A and B
EP = 12
angle AEP=30º
angle BEP = 15º
in top triangle:
AP/12 = tan 30
AP = 12tan30 = 6.928
in bottom triangle
BP/12 = tan15
BP = 12tan15 = 3.215
AB = 10.14
#1 My diagram has two right angle triangles with a common right angle, the one with the 53 angle embedded within the larger one.
label the distance from the nearer boat to the lighthouse x, label the distance from the farther boat to the lighthouse y
then tan53 = 21/x
x = 21/tan53
and tan 27 = y/21
distance between the two boats = y-x
#1 (similar to #2)
my diagram has the man standing 12 to the left of the statue, A the top, B the bottom, the line to the top makes an angle of 30º and to the bottom of the statue 15º
draw a horizontalline from the "eye" to the statue meeting it at P
use the tangent ratio twice to find AP and BP
height of statue = AP + BP
1. Well, let's see here. The man is standing 12 feet away from the statue and we know the angle of elevation to the top is 30 degrees. So if we use a little bit of trigonometry, we can determine the height of the statue. But instead of doing that, let's just pretend the statue is as tall as me when I'm wearing stilts. So, the statue is obviously 6 feet tall because 6 feet is the perfect height for a statue, don't you think?
2. Ah, a lighthouse, some boats, and some angles to calculate. Sounds like a real triangle party! So, the lighthouse is 21 meters above water level, and the angle of depression to the nearest boat is 53 degrees, and to the farthest boat, it's 27 degrees. Now, if we had Moana's navigational skills, we could probably just sail between the two boats and measure the distance, but unfortunately, we don't. So let's crunch some numbers. Using those angles, we can determine the distances from the lighthouse to each boat, and then, voila, we'll know the distance between the two boats. But honestly, who needs fancy calculations when we can just shout really loudly and ask the boats how far apart they are? Boats can be pretty chatty, you know!
1. To find the height of the statue, we can use trigonometry. Let's break down the information given:
- The man stands 12 feet away from the statue, and the angle of elevation from the eye level to the top of the statue is 30 degrees.
- The angle of depression from the same eye level to the base of the statue is 15 degrees.
To solve this, we can use the tangent function, as tangent is defined as the opposite side divided by the adjacent side of a right triangle.
First, let's find the height of the statue using the angle of elevation. Since we have the distance (12 feet) and the angle of elevation (30 degrees), we can set up the equation:
tan(30 degrees) = height of the statue / 12 feet
Using a scientific calculator, we can find that the tangent of 30 degrees is approximately 0.5774. To solve for the height of the statue, we can multiply both sides of the equation by 12 feet:
0.5774 * 12 = height of the statue
This gives us a height of approximately 6.9288 feet.
Next, let's find the distance from the eye level to the base of the statue. Note that the angle of depression is measured from the same eye level, so we can set up a similar equation:
tan(15 degrees) = distance to the base / 12 feet
Again, using a scientific calculator, we can find that the tangent of 15 degrees is approximately 0.2679. To solve for the distance to the base of the statue, we can multiply both sides of the equation by 12 feet:
0.2679 * 12 = distance to the base
This gives us a distance of approximately 3.2148 feet.
Finally, to find the total height of the statue, we add the height from the eye level to the base of the statue:
total height = height of the statue + distance to the base
= 6.9288 feet + 3.2148 feet
= 10.1436 feet
Therefore, the height of the statue is approximately 10.1436 feet.
2. To find the distance between the two boats, we can use trigonometry and the concept of similar triangles. Let's break down the information given:
- The lighthouse is 21 meters above water level.
- The angle of depression from the top of the lighthouse to the nearest boat is 53 degrees.
- The angle of depression from the top of the lighthouse to the farthest boat is 27 degrees.
Using the concept of similar triangles, we can set up the following proportion:
(height of the lighthouse) / (distance between the boats) = (tan(53 degrees) - tan(27 degrees))
We can simplify this equation by calculating the tangent of 53 degrees and 27 degrees using a scientific calculator. Let's assume they are approximately 1.327 and 0.509 respectively.
Now we can rewrite the equation:
21 / (distance between the boats) = 1.327 - 0.509
To find the distance between the boats, we can isolate the variable:
(distance between the boats) = 21 / (1.327 - 0.509)
Using the values we have, substituting them into the equation, and performing the calculations, we find:
(distance between the boats) ≈ 21 / 0.818
This gives us a distance of approximately 25.67 meters.
Therefore, the distance between the two boats is approximately 25.67 meters.