Ferris Wheel Height Change Calculation Explained
Hey guys! Today, we're diving into a super cool math problem involving a Ferris wheel. Imagine you're on this giant wheel, and you start at a point level with the center. The wheel spins, and you move to a new height. The challenge? Figuring out exactly how much your height changes. This isn't just a fun thought experiment; it's a practical application of trigonometry and geometry that you might encounter in real-world scenarios. We'll break down the problem step by step, so by the end, you'll be a Ferris wheel height change master!
Let's get the ball rolling with the problem statement. We're told that a seat on a Ferris wheel starts level with the center of the wheel. This is our starting point, our baseline. The wheel itself has a whopping diameter of 210 feet. That's huge! Now, here's the twist: the wheel rotates 45 degrees counterclockwise and then stops. Our mission, should we choose to accept it (and we do!), is to calculate how much the height of the seat changes due to this rotation. To tackle this, we'll need to dust off some of our geometry and trigonometry skills, particularly how angles and circles interact, and how we can use trigonometric functions to find distances.
To make this problem less intimidating, let's break it down into smaller, more manageable parts. First, we need to visualize the Ferris wheel. Think of it as a big circle. The diameter, which is 210 feet, is the distance across the circle through the center. This means the radius (the distance from the center to any point on the circle) is half of that, which is 105 feet. Got it? Great! Now, imagine the seat's starting position, level with the center. This means it's at the same height as the center. When the wheel rotates 45 degrees, the seat moves along the circumference of the circle. The crucial part is to figure out the new vertical distance of the seat from the center. This change in vertical distance will tell us how much the height of the seat has changed. We'll be using trigonometric functions, specifically the sine function, to calculate this vertical distance. Remember, sine relates an angle to the opposite side in a right triangle, and that's exactly the relationship we need here!
Now for the fun part: applying trigonometry! Remember that the height change we're looking for is the vertical distance the seat moves from its starting position. If we draw a line from the new seat position straight down to the horizontal line that represents the center of the wheel, we form a right triangle. The hypotenuse of this triangle is the radius of the Ferris wheel (105 feet), and the angle at the center of the wheel is the rotation angle (45 degrees). The side opposite this angle is the height change we want to find. This is where the sine function comes to the rescue! The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. So, in our case: sin(45 degrees) = (height change) / (radius). We know the radius is 105 feet, and we can easily find the sine of 45 degrees (it's a common angle, and sin(45°) = $\frac\sqrt{2}}{2}$). Plugging these values into our equation, we get}{2}$ = (height change) / 105 feet. To find the height change, we simply multiply both sides of the equation by 105 feet. This gives us the height change = 105 * $\frac{\sqrt{2}}{2}$ feet. Calculating this value will give us the numerical answer to our problem.
Alright, let's crunch the numbers and find the height change! We've established that the height change is equal to 105 * $\frac{\sqrt{2}}{2}$ feet. Now, we need to calculate this value. The square root of 2 is approximately 1.414. So, our equation becomes: Height change = 105 * (1.414 / 2) feet. First, divide 1.414 by 2, which gives us 0.707. Then, multiply 105 by 0.707. This gives us approximately 74.235 feet. So, the height of the seat changes by about 74.235 feet when the Ferris wheel rotates 45 degrees. This is a significant change in height, highlighting the scale of these massive Ferris wheels!
So, there you have it! After working through the problem step-by-step, we've found that the height of the seat changes by approximately 74.235 feet when the Ferris wheel rotates 45 degrees counterclockwise. This answer makes sense when you think about the geometry of the situation. A 45-degree rotation is halfway between the starting position (level with the center) and the highest point on the wheel (105 feet above the center). Therefore, a height change of around 74 feet feels like a reasonable result. More importantly, this problem demonstrates how we can use trigonometry to solve real-world problems involving circles and angles. By breaking down the problem into smaller parts, visualizing the geometry, and applying the appropriate trigonometric functions, we were able to find the solution. This is the power of mathematics in action!
Awesome work, guys! We've successfully navigated the Ferris wheel problem, using our math skills to figure out the height change after a rotation. This wasn't just about finding a number; it was about understanding the principles behind the problem and applying the right tools to solve it. We explored the relationship between angles, circles, and trigonometric functions, and saw how they can help us understand and quantify real-world scenarios. So, the next time you're on a Ferris wheel, you'll have a whole new appreciation for the math that's involved. Keep practicing, keep exploring, and keep those math skills sharp! You never know when they'll come in handy.
The final answer is $\frac{105\sqrt{2}}{1}$ feet. This corresponds to option A in the given choices.
Ferris wheel, height change, trigonometry, diameter, radius, rotation angle, sine function, mathematics, problem-solving, geometry, vertical distance, right triangle, hypotenuse, opposite side, calculation, solution, interpretation, real-world applications
How much does the height of the seat change after the Ferris wheel rotates $45^{\circ}$ counterclockwise, given the wheel's diameter is 210 feet and the seat starts level with the center?
