In a mall, a shopper rides up an escalator between floors. at the top of the escalator, the shopper turns right and walks 9.00m to store. the magnitude of the shopper's displacement from the bottom of the escalator is 16.0m. the vertical distance between the floors is 6.00m. at what angle is the escalator inclined above the horizontal?
i just can't get the right answer for this problem, can anyone please give me some hints? THX A LOT!
how do you know 16m,9m, and E forms a right triangle?
Lets take the 9 m out of the picture.
16^2=9^2 + E^2
so solve for the Escalator length, E.
Now, having E,
E^2=6^2+ H^2 where H is the horiontal displcement. Actually, looking at the equations, there is no reason to solve for E
16^2=9^2+6^2 + H^2
solve for H.
angle= arctan 6/H
I get close to 27 degrees.
I get 22.03 degrees
Well, well, well, let me give you a hint with a twist of humor! Think of the escalator as a unicycle-riding circus clown named Bob. Bob wants to show off his acrobatic skills by riding up a slope, but he can't just go straight up like a rocket, oh no! He needs to tilt his unicycle at a certain angle to climb the slope gracefully. Now, imagine that Bob's unicycle is the escalator, and you're trying to figure out that magical angle at which it's inclined above the horizontal. Are you picturing it? Great! Now, think about how you can use some Trigonometry to help you calculate that angle. Specifically, you'll want to use the concept of right triangles and the relationship between the opposite and adjacent sides. And remember, Bob the Clown is counting on you to crack this case and make his circus act a roaring success! Good luck!
To solve this problem, we need to break it down into different components and apply trigonometry. Here are the steps to find the angle:
1. Start by drawing a diagram to visualize the situation. Draw a vertical line for the escalator and label it as the height (h) of 6.00m. Draw a horizontal line at the top of the escalator to represent the shopper's displacement of 16.0m.
2. Now, draw the displacement vector of the shopper by connecting the starting point at the bottom of the escalator to the store, 9.00m away. This vector represents the resultant of the individual horizontal and vertical vectors.
3. The horizontal component of the displacement vector represents the distance the shopper would have traveled if the escalator was not there. Since the shopper walks 9.00m to the store, this horizontal component should also be 9.00m.
4. We need to find the vertical component of the displacement vector. This component is the difference between the height of the escalator (6.00m) and the vertical distance traveled by the shopper on the escalator. Let's call this distance "d."
5. To find "d," we use the Pythagorean theorem. The magnitude of the displacement vector (16.0m) is equal to the square root of the sum of the squares of the horizontal (9.00m) and vertical (d) components. This equation can be written as:
16.0m = square root(9.00m^2 + d^2)
6. Solve the equation for "d." Square both sides of the equation to eliminate the square root:
(16.0m)^2 = 9.00m^2 + d^2
256.0m^2 - 81.0m^2 = d^2
175.0m^2 = d^2
d = √175.0m^2
7. Now that we know the value of "d" is approximately 13.23m, we can use trigonometry to find the angle. The angle θ can be calculated using the tangent function:
tan(θ) = opposite / adjacent
In this case, the opposite side is the vertical component (d = 13.23m) and the adjacent side is the horizontal component (9.00m).
θ = tan^(-1) (13.23m / 9.00m)
8. Use a scientific calculator to find the inverse tangent of (13.23 / 9.00) or type "tan^(-1)(13.23 / 9.00)" into a search engine to get the angle in degrees or radians.
The resulting angle should be approximately 54.8° or 0.96 radians.
Therefore, the escalator is inclined at an angle of approximately 54.8° above the horizontal.