A kite flying in the air has a
94
-
ft
string attached to it, and the string is pulled taut. The angle of elevation of the kite is
61°
. Find the height of the kite. Round your answer to the nearest tenth.
To find the height of the kite, we can use trigonometric functions. In this case, we can use the sine function.
The sine function relates the side opposite the angle to the hypotenuse of a right triangle. In this case, the height of the kite is the side opposite the angle, and the hypotenuse is the string length.
So, we can write:
sin(61°) = opposite/hypotenuse
sin(61°) = height/94ft
To solve for the height, we can rearrange the equation:
height = sin(61°) * 94ft
Using a calculator, we evaluate sin(61°) to be approximately 0.8681.
height = 0.8681 * 94ft
height ≈ 81.7ft
Therefore, the height of the kite is approximately 81.7 feet.
Nope
I apologize for the error. Let's redo the calculation.
To find the height of the kite, we can use trigonometric functions. In this case, we can use the sine function.
Since the string is pulled taut and the angle of elevation of the kite is 61°, we can consider a right triangle where the vertical leg represents the height of the kite and the hypotenuse represents the string length.
So, we can write:
sin(61°) = height/94ft
To solve for the height, we can rearrange the equation:
height = sin(61°) * 94ft
Using a calculator, we evaluate sin(61°) to be approximately 0.8746.
height = 0.8746 * 94ft
height ≈ 82.2ft
Therefore, the height of the kite is approximately 82.2 feet, rounded to the nearest tenth.
A surveyor wants to know the length of a tunnel built through a mountain. According to his equipment, he is located
54
meters from one entrance of the tunnel, at an angle of
56°
to the perpendicular. Also according to his equipment, he is
31
meters from the other entrance of the tunnel, at an angle of
13°
to the perpendicular. Based on these measurements, find the length of the entire tunnel.
Do not round any intermediate computations. Round your answer to the nearest tenth.
Note that the figure below is not drawn to scale.
To find the length of the entire tunnel, we can use trigonometric functions and the Law of Sines. Let's label the unknown length of the tunnel as "x".
From one entrance of the tunnel, the surveyor is located 54 meters away at an angle of 56° to the perpendicular. This creates a right triangle where the opposite side is x and the adjacent side is 54 meters.
Using the tangent function, we can write:
tan(56°) = opposite/adjacent
tan(56°) = x/54m
Rearranging the equation, we have:
x = tan(56°) * 54m
Using a calculator, we evaluate tan(56°) to be approximately 1.4983.
x = 1.4983 * 54m
x ≈ 80.9m
So, the length of the tunnel from one end is approximately 80.9 meters.
From the other entrance of the tunnel, the surveyor is located 31 meters away at an angle of 13° to the perpendicular. Again, we can use the tangent function to find the length of the tunnel.
tan(13°) = opposite/adjacent
tan(13°) = x/31m
Rearranging the equation, we have:
x = tan(13°) * 31m
Using a calculator, we evaluate tan(13°) to be approximately 0.2376.
x = 0.2376 * 31m
x ≈ 7.4m
So, the length of the tunnel from the other end is approximately 7.4 meters.
To find the length of the entire tunnel, we add the lengths from both ends:
Total length = 80.9m + 7.4m
Total length ≈ 88.3m
Therefore, the length of the entire tunnel is approximately 88.3 meters, rounded to the nearest tenth.
WRONG
I apologize for the incorrect response. Let's try to solve the problem again.
From the given information, we have the following diagram:
A B
/| |\
/ | | \
/ | | \
/ | | \
-------------------------
54m 31m
Let's assume that the length of the tunnel is "x" meters.
Using the Law of Sines, we can set up the following equations:
sin(56°) = x / 54m (1)
sin(13°) = x / 31m (2)
To find "x", we can solve equations (1) and (2) simultaneously.
Rearranging equation (1), we have:
x = 54m * sin(56°)
Using a calculator, we evaluate sin(56°) to be approximately 0.8290.
x = 54m * 0.8290
x ≈ 44.7m
Similarly, rearranging equation (2), we have:
x = 31m * sin(13°)
Using a calculator, we evaluate sin(13°) to be approximately 0.2249.
x = 31m * 0.2249
x ≈ 6.97m
Now, we need to find the total length of the tunnel by adding the lengths from both ends:
Total length = 44.7m + 6.97m
Total length ≈ 51.67m
Therefore, the length of the entire tunnel is approximately 51.67 meters, rounded to the nearest tenth.
NOPE WRONG
Apologies for the confusion. Let's approach the problem again.
From the given information, we have the following diagram:
A B
/| |\
/ | | \
/ | | \
/ | | \
-------------------------
54m 31m
Let's assume that the length of the tunnel is "x" meters, and let's calculate the height of the tunnel's midpoint as "h" meters.
Using the tangent function, we can set up the following equations:
tan(56°) = h / (54m / 2) (1)
tan(13°) = h / (31m / 2) (2)
Simplifying equations (1) and (2), we have:
h = tan(56°) * (54m / 2)
h = tan(13°) * (31m / 2)
Using a calculator, we evaluate tan(56°) to be approximately 1.4983 and tan(13°) to be approximately 0.2249.
Plugging in these values, we have:
h = 1.4983 * (54m / 2)
h = 40.45m
h = 0.2249 * (31m / 2)
h = 3.48m
Now, we can calculate the length of the tunnel by adding the lengths from both ends:
Total length = 2 * h
Total length = 2 * (40.45m + 3.48m)
Total length = 2 * 43.93m
Total length = 87.86m
Therefore, the length of the entire tunnel is approximately 87.86 meters, rounded to the nearest tenth.