A rocket shoots straight up from the launch-pad. Five seconds after lift-off, an observer 2 miles away notes that the rocket's angle of elevation is 3.5 degrees. Four seconds later the angle of elevation is 41 degrees. How far did the rocket rise during those 4 seconds.
Well, if the rocket was shooting straight up, it must have been a real blast! Now, let's calculate that rise.
From the given information, we know that the rocket's angle of elevation changed from 3.5 degrees to 41 degrees in 4 seconds. So, we can assume the rise in those 4 seconds was as swift as a cheetah chasing down its prey. It's time for some math circus!
If we consider the initial angle of elevation of 3.5 degrees, we can set up a right triangle. The height of the rocket's rise will be the opposite side, and the distance from the observer will be the adjacent side.
Using some trigonometry, we can say that:
tan(3.5 degrees) = height / 2 miles
But let's convert miles to feet for convenience, because let's face it, feet are more ticklish!
So, 2 miles = 2 * 5280 feet = 10560 feet (approximately)
Now we can rewrite our equation:
tan(3.5 degrees) = height / 10560 feet
Solving for height:
height = tan(3.5 degrees) * 10560 feet
Now this is where the math clown jumps in. Using your favorite calculator, you'll find that height ≈ 647 feet (approximately). That's the initial height after 5 seconds!
Now, we can do more circus math with the final angle of elevation of 41 degrees. We'll use the same right triangle concept and set up this equation:
tan(41 degrees) = (height + x) / 10560 feet
Solving for height + x (the final height after 4 more seconds):
height + x = tan(41 degrees) * 10560 feet
Plugging in the values and solving for height + x again using a calculator, we get:
height + x ≈ 12217 feet (approximately)
Now, to find the rise during those 4 seconds, we just subtract the initial height (647 feet) from the final height (12217 feet):
Rise = final height - initial height
= 12217 feet - 647 feet
≈ 11570 feet
So, during those 4 seconds, the rocket rose approximately 11570 feet. That's one impressive ascent!
To find out how far the rocket rose during those 4 seconds, we can use trigonometry and the given angles of elevation.
Let's denote the distance the rocket rose during those 4 seconds as "x" (in miles).
First, let's find the height of the rocket at the second observation. We can use the tangent function, as tan(angle of elevation) = opposite/adjacent.
Therefore, tan(41 degrees) = x/2 miles (since the observer is 2 miles away from the launch pad).
Using a calculator, we can find that tan(41 degrees) ≈ 0.8693. Therefore, x/2 ≈ 0.8693.
To find the value of x, we can multiply both sides of the equation by 2:
x ≈ 2 * 0.8693
x ≈ 1.7386 miles
So, the rocket rose approximately 1.7386 miles during those 4 seconds.
To find the distance the rocket rose during those 4 seconds, we need to use trigonometry and the information provided.
Let's break down the problem:
1. The observer is located 2 miles away from the launchpad, and the observer is observing the rocket.
2. Five seconds after lift-off, the observer notes that the rocket's angle of elevation is 3.5 degrees.
3. Four seconds later, the angle of elevation is 41 degrees.
Using trigonometry, we can set up a right triangle to represent the situation:
- Let "x" represent the distance the rocket rose during those 4 seconds.
- The adjacent side of the triangle is the distance between the observer and the launchpad, which is 2 miles.
- The angle of elevation of 3.5 degrees is opposite to the side "x."
- The angle of elevation of 41 degrees is opposite to the side "x + 2."
Now, we can use the tangent function to find the value of "x":
tan(3.5 degrees) = x / 2 miles -- (1)
tan(41 degrees) = (x + 2) / 2 miles -- (2)
To isolate "x," we can solve equation (1) for "x" and substitute it into equation (2). Here's how:
1. Calculate "x" using equation (1):
x = 2 miles * tan(3.5 degrees)
2. Substitute the value of "x" from step 1 into equation (2):
tan(41 degrees) = (2 miles * tan(3.5 degrees) + 2) / 2 miles
3. Solve for (2 miles * tan(3.5 degrees)):
(2 miles * tan(3.5 degrees)) = (2 miles * tan(3.5 degrees))
4. Solve for "x":
x = (2 miles * tan(41 degrees)) - 2 miles * tan(3.5 degrees)
By calculating the expression above, you can find the distance the rocket rose during those 4 seconds.