The two hot air balloons in the drawing are 48.2 and h = 63.0 m above the ground. A person in the left balloon observes that the right balloon is è = 11.3° above the horzontal. What is the horizontal distance x between the two balloons?
tan 11.3° = (63.0-48.2)/x
x = 74.1 m
To find the horizontal distance (x) between the two balloons, we can use trigonometry. Let's break down the problem and solve it step by step:
Step 1: Understand the problem and draw a diagram.
We have two hot air balloons, one on the left and one on the right. The left balloon has a height (h) of 48.2 meters above the ground. The right balloon is observed to be at an angle (è) of 11.3° above the horizontal line.
We need to find the horizontal distance (x) between the two balloons.
Let's draw a diagram to visualize the situation. Draw a horizontal line at the bottom to represent the ground. Mark a point on the left side for the left balloon and another point on the right side for the right balloon. Connect these two points with a straight line to represent the horizontal distance between the balloons. Label the height of the left balloon as 48.2 m and the angle above the horizontal line as 11.3°.
Step 2: Identify the right triangle and apply trigonometry.
Looking at the diagram, we can identify a right triangle formed by the left balloon, the horizontal distance (x), and a vertical segment connecting the right balloon to the ground.
Here, the opposite side is the vertical segment (h), adjacent side is the horizontal distance (x), and the angle (è) is the angle between the horizontal line and the line connecting the two balloons.
We can use the tangent function to relate these sides and calculate the value of x.
Step 3: Apply the tangent function.
The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, we have:
tan(è) = h / x
To find the value of x, we rearrange the equation to isolate x:
x = h / tan(è)
Plugging in the given values:
x = 48.2 m / tan(11.3°)
Step 4: Calculate the value of x.
Using a calculator, we can find the tangent of 11.3° and calculate the value of x:
x = 48.2 m / tan(11.3°)
x ≈ 48.2 m / 0.1971
x ≈ 244.3 m
Therefore, the horizontal distance (x) between the two balloons is approximately 244.3 meters.