A person flying a kite holds the string 4 feet above ground level. The string of the kite is taut and makes an angle of 60° with the horizontal (see the figure). Approximate the height of the kite above level ground if 700 feet of string is payed out. (Round your answer to one decimal place.)
A guy wire is attached to the top of a radio antenna and to a point on horizontal ground that is 30.0 meters from the base of the antenna. If the wire makes an angle of 55°10' with the ground, approximate the length l of the wire. (Round your answer to the nearest tenth.)
A rocket is fired at sea level and climbs at a constant angle of 65° through a distance of 20,000 feet. Approximate its altitude to the nearest foot.
Each of these involves drawing a triangle with its base parallel to the ground, with a given angle. Then, either a side or the hypotenuse is given. If b is the base and h is the height, c is the hypotenuse, and a is the angle
tan(a) = h/b
cos(a) = b/c
sin(a) = h/c
For example, in the first problem,
(h-4)/700 = sin(60°)
In the second,
30/L = cos(55°10')
In the third,
h/20000 = sin(65°)
To find the height of the kite, we can use trigonometry. In the given figure, we have a right triangle formed by the string of the kite, the height of the kite above ground level, and the horizontal distance from the person to the kite.
We are given that the angle made by the string with the horizontal is 60° and the string length is 700 feet. Let's denote the height of the kite above ground level as h.
Using the trigonometric function sine (sin), we can write:
sin(60°) = h / 700
Solving for h:
h = 700 * sin(60°)
Using a calculator, we find that sin(60°) is approximately 0.866. Now we can substitute this value:
h = 700 * 0.866
Calculating this expression, we find that the height of the kite above ground level is approximately 606.2 feet.
Now let's move on to the second problem.
To find the length of the guy wire, we can again use trigonometry. In the given figure, we have a right triangle formed by the guy wire, the horizontal distance from the base of the antenna to the attaching point of the guy wire, and the angle made by the wire with the ground.
We are given that the angle is 55°10'. To work with this angle, we need to convert it to decimal degrees. We can do this by dividing the minutes (10') by 60 and adding it to the degrees (55):
55 + (10 / 60) = 55.1667°
Now, let's denote the length of the guy wire as l. Using the trigonometric function sine (sin), we can write:
sin(55.1667°) = l / 30.0
Solving for l:
l = 30.0 * sin(55.1667°)
Using a calculator, we find that sin(55.1667°) is approximately 0.823. Now we can substitute this value:
l = 30.0 * 0.823
Calculating this expression, we find that the length of the guy wire is approximately 24.7 meters.
Finally, let's move on to the third problem.
To find the altitude of the rocket, we can again use trigonometry. We have a right triangle formed by the altitude of the rocket, the distance traveled by the rocket, and the angle of ascent (65°).
We are given that the rocket climbs through a distance of 20,000 feet and makes an angle of 65° with the ground. Let's denote the altitude of the rocket as a.
Using the trigonometric function cosine (cos), we can write:
cos(65°) = a / 20,000
Solving for a:
a = 20,000 * cos(65°)
Using a calculator, we find that cos(65°) is approximately 0.4226. Now we can substitute this value:
a = 20,000 * 0.4226
Calculating this expression, we find that the altitude of the rocket is approximately 8,452 feet (rounded to the nearest foot).
To find the height of the kite above ground level, we can use trigonometry.
Step 1: We need to find the length of the horizontal component of the string (x). To do this, we can use the sine function:
sin(60°) = x / 700 feet
x = 700 feet * sin(60°)
x ≈ 700 feet * 0.866
x ≈ 606.2 feet
Step 2: The height of the kite above ground level is the length of the vertical component of the string. To find this, we can use the cosine function:
cos(60°) = height / 700 feet
height = 700 feet * cos(60°)
height ≈ 700 feet * 0.5
height ≈ 350 feet
Therefore, the approximate height of the kite above ground level is 350 feet.
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To find the length of the guy wire, we can use trigonometry again.
Step 1: We need to find the length of the vertical component of the guy wire (y). To do this, we can use the sine function:
sin(55°10') = y / l
y = l * sin(55°10')
y ≈ l * 0.82462
Step 2: The length of the guy wire is the hypotenuse of a right triangle. We already have the vertical component (y) and the horizontal component (30.0 meters), so we can use the Pythagorean theorem to find the length (l):
l^2 = (30.0 meters)^2 + y^2
l^2 ≈ (30.0 meters)^2 + (l * 0.82462)^2
l^2 ≈ 900 + 0.6804584l^2
0.3195416l^2 ≈ 900
l^2 ≈ 900 / 0.3195416
l^2 ≈ 2816.1892
l ≈ sqrt(2816.1892)
l ≈ 53.0 meters
Therefore, the approximate length of the guy wire is 53.0 meters.
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To find the altitude of the rocket, we can use trigonometry once again.
Step 1: We need to find the length of the vertical component of the rocket's path. To do this, we can use the sine function:
sin(65°) = altitude / 20,000 feet
altitude = 20,000 feet * sin(65°)
altitude ≈ 20,000 feet * 0.90631
altitude ≈ 18,126 feet
Therefore, the approximate altitude of the rocket is 18,126 feet.