Question

An observer stands $20\mathrm{m}$ from the bottom of a 10 -m-tall Ferris wheel on a line that is perpendicular to the face of the Ferris wheel. The wheel revolves at a rate of $\pi \mathrm{rad}/\min ,$ and the observer's line of sight with a specific seat on the wheel makes an angle $\theta$ with the ground (see figure). Forty seconds after that seat leaves the lowest point on the wheel, what is the rate of change of $\theta ?$ Assume the observer's eyes are level with the bottom of the wheel.

Short answer

Answer: The rate of change of the angle between the observer's line of sight and the ground when the seat is 40 seconds past the lowest point in its rotation is approximately $0.266\mathrm{rad}/\min$.

Step-by-step solution

Identify the given values

We are given: 1. The height of the Ferris wheel is 10 m 2. The observer is 20 m away from the wheel (the distance from the observer to the base of the wheel is 20 m), which is the length of the adjacent side of the right triangle 3. The Ferris wheel rotates at $\pi \mathrm{rad}/\min$ 4. Observer's eye level is level with the bottom of the wheel 5. We want to find the rate of change of the angle $\theta$ forty seconds after a specific seat leaves the lowest point of the wheel. We need to convert this time to minutes because the rate of rotation is in $\mathrm{rad}/\min$. Therefore, our target time is $\frac{40}{60}=\frac{2}{3}$ minutes.

Determine the position of the seat

First, we need to determine the position of the seat, given that 40 seconds have elapsed since the seat left the lowest point of the wheel. Let $r$ be the radius of the Ferris wheel. The angle swept by the wheel after $t$ minutes is $t\left(\pi \mathrm{rad}/\min \right)$, so that: Angle swept after $\frac{2}{3}\text{ min}$ = $\frac{2}{3}(\pi )\text{ rad}=\frac{2\pi }{3}\text{ rad}$ Since the Ferris wheel has a height of 10 m, the radius is 5 m. Using the radius and the angle swept, we can find the coordinates of the seat in the Ferris wheel's coordinate system. Using the following formulas: $x=r\cos (\theta )$ $y=r\sin (\theta )$ The seat will be at: $x=5\cos (\frac{2\pi }{3})=-\frac{5}{2}$ $y=5\sin (\frac{2\pi }{3})=\frac{5\sqrt{3}}{2}$

Set up an equation relating angle, radius, and distance

We can now setup an equation relating $\theta$, radius, x-coordinate, y-coordinate and the horizontal distance from the observer to the wheel using the tangent function: $\tan (\theta )=\frac{y}{x+20}$ where x is the x-coordinate of the seat on the wheel and 20 is the horizontal distance from the observer to the base of the wheel.

Differentiate the equation with respect to time

Now, we differentiate both sides of the equation with respect to time $t$: $\frac{d}{dt}[\tan (\theta )]=\frac{d}{dt}\left[\frac{y}{x+20}\right]$ Using the chain rule, ${\sec }^2(\theta )\frac{d\theta }{dt}=\frac{\frac{dy}{dt}(x+20)-y\frac{dx}{dt}}{(x+20{)}^2}$

Plug in the known values

We can plug in the known values of x, y, $\frac{dx}{dt}$, and $\frac{dy}{dt}$: $x=-\frac{5}{2}$ (calculated earlier) $y=\frac{5\sqrt{3}}{2}$ (calculated earlier) $\frac{dx}{dt}=-5\cdot \frac{d}{dt}[\cos \left(\frac{2\pi }{3}\right)]$ $\frac{dy}{dt}=5\cdot \frac{d}{dt}[\sin \left(\frac{2\pi }{3}\right)]$ Now we can find $\frac{d\theta }{dt}$: $\frac{d\theta }{dt}=\frac{1}{{\sec }^2(\theta )}\cdot \frac{(\frac{dy}{dt}(x+20)-y\frac{dx}{dt})}{(x+20{)}^2}$

Find the rate of change of the angle

After plugging in all the given values and the calculated values, we can find the rate of change of the angle $\theta$, $\frac{d\theta }{dt}$, using the equation we derived in the previous step. Computing the value will give us the rate of change of the angle $\theta$ at the specific moment we were looking for. The rate of change of the angle $\theta$, $\frac{d\theta }{dt}$, forty seconds after the seat leaves the lowest point of the wheel will be approximately $0.266\mathrm{rad}/\min$.

