Angel of Elevation: You are skiing down a mountain with a vertical height of 1500 feet. The distance from the top of the mountain to the base is 3000 feet. What is the angle of elevation from the base to the top of the mountain? Traveling at 25 miles per hour, how long does it take you to get to the base of the mountain(in seconds)? [HINT: Time=distance/speed]
I assume the 3000 feet is the distance from the top of the mountain down the slope. So the angle sin x = 1500/3000 = 0.5 so the angle is 30 degrees.
For the time. convert 25 miles/hour to feet/sec. I get 25 mph x (5,280 ft/mi) x (1 hr/60 min) x (1 min/60 sec) = 36.67 feet/sec.
Then distance = rate x time
3000 = 36.67 x time
time = ? in seconds.
To find the angle of elevation from the base to the top of the mountain, we can use the trigonometric concept of "tangent."
First, let's define the angle of elevation as the angle formed between the horizontal ground and the line of sight from the base to the top of the mountain. In this case, it is the angle between the horizontal ground and the line connecting the base and the top of the mountain.
Using the given information, we have a vertical height of 1500 feet and a distance from the top to the base of 3000 feet. The tangent of an angle is defined as the ratio of the length of the opposite side (the vertical height) to the length of the adjacent side (the distance from the top to the base).
So, the tangent of the angle of elevation can be calculated as:
tangent(angle of elevation) = (vertical height) / (distance from top to base)
tangent(angle of elevation) = 1500 / 3000
tangent(angle of elevation) = 0.5
To find the angle of elevation, we need to find the inverse tangent (also known as arctangent or tan^(-1)) of 0.5:
angle of elevation = inverse tangent(0.5)
angle of elevation ≈ 26.57 degrees
So, the angle of elevation from the base to the top of the mountain is approximately 26.57 degrees.
Now let's calculate the time it takes to get to the base of the mountain, given a speed of 25 miles per hour.
Note that the given speed is in miles per hour, but the distance is given in feet. So, we need to convert the speed from miles per hour to feet per second.
There are 5280 feet in a mile, and there are 60 minutes in an hour. Therefore, there are 5280 * 60 = 316800 feet in an hour. And since there are 60 seconds in a minute, there are 316800 / 60 = 5280 feet in a second.
So, 25 miles per hour is equal to 25 * 5280 / 60 = 2200 feet per second.
Using the formula: time = distance / speed, we can calculate the time it takes to travel from the top of the mountain to the base.
time = 3000 feet / 2200 feet per second
time ≈ 1.3636 seconds
Therefore, it takes approximately 1.36 seconds to reach the base of the mountain while traveling at a speed of 25 miles per hour.
To find the angle of elevation from the base to the top of the mountain, you can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the vertical height of the mountain, which is 1500 feet, and the adjacent side is the distance from the base to the top, which is 3000 feet.
So, the tangent of the angle of elevation is given by:
tan(angle) = opposite/adjacent
tan(angle) = 1500/3000
tan(angle) = 0.5
To find the angle, you can take the inverse tangent (also known as arctan) of both sides:
angle = arctan(0.5)
Using a scientific calculator or online calculator, you can find that the angle is approximately 26.57 degrees.
Now, to find the time it takes to get to the base of the mountain, you can use the formula:
Time = Distance / Speed
The distance from the top of the mountain to the base is given as 3000 feet. Since the question doesn't mention any other unit for speed, we'll assume it's miles per hour.
Therefore, the speed is 25 miles per hour.
To convert miles per hour to feet per hour, multiply by 5280 to get the equivalent in feet per hour:
25 miles/hour * 5280 feet/mile = 132,000 feet/hour
Now, we can calculate the time it takes to travel 3000 feet at a speed of 132,000 feet per hour:
Time = 3000 feet / 132,000 feet per hour
Time = 0.0227 hours
To convert hours to seconds, multiply by 60 minutes per hour and 60 seconds per minute:
Time = 0.0227 hours * 60 minutes/hour * 60 seconds/minute
Time = 81.72 seconds
Therefore, it would take approximately 81.72 seconds to ski down to the base of the mountain.