Twenty minutes after being launched, a hot-air balloon has risen to an altitude of 300 m. The pilot can still see the starting point on the ground at a 25° angle of depression. How many meters is the balloon from the starting point?

Angle of depression is the angle below the horizontal that an observer must look in order to see an object below him. If you draw the figure, you can actually see that the hot-air balloon is located 25° above the horizontal.

We actually formed a triangle here, with height equal to 300, unknown base and 25° angle between the hypotenuse and base. What we need to find here is the length of hypotenuse (which is also the distance from the starting point to the balloon). Let x = hypotenuse. We shall use sine:
sin (angle) = opposite/hypotenuse
sin 25° = 300 / x
x = 300 / sin 25°
x = 709.9 m

Hope this helps~ :3

To solve this problem, we can use trigonometry and the concept of tangent.

The angle of depression is the angle between the line of sight to the object and the horizontal line. In this case, the angle of depression is 25°.

Let's use the tangent function to find the distance of the balloon from the starting point.

tangent(angle) = opposite / adjacent

We know the opposite side, which is the altitude of the balloon (300 m). We can find the adjacent side, which is the distance of the balloon from the starting point.

tangent(25°) = 300 m / adjacent

To find the adjacent side:

adjacent = 300 m / tangent(25°)

Using a calculator, the tangent of 25° is approximately 0.4663.

adjacent = 300 m / 0.4663 ≈ 643.01 m

Therefore, the balloon is approximately 643.01 meters away from the starting point.

To find the distance of the balloon from the starting point, we can use trigonometry. Let's break down the given information:

1. The altitude of the balloon is 300 m.
2. The pilot can see the starting point at a 25° angle of depression.

An angle of depression is the angle formed between a line of sight from an observer downward to an object and a horizontal line. In this case, the starting point on the ground is the object being observed, and the line of sight is the line connecting the starting point to the balloon.

To find the distance to the starting point, we need to find the length of the line connecting the starting point to the balloon. We can use trigonometry, specifically the tangent function, to solve this.

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is the altitude (300 m), and the adjacent side is the distance we want to find.

Using the tangent function, we can set up the following equation:

tan(25°) = opposite side / adjacent side

Rearranging the equation to solve for the adjacent side (distance), we get:

adjacent side = opposite side / tan(25°)

Plugging in the values we know, we get:

adjacent side = 300 m / tan(25°)

Now, we can use a scientific calculator to find the tangent of 25° (approximately 0.4663) and substitute it into the equation:

adjacent side = 300 m / 0.4663

Calculating this, we find:

adjacent side ≈ 643.36 m

Therefore, the balloon is approximately 643.36 meters from the starting point.