A weather balloon is sighted between points A and B which are 5 miles apart on level ground. The angle of elevation of the balloon from A is 37 degrees and it's angle of elevation from B is 58 degrees. Find the height, in feet, of the balloon above the ground.
tan 37 = h/x
tan 58 = h/(5-x)
x tan 37 = (5-x) tan 58
x (tan 37+tan 58) = 5 tan 58
x = 5 tan 58/ (tan 58 + tan 37)
then
h = x tan 37
ok.... so what is the answer?
To find the height of the balloon above the ground, we can use trigonometry. Let's call the height of the balloon h.
First, we need to determine the distances from points A and B to the balloon. We can use the tangent function, which relates the angle of elevation to the height and distance.
From point A, we have:
tan(37 degrees) = h / x1,
where x1 is the distance from point A to the balloon.
From point B, we have:
tan(58 degrees) = h / x2,
where x2 is the distance from point B to the balloon.
To solve for the distances x1 and x2, we can rearrange the equations as follows:
x1 = h / tan(37 degrees),
x2 = h / tan(58 degrees).
Now, we know that the total distance between points A and B is 5 miles. Therefore, the sum of the distances x1 and x2 should be equal to 5 miles.
x1 + x2 = 5.
Substituting the expressions for x1 and x2 in terms of h:
(h / tan(37 degrees)) + (h / tan(58 degrees)) = 5.
Now, we can solve this equation for h.
1. Calculate the values of tan(37 degrees) and tan(58 degrees) using a calculator or relevant tables.
2. Substitute these values into the equation:
(h / tan(37 degrees)) + (h / tan(58 degrees)) = 5.
3. Solve the equation for h. Multiply through by the product of the denominators:
h * (tan(58 degrees) + tan(37 degrees)) = 5 * tan(37 degrees) * tan(58 degrees).
4. Divide both sides by (tan(58 degrees) + tan(37 degrees)) to solve for h:
h = (5 * tan(37 degrees) * tan(58 degrees)) / (tan(58 degrees) + tan(37 degrees)).
5. Calculate the value of h using the equation from step 4.
The result will give you the height of the balloon above the ground in feet.