Math / Algebra

QuestionThe wheel and piston device shown above consists of a wheel of radius 1 foot that is connected at the point $\mathbf{W}$ to a piston at $\mathbf{P}$ by a connecting rod (represented by the segment WP in the diagram) of length 8 feet. The wheel rotates counterclockwise at a rate of of 4 radians per second as the piston moves up and down along the $y$ -axis. (Click the hint to see animation). The point $\mathbf{W}$ is at $(1,0)$ at $t=0$ seconds. a) What is the measure of angle $\theta$ after $t$ seconds? $\theta=$

Studdy Solution

STEP 1

What is this asking? We need to find the angle of the wheel after a certain amount of time, knowing how fast it's spinning. Watch out! Make sure to use radians, not degrees, since the rotation speed is given in radians per second.

STEP 2

1. Relate angle to time

STEP 3

We're given that the wheel rotates at a rate of 4 radians per second.
This is our rate of change.

STEP 4

We can write this relationship mathematically as: $\text{Angle} = \text{Rate} \cdot \text{Time}$

STEP 5

Let $\theta$ be the angle.
The rate is given as 4 radians per second, and the time is represented by $t$ seconds.
So, after $t$ seconds, the angle $\theta$ is: $\theta = 4 \cdot t$ Where $\theta$ is in radians.

STEP 6

At $t = 0$ , the point $\mathbf{W}$ is at $(1, 0)$ , which means the initial angle is 0 radians.
Our formula already accounts for this, since when $t = 0$ , $\theta = 4 \cdot 0 = 0$ .
So, we're good to go!

STEP 7

The angle $\theta$ after $t$ seconds is given by: $\theta = 4t$

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Studdy Solution

STEP 1

What is this asking? We need to find the angle of the wheel after a certain amount of time, knowing how fast it's spinning. Watch out! Make sure to use radians, not degrees, since the rotation speed is given in radians per second.

STEP 2

1. Relate angle to time

STEP 3

We're given that the wheel rotates at a rate of 4 radians per second.
This is our rate of change.

STEP 4

We can write this relationship mathematically as: $\text{Angle} = \text{Rate} \cdot \text{Time}$

STEP 5

Let $\theta$ be the angle.
The rate is given as 4 radians per second, and the time is represented by $t$ seconds.
So, after $t$ seconds, the angle $\theta$ is: $\theta = 4 \cdot t$ Where $\theta$ is in radians.

STEP 6

At $t = 0$ , the point $\mathbf{W}$ is at $(1, 0)$ , which means the initial angle is 0 radians.
Our formula already accounts for this, since when $t = 0$ , $\theta = 4 \cdot 0 = 0$ .
So, we're good to go!

STEP 7

The angle $\theta$ after $t$ seconds is given by: $\theta = 4t$

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Studdy Solution

STEP 1

What is this asking? We need to find the angle of the wheel after a certain amount of time, knowing how fast it's spinning. Watch out! Make sure to use radians, not degrees, since the rotation speed is given in radians per second.

STEP 2

1. Relate angle to time

STEP 3

We're given that the wheel rotates at a rate of **4 radians per second**.This is our **rate of change**.

STEP 4

We can write this relationship mathematically as: Angle=Rate⋅Time\text{Angle} = \text{Rate} \cdot \text{Time}Angle=Rate⋅Time

STEP 5

Let θ\thetaθ be the angle.The rate is given as **4 radians per second**, and the time is represented by ttt seconds.So, after ttt seconds, the angle θ\thetaθ is: θ=4⋅t\theta = 4 \cdot tθ=4⋅t Where θ\thetaθ is in radians.

STEP 6

At t=0t = 0t=0, the point W\mathbf{W}W is at (1,0)(1, 0)(1,0), which means the initial angle is **0 radians**.Our formula already accounts for this, since when t=0t = 0t=0, θ=4⋅0=0\theta = 4 \cdot 0 = 0θ=4⋅0=0.So, we're good to go!

STEP 7

The angle θ\thetaθ after ttt seconds is given by: θ=4t\theta = 4tθ=4t

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

We're given that the wheel rotates at a rate of 4 radians per second.
This is our rate of change.

We can write this relationship mathematically as: $\text{Angle} = \text{Rate} \cdot \text{Time}$

Let $\theta$ be the angle.
The rate is given as 4 radians per second, and the time is represented by $t$ seconds.
So, after $t$ seconds, the angle $\theta$ is: $\theta = 4 \cdot t$ Where $\theta$ is in radians.

At $t = 0$ , the point $\mathbf{W}$ is at $(1, 0)$ , which means the initial angle is 0 radians.
Our formula already accounts for this, since when $t = 0$ , $\theta = 4 \cdot 0 = 0$ .
So, we're good to go!

The angle $\theta$ after $t$ seconds is given by: $\theta = 4t$