A plane flies towards a ground-based radar dish. Radar locates the plane at a distance D = 41 km from the dish, at an angle of 52 degrees above horizontal. What is the plane's horizontal distance, from the radar dish.
Well, since the plane is flying towards the ground-based radar dish, it's probably trying to get a closer look at what's on TV. But I'm just a clown bot, so I'll do my best to help with some math instead of cracking jokes.
To find the plane's horizontal distance from the radar dish, we can use some trigonometry. Remember that the angle of 52 degrees above horizontal is measured from the ground to the plane.
We can use the sine of the angle to find the vertical distance, and then use cosine of the angle to find the horizontal distance.
Let's call the horizontal distance from the radar dish "x". We can set up a trigonometric equation:
sin(52 degrees) = vertical distance / 41 km
cos(52 degrees) = x / 41 km
To find the plane's horizontal distance, we can rearrange the equation:
x = cos(52 degrees) * 41 km
Using a calculator, we can find that cos(52 degrees) is approximately 0.6157. So, the plane's horizontal distance is:
x = 0.6157 * 41 km
x ≈ 25.24037 km
So, the plane's horizontal distance from the radar dish is approximately 25.24 kilometers.
To find the plane's horizontal distance from the radar dish, we can use trigonometric relationships.
First, we need to determine the vertical distance between the plane and the radar dish. We can use the sine function, which relates the opposite side to the hypotenuse of a right triangle. In this case, the opposite side is the vertical distance, and the hypotenuse is the total distance D.
sin(θ) = opposite / hypotenuse
sin(52°) = vertical distance / 41 km
To isolate the vertical distance, we can rearrange the equation:
vertical distance = sin(52°) * 41 km
Next, we can calculate the horizontal distance using the cosine function. The cosine function relates the adjacent side to the hypotenuse of a right triangle. In this case, the adjacent side is the horizontal distance, and the hypotenuse is still the total distance D.
cos(θ) = adjacent / hypotenuse
cos(52°) = horizontal distance / 41 km
Again, let's isolate the horizontal distance:
horizontal distance = cos(52°) * 41 km
Using a calculator or trigonometric table, we can find the value of sin(52°) and cos(52°) to substitute into the equations:
sin(52°) ≈ 0.788
cos(52°) ≈ 0.616
Now, we can calculate the vertical and horizontal distances:
vertical distance = 0.788 * 41 km ≈ 32.308 km
horizontal distance = 0.616 * 41 km ≈ 25.256 km
Therefore, the plane's horizontal distance from the radar dish is approximately 25.256 km.
To find the plane's horizontal distance from the radar dish, we can use trigonometry. Here's how you can solve it step by step:
1. Start by visualizing the problem. The radar dish acts as the origin, and the plane's position is given as a distance of 41 km and an angle of 52 degrees above horizontal.
2. Identify the angle given as the angle of elevation (θ).
3. Split the problem into two components: the horizontal (x) and vertical (y) distances.
4. Use the trigonometric function sine (sin) to find the vertical distance (y). In this case, sin(θ) = opposite/hypotenuse, which means sin(52°) = y/41 km.
Rearranging the equation gives us: y = 41 km * sin(52°).
5. Calculate the vertical distance (y) by plugging in the values: y = 41 km * sin(52°) ≈ 31.62 km.
6. Use the trigonometric function cosine (cos) to find the horizontal distance (x). In this case, cos(θ) = adjacent/hypotenuse, which means cos(52°) = x/41 km.
Rearranging the equation gives us: x = 41 km * cos(52°).
7. Calculate the horizontal distance (x) by plugging in the values: x = 41 km * cos(52°) ≈ 25.03 km.
So, the plane's horizontal distance from the radar dish is approximately 25.03 km.