You are flying a kite while standing 60 ft from a fence. If the kite string makes a 40 degree angle with the ground, how much string, in feet, is let out when the kite is directly above the fence?
Make a diagram of the right-angled triangle to see that
tan 40° =height/60
height = 60sin40°
= ...
To find out how much string is let out when the kite is directly above the fence, we can use trigonometry.
Let's start by drawing a diagram to better understand the situation.
```
.
\
\ |\
\ | \
\ | \
\ | \
\ | \
--------------+-----\
60 ft |
|
```
In the diagram above, the line represents the ground, and the diagonal line with the angle marked represents the kite string. The fence is located where the diagonal line intersects the ground.
To solve the problem, we need to find the length of the diagonal line (the kite string) when the point directly above the fence is reached.
Using trigonometry, we can determine the length of the kite string by using the sine function. The sine of an angle is the ratio of the opposite side to the hypotenuse.
In this case, the opposite side is the height of the kite above the ground, and the hypotenuse is the length of the kite string.
Let's calculate the height of the kite above the ground using the sine function:
sin(40°) = height / 60 ft
To isolate the height, we multiply both sides of the equation by 60 ft:
height = 60 ft * sin(40°)
Using a scientific calculator, we can find that sin(40°) is approximately 0.6428.
height ≈ 60 ft * 0.6428
height ≈ 38.57 ft
So, the height of the kite above the ground when it is directly above the fence is approximately 38.57 feet.
Finally, to find the length of the kite string, we can use the Pythagorean theorem. The hypotenuse (kite string) is the square root of the sum of the squares of the two shorter sides (60 ft and the height):
kite string = √(60 ft)^2 + (38.57 ft)^2
Using a calculator, we can solve for the kite string:
kite string ≈ √(3600 ft^2 + 1488.6449 ft^2)
kite string ≈ √(5088.6449 ft^2)
kite string ≈ 71.40 ft
Therefore, when the kite is directly above the fence, approximately 71.40 feet of string is let out.