A 40 foot long escalator rises from the first floor to the second of a shopping mall. The escalator makes a 30 degree angle with the horizontal. How high above the first floor is the second floor?
Is it 20 feet
sin 30° = height/40
height = 40sin30
= ...
We never learned sin We use a different formula. I am using 30 60 90 triangle formula
Well, let's see. With a 30 degree angle, it seems like the escalator is trying to show off its flexibility. But let's not be envious of an inanimate object. So, to find out how high the second floor is above the first, we can use a little trigonometry.
The escalator's rise can be determined by finding the height of a right triangle where the hypotenuse is 40 feet (the length of the escalator) and the angle opposite the height is 30 degrees.
Using a bit of math magic, we can say that the height of the second floor is approximately 20.67 feet. So, make sure you take some extra steps when ascending that escalator. It's got some spring in its step!
To find the height of the second floor, we can break down the problem into two components: the vertical height and the horizontal distance.
First, let's calculate the vertical height. We can use trigonometry, specifically the sine function, to determine the height.
In this case, the angle between the escalator and the horizontal floor is 30 degrees. We know that the length of the escalator is 40 feet.
Using the formula: vertical height = length of the escalator * sine(angle)
vertical height = 40 feet * sine(30 degrees)
To calculate the sine of 30 degrees, we can use a scientific calculator or reference table. The sine of 30 degrees is 0.5.
vertical height = 40 feet * 0.5
vertical height = 20 feet
Therefore, the second floor is 20 feet above the first floor.
Now, let's calculate the horizontal distance.
Using the same 30-degree angle, we can use the cosine function to determine the horizontal distance.
Using the formula: horizontal distance = length of the escalator * cosine(angle)
horizontal distance = 40 feet * cosine(30 degrees)
To calculate the cosine of 30 degrees, we can use a scientific calculator or reference table. The cosine of 30 degrees is approximately 0.866.
horizontal distance = 40 feet * 0.866
horizontal distance ≈ 34.64 feet
Therefore, the second floor is approximately 20 feet above the first floor, and the horizontal distance between the two floors is approximately 34.64 feet.