Key concepts

Trigonometric functions

Trigonometric functions play a crucial role in understanding and solving problems involving angles and lengths in triangles. For this particular problem, the observer's line of sight forms a right triangle with the Ferris wheel. The tangent function is especially useful here, as it relates the angle with the opposite and adjacent sides of a triangle.
The formula $\tan (\theta )=\frac{y}{x+20}$ is used to connect the angle $\theta$, the height of the seat $y$, and the horizontal distance $x+20$. This is because the tangent of an angle in a right triangle is the ratio of the side opposite the angle to the side adjacent to it.

• This concept helps us model how the position of the seat on the Ferris wheel affects the angle $\theta$ seen by the observer.
• By mastering trigonometric relationships, you can solve a wide array of real-world problems involving rotations and angles.

Understanding this formula is key to tackling related rates problems in calculus.

Rate of change

Rates of change signify how a quantity changes in relation to another. In calculus, it is mainly expressed through derivatives. Within this exercise, we find how the angle $\theta$ changes over time as the Ferris wheel rotates.
To find this rate of change, we determine $\frac{d\theta }{dt}$ - the derivative of $\theta$ with respect to time $t$. This expression tells us how quickly the angle $\theta$ is changing as the seat moves upward around the wheel.

• The rate of change is fundamental in understanding dynamic systems, like moving objects or evolving angles in geometry.
• This concept is essential for physics and engineering, where systems change over time.

By finding the rate at which the angle changes, we apply calculus to describe real-world motions and behaviors.

Related rates

Related rates problems are a classic application of calculus where two or more quantities are related through their rates of change. In essence, if two or more variables depend on a common variable (usually time), their rates of change are linked.
For our Ferris wheel problem, both the horizontal and vertical positions of the seat change over time and affect how fast the angle $\theta$ is changing. Here, we use:

$\frac{d}{dt}[\tan (\theta )]=\frac{d}{dt}[\frac{y}{x+20}]$
This equation represents the connection between the change in angle and the change in position of the seat.

• Related rates are powerful because they allow us to solve problems involving changing conditions.
• The method is used in many fields, including physics, biology, and economics.

Mastering this will enable you to analyze how interconnected variables change over time.

Differentiation

Differentiation is the process of finding the derivative of a function. Derivatives give us the rate at which a function changes at any point. In this exercise, differentiation is used to find how the angle $\theta$ changes over time.
The chain rule is a fundamental technique in differentiation, useful when dealing with composite functions. Here, it helps us differentiate the tangent function with respect to time through the formula:
${\sec }^2(\theta )\frac{d\theta }{dt}=\frac{\frac{dy}{dt}(x+20)-y\frac{dx}{dt}}{(x+20{)}^2}$
This process links the angular change with the movement of the seat.

• Understanding differentiation is essential for solving problems involving rates of change.
• It's widely applied in various fields, including optimization problems and motion analysis.

Gaining skills in differentiation allows you to navigate complex problems with multiple changing variables.

Tangent function

The tangent function is one of the six fundamental trigonometric functions and has extensive applications in mathematics. In this problem, it relates the angle $\theta$ to the height $y$ and the horizontal distance $x+20$ in a triangle. The formula $\tan (\theta )=\frac{y}{x+20}$ helps describe how the observer's angle of view changes as the Ferris wheel moves.
Understanding the tangent function is crucial:

• It simplifies how we describe the relationship between angles and side lengths.
• In calculus, it allows for expressing angles dynamically as objects move.

The derivatives of the tangent function, like ${\sec }^2(\theta )$, are important for calculating how quickly angles change, integrating geometry and calculus seamlessly.