A kite has a string of length 60 metres. On a windy day all the string is let out and makes an angle of between 20 and 36 with the ground. Find the minimum and maximum heights of the kite
h/60 = sinθ, so
h = 60 sinθ
60 sin20 <= h <= 60 sin36
now just plug and chug
20,4 35,4
Well, someone's taking kite flying to new heights! Let's determine the minimum and maximum heights of the kite, shall we?
To do so, we need to find the lengths of the sides of a right triangle formed by the string and the height of the kite.
Let's call the height h and the angle θ. Now, using a little trigonometry magic, we can say that:
sin(θ) = h / 60
To find the minimum height, we'll use the smallest angle, which is 20 degrees:
sin(20) = h / 60
h = 60 * sin(20)
Now, to find the maximum height, we'll use the largest angle, which is 36 degrees:
sin(36) = h / 60
h = 60 * sin(36)
Plug those values into your calculator and viola! You'll have the minimum and maximum heights of your splendidly flying kite. Happy kite adventures!
To find the minimum and maximum heights of the kite, we can use trigonometry.
Let's start by drawing a right-angled triangle representing the situation. The string of the kite forms the hypotenuse of the triangle, and the vertical component represents the height of the kite.
Let's denote the angle between the string and the ground as θ, the length of the vertical component (height) as h, and the length of the horizontal component as x.
Since we know the length of the string is 60 meters, we can use trigonometry to relate the height to the angle θ:
sin(θ) = h / 60
To find the maximum and minimum heights, we need to find the maximum and minimum values for θ between 20° and 36°.
First, let's calculate the maximum height:
For the maximum height, we want the maximum value for sin(θ). So, we need to find the angle that gives the maximum sine value within the given range (20° to 36°). The angle that gives the maximum sine value is 90°. However, since we know that the angle is between 20° and 36°, the maximum value for sin(θ) within this range is 1.
So, for the maximum height:
sin(θ_max) = 1
h_max / 60 = 1
h_max = 60 meters
Therefore, the maximum height of the kite is 60 meters.
Now, let's calculate the minimum height:
For the minimum height, we want the minimum value for sin(θ). So, we need to find the angle that gives the minimum sine value within the given range (20° to 36°). The angle that gives the minimum sine value is 0°.
So, for the minimum height:
sin(θ_min) = 0
h_min / 60 = 0
h_min = 0 meters
Therefore, the minimum height of the kite is 0 meters.
In conclusion, the minimum height of the kite is 0 meters, and the maximum height is 60 meters.
To find the minimum and maximum heights of the kite, we can make use of basic trigonometry.
Let's start by drawing a diagram to represent the situation. Assume the string of length 60 meters is tied to the ground at point A, and the kite is at point K, forming an angle θ with the ground.
A
|\
|θ\
| \
| \
K
We can see that the height of the kite is represented by side AK, which can be divided into two components: the vertical component (h) and the horizontal component (x).
Using trigonometry, we can relate the angles and sides as follows:
1. sine(θ) = h / 60
2. cosine(θ) = x / 60
To find the minimum and maximum heights, we need to consider the range of possible angles θ mentioned in the question (between 20 and 36 degrees).
Let's calculate the heights for both the minimum (20 degrees) and maximum (36 degrees) angles:
For the minimum angle (20 degrees):
1. sine(20) = h / 60
h = 60 * sine(20)
h ≈ 20.392 meters
For the maximum angle (36 degrees):
1. sine(36) = h / 60
h = 60 * sine(36)
h ≈ 35.074 meters
Therefore, the minimum height of the kite is approximately 20.392 meters, and the maximum height is approximately 35.074 meters.