bystander at the Pegasus Parade, is lying on his back and observing a large balloon of a cartoon character, floating directly above Broadway Street. He is located 32 feet from a point on the street directly beneath the balloon. To see the top of the balloon, he looks up at an angle of 56°. To see the bottom of the balloon, he looks up at an angle of 46°. To the nearest tenth, how tall, in feet, is the balloon?
any answers on how to set this up
Draw a diagram. If the bottom of the balloon is at height x and the top of the balloon is at height y, then
x/32 = tan46°
y/32 = tan56°
The height of the balloon is just y-x, or
32tan56° - 32tan46° = 32(tan56°-tan46°)
To solve this problem, we can use trigonometry and set up two right triangles. Let's label the required height of the balloon as 'h.'
In the first triangle, the observer's line of sight to the top of the balloon forms an angle of 56° with the ground. The length of the adjacent side in this triangle is 32 feet, representing the distance from the observer to the point beneath the balloon.
In the second triangle, the observer's line of sight to the bottom of the balloon forms an angle of 46° with the ground. Similar to the first triangle, the adjacent side is also 32 feet.
We can now apply the tangent function to these triangles:
In the first triangle:
tan(56°) = h/32
In the second triangle:
tan(46°) = h/32
Now, we can solve these equations simultaneously to find the value of 'h.'
First, rearrange the equation for the first triangle to solve for 'h':
h = 32 * tan(56°)
h ≈ 49.8 feet
Next, rearrange the equation for the second triangle to solve for 'h':
h = 32 * tan(46°)
h ≈ 38.7 feet
Since the balloon's height is the sum of these two values:
Balloon height ≈ 49.8 + 38.7 ≈ 88.5 feet
Therefore, the balloon is approximately 88.5 feet tall.