The angle of depression from a hot air ballon in the air to a person on the ground is 36 degrees. If the person steps back 10ft the new angle of depression is 25 degrees. If the person is 6ft tall, how far off the ground is the hot air ballon?
add six feet at the end
tan 25 = h/(x+10)
tan 36 = h/x
h = (x+10)(.466)
h = x(.727)
so
.727 x = .466 x + 4.66
.261 x = 4.66
x = 17.9 ft
so
h = .727 (17.9) = 13
13 + 6 = 19 ft
Well, well, let's solve this puzzle with a dash of humor, shall we?
First, let's set the stage. We have a person standing on the ground, stepping back 10ft, and looking up at a hot air balloon in the sky. Quite the sight, I must say!
Now, remember that an angle of depression simply means the angle between the person's line of sight and the horizontal. The larger the angle, the "deeper" the object appears to be in relation to the person.
We have two angles of depression here: 36 degrees originally and 25 degrees after stepping back. This person must have some top-notch vision to measure angles accurately while taking a step back!
Let's assume that when the person steps back, they are looking directly at the balloon. This forms a nice little right triangle with the person's height (6ft) as one side, the distance they stepped back (10ft) as the adjacent side, and the vertical distance from the balloon to the person as the opposite side.
Now let's use our trigonometry skills to find the height of the balloon.
In the original position, the tangent of 36 degrees is equal to the opposite side (height of balloon) divided by the adjacent side (distance between person and balloon).
So, tan(36) = height of balloon / distance to the balloon.
Now, in the new position, the tangent of 25 degrees is equal to the same height of the balloon, but with a new distance between the person and the balloon (distance to the balloon plus the 10ft they stepped back).
So, tan(25) = height of balloon / (distance to the balloon + 10ft).
Now if we equate these two expressions, we can solve for the height of the balloon:
(tan(36)) / (tan(25)) = (height of balloon) / (distance to the balloon + 10ft).
And if we substitute the values of the known angles into our equation, we can calculate the height of the balloon:
(tan(36)) / (tan(25)) = (height of balloon) / (distance to the balloon + 10ft).
(0.7265) / (0.4663) = (height of balloon) / (distance to the balloon + 10ft).
Now, to find the height of the balloon, we can rearrange the equation and solve for it:
(height of balloon) = (distance to the balloon + 10ft) * (0.7265 / 0.4663).
Calculating this delightful equation should give us the height of the balloon, which will tell us how far off the ground it is. Enjoy the math, my friend!
To solve this problem, we can use trigonometry. Let's assume the height of the hot air balloon is h feet.
Step 1: Set up a right triangle
Draw a right triangle, where the person's height is the vertical side, the distance from the person to the original position is the horizontal side, and the hypotenuse is the distance from the hot air balloon to the person.
Step 2: Label the known values
The angle of depression from the balloon to the person on the ground is 36 degrees, and when the person steps back, the new angle of depression is 25 degrees. The person's height is 6 feet, and the distance the person stepped back is 10 feet.
Step 3: Identify the trigonometric ratio
Since we have the angle of depression and the lengths of the sides, we can use the tangent ratio, which is defined as the ratio of the length of the opposite side to the length of the adjacent side.
Step 4: Set up the equation
Using the tangent ratio, we can set up the following equation:
tan(36) = 6/h (for the original position)
tan(25) = 6/(h+10) (for the new position)
Step 5: Solve the equation
Let's solve the first equation for h:
h = 6/tan(36)
Using a calculator, we can find:
h ≈ 6/0.7265 ≈ 8.26 feet
Now, let's solve the second equation for h:
h + 10 = 6/tan(25)
h = 6/tan(25) - 10
Using a calculator, we can find:
h ≈ 6/0.4663 - 10 ≈ 12.88 feet
Step 6: Determine the height of the hot air balloon
Since the height of the hot air balloon is the same in both cases, we can take the average of the two values we found:
(height from the original position + height from the new position)/2 = (8.26 + 12.88)/2 = 10.57 feet
Therefore, the hot air balloon is approximately 10.57 feet off the ground.
To find the height of the hot air balloon, we can use trigonometry. Let's denote the height of the hot air balloon as "h" and the distance between the person and the balloon as "x".
First, let's consider the situation where the angle of depression is 36 degrees. We can create a right triangle with the vertical height of the person being 6ft, the distance between the person and the balloon being "x", and the angle of depression being 36 degrees.
Using trigonometry, we know that:
tan(angle of depression) = vertical height / distance
So, for the first situation:
tan(36) = 6 / x
Now, let's consider the situation where the person steps back 10ft, and the new angle of depression is 25 degrees. In this case, the distance between the person and the balloon becomes "x + 10ft". We can create another right triangle with the vertical height of the person being 6ft, the distance between the person and the balloon being "x + 10ft", and the new angle of depression being 25 degrees.
Using trigonometry again, we have:
tan(25) = 6 / (x + 10)
Now we have a system of two equations:
1) tan(36) = 6 / x
2) tan(25) = 6 / (x + 10)
To find the value of "x", we can solve this system of equations. By isolating "x" in the first equation, we get:
x = 6 / tan(36)
Once we find the value of "x", we can plug it into the second equation to find the height of the hot air balloon.
Keep in mind that trigonometric functions work with radian values, so we need to convert the given angles from degrees to radians.
Now we can calculate the height of the hot air balloon.