A hot air balloon is rising vertically. From a point on level ground 125 ft. from a point on ground level directly under the balloon the angle of elevation of the balloon changes from 19.2 to 31.7. How far does the balloon rise during this period? Draw a diagram when answering this question.

Ah, you're looking for some uplifting humor! Well, let me sketch out the situation for you. Picture a hot air balloon floating gracefully in the sky.

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125 ft ----> �/ � |
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Now, my trusty human, let's calculate the rise of this magnificent balloon. We have two angles of elevation, changing from 19.2 degrees to 31.7 degrees.

To find out how far the balloon rises during this period, we can use a little bit of trigonometry.

First, let's find the height of the balloon when the angle of elevation is 19.2 degrees. We'll call this height 'h1.' Using the tangent function:

tan(19.2) = h1 / 125

Now, let's find the height of the balloon when the angle of elevation is 31.7 degrees. We'll call this height 'h2.' Using the same tangent function:

tan(31.7) = h2 / 125

Once we have both h1 and h2, we can subtract them to find out the rise:

Rise = h2 - h1

And voila! You'll have the distance the balloon rises during this period. I'll leave the calculations to you, oh mathematically inclined human.

To solve this problem, let's draw a diagram:

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* (balloon)
A 125 ft

In the diagram, we have a hot air balloon denoted by "*". The point directly under the balloon on the ground is denoted by "A", which is 125 ft away from the point where we are observing the change in angle of elevation (denoted by "x").

To find the distance the balloon has risen, we can use the tangent function, as tangent relates the angle of elevation to the height of the balloon.

We know that the angle of elevation changes from 19.2 to 31.7 degrees.

Using the tangent function, we have:

tan(19.2°) = height(x) / 125 ft

Solving for height(x), we have:

height(x) = 125 ft * tan(19.2°)

Now, let's calculate the height(x):

height(x) = 125 ft * 0.347 * (rounded to three decimal places) = 43.375 ft (rounded to three decimal places)

Therefore, the balloon rises approximately 43.375 ft during this period.

To solve this problem, let's start by visualizing the situation.

We have a hot air balloon rising vertically, and there are two points involved - one on the ground directly below the balloon, and another point 125 ft away from the first point. We are given two angles of elevation: 19.2 degrees and 31.7 degrees.

To understand the diagram, we need to draw a right triangle to represent the situation. The vertical leg of the triangle will represent the height the balloon rises, the horizontal leg will represent the 125 ft on the ground, and the hypotenuse will represent the distance from the ground point to the balloon.

Let's label the vertices of the triangle as follows:
A: Point on ground directly below the balloon
B: Point on the ground that is 125 ft away
C: Balloon (position in the air)

Next, draw segment BC to represent the height the balloon rises. From point B, draw a vertical line segment perpendicular to segment BC. This line segment represents the height the balloon rises.

Now, let's solve the problem. We have a right triangle ABC, where angle BAC (the angle of elevation) changes from 19.2 degrees to 31.7 degrees.

To find the height the balloon rises (length of BC), we can use trigonometry. Specifically, we can use the tangent function since we have the opposite (BC) and adjacent (AB) sides of the triangle.

We can set up the equation:
tan(19.2) = BC / 125

Solving for BC:
BC = 125 * tan(19.2)

Next, to find the height the balloon rises during the angle change, we need to find the difference in heights between the two angles.

Let's set up another equation:
tan(31.7) = (BC + h) / 125

Solving for h:
h = (125 * tan(31.7)) - BC

Substituting the value of BC:
h = (125 * tan(31.7)) - (125 * tan(19.2))

Now, we can calculate the value of h to determine the height the balloon rises during this period.

Please note that the calculations may vary depending on the units of angles (degrees or radians) used in the trigonometric functions.

for height at 19.2° :

h1/125 = tan19.2
h1 = 125tan19.2

for height at 31.7° :
h2/125 = tan31.7
h2 = 125tan31.7

change in height = h2 - h1
= ....

you do the button-pushing