From a weather balloon high in the air, the angle of depression of a ground marker is 27.8 degrees. If the balloon rises 345 m straight up the angle of depression from this higher point is 46.2 degrees. How high above the ground is the balloon?
@steve what how do you do that
You can just use the sine law.
To solve this problem, we can use trigonometry. Let's break it down step by step:
1. First, let's visualize the situation. Imagine a right triangle formed by the weather balloon, the ground marker, and a point directly below the balloon on the ground.
2. We are given that the angle of depression of the ground marker from the balloon is 27.8 degrees. This means that the angle between the line connecting the balloon and the ground marker and the horizontal line is 27.8 degrees.
3. Since we know one angle in a right triangle, we can use the concept of complementary angles. The complementary angle to 27.8 degrees is 90 degrees - 27.8 degrees = 62.2 degrees. Therefore, we have a right triangle with one angle of 62.2 degrees and one right angle.
4. After the balloon rises 345 m straight up, we now have a new angle of depression from this higher point, which is 46.2 degrees. This means that the angle between the line connecting the new higher point and the ground marker and the horizontal line is 46.2 degrees.
5. Again, we can use the concept of complementary angles. The complementary angle to 46.2 degrees is 90 degrees - 46.2 degrees = 43.8 degrees. So, we have a new right triangle with one angle of 43.8 degrees and one right angle.
6. Now, we can use trigonometric ratios to find the height of the balloon. In this case, we can use the tangent function, which relates the opposite side to the adjacent side in a right triangle.
Using the first right triangle, we have the equation:
tan(27.8 degrees) = opposite side (height of the balloon) / adjacent side (distance between the balloon and the ground marker)
Using the second right triangle, we have the equation:
tan(43.8 degrees) = opposite side (height of the balloon + 345 m) / adjacent side (distance between the balloon's higher point and the ground marker)
7. Now we can solve these equations simultaneously to find the height of the balloon.
Let's start with the first equation:
tan(27.8 degrees) = opposite side / adjacent side
tan(27.8 degrees) = height of the balloon / distance between the balloon and the ground marker
Rearranging the equation, we get:
height of the balloon = tan(27.8 degrees) * distance between the balloon and the ground marker
8. Now, let's focus on the second equation:
tan(43.8 degrees) = opposite side / adjacent side
tan(43.8 degrees) = (height of the balloon + 345 m) / distance between the balloon's higher point and the ground marker
Rearranging the equation, we get:
height of the balloon + 345 m = tan(43.8 degrees) * distance between the balloon's higher point and the ground marker
9. Now we can substitute the value we found in step 7 into step 8:
(tan(27.8 degrees) * distance between the balloon and the ground marker) + 345 m = tan(43.8 degrees) * distance between the balloon's higher point and the ground marker
10. Solving this equation will give us the height of the balloon above the ground.
If the marker is at distance x away from the spot just below the balloon, we have
h/x = tan 27.8
(h+345)/x = tan 46.2
Eliminate x to get
h/tan 27.8 = (h+345)/tan 46.2
Now just solve for h.