Math / Algebra

Question8. If the rotation angle $\theta=\frac{5 \pi}{2}$ radians and the angular speed $\omega=\frac{5 \pi}{16}$ radians per minute, find the rotation time $t$ in minut
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Studdy Solution

STEP 1

What is this asking? How long does it take to rotate $\frac{5\pi}{2}$ radians if you're rotating at a speed of $\frac{5\pi}{16}$ radians per minute? Watch out! Make sure your units match up – we're dealing with radians and minutes here!

STEP 2

1. Set up the relationship between angle, angular speed, and time.
2. Solve for time.

STEP 3

Remember the core relationship: angle equals angular speed multiplied by time.
Think of it like distance equals speed multiplied by time, but for angles!
We can write this as: $\theta = \omega \cdot t$ Where: $\theta$ is the rotation angle, $\omega$ is the angular speed, and $t$ is the time.

STEP 4

We know $\theta = \frac{5\pi}{2}$ radians and $\omega = \frac{5\pi}{16}$ radians per minute.
Let's plug these values into our formula: $\frac{5\pi}{2} = \frac{5\pi}{16} \cdot t$

STEP 5

We want to get t by itself.
To do this, we can multiply both sides of the equation by the reciprocal of $\frac{5\pi}{16}$ , which is $\frac{16}{5\pi}$ .
This is like dividing both sides by $\frac{5\pi}{16}$ , but thinking about it as multiplying by the reciprocal helps us see how things simplify nicely! $\frac{16}{5\pi} \cdot \frac{5\pi}{2} = \frac{16}{5\pi} \cdot \frac{5\pi}{16} \cdot t$

STEP 6

On the right side, $\frac{16}{5\pi} \cdot \frac{5\pi}{16}$ becomes 1, effectively isolating t.
On the left side, the $\pi$ and $5$ in the numerator and denominator divide to one, leaving us with: $\frac{16}{2} = t$

STEP 7

Finally, we divide 16 by 2 to get our final answer: $t = 8$

STEP 8

The rotation time $t$ is 8 minutes.

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Question8. If the rotation angle $\theta=\frac{5 \pi}{2}$ radians and the angular speed $\omega=\frac{5 \pi}{16}$ radians per minute, find the rotation time $t$ in minut
Enter your next step here

Studdy Solution

STEP 1

What is this asking? How long does it take to rotate $\frac{5\pi}{2}$ radians if you're rotating at a speed of $\frac{5\pi}{16}$ radians per minute? Watch out! Make sure your units match up – we're dealing with radians and minutes here!

STEP 2

1. Set up the relationship between angle, angular speed, and time.
2. Solve for time.

STEP 3

Remember the core relationship: angle equals angular speed multiplied by time.
Think of it like distance equals speed multiplied by time, but for angles!
We can write this as: $\theta = \omega \cdot t$ Where: $\theta$ is the rotation angle, $\omega$ is the angular speed, and $t$ is the time.

STEP 4

We know $\theta = \frac{5\pi}{2}$ radians and $\omega = \frac{5\pi}{16}$ radians per minute.
Let's plug these values into our formula: $\frac{5\pi}{2} = \frac{5\pi}{16} \cdot t$

STEP 5

STEP 6

On the right side, $\frac{16}{5\pi} \cdot \frac{5\pi}{16}$ becomes 1, effectively isolating t.
On the left side, the $\pi$ and $5$ in the numerator and denominator divide to one, leaving us with: $\frac{16}{2} = t$

STEP 7

Finally, we divide 16 by 2 to get our final answer: $t = 8$

STEP 8

The rotation time $t$ is 8 minutes.

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Question8. If the rotation angle θ=5π2\theta=\frac{5 \pi}{2}θ=25π​ radians and the angular speed ω=5π16\omega=\frac{5 \pi}{16}ω=165π​ radians per minute, find the rotation time ttt in minutEnter your next step here

Studdy Solution

STEP 1

What is this asking? How long does it take to rotate 5π2\frac{5\pi}{2}25π​ radians if you're rotating at a speed of 5π16\frac{5\pi}{16}165π​ radians per minute? Watch out! Make sure your units match up – we're dealing with radians and minutes here!

STEP 2

1. Set up the relationship between angle, angular speed, and time.2. Solve for time.

STEP 3

STEP 4

We know θ=5π2\theta = \frac{5\pi}{2}θ=25π​ radians and ω=5π16\omega = \frac{5\pi}{16}ω=165π​ radians per minute.Let's **plug these values** into our formula: 5π2=5π16⋅t\frac{5\pi}{2} = \frac{5\pi}{16} \cdot t25π​=165π​⋅t

STEP 5

STEP 6

On the right side, 165π⋅5π16\frac{16}{5\pi} \cdot \frac{5\pi}{16}5π16​⋅165π​ becomes 1, effectively isolating *t*.On the left side, the π\piπ and 555 in the numerator and denominator divide to one, leaving us with: 162=t\frac{16}{2} = t216​=t

STEP 7

Finally, we **divide 16 by 2** to get our **final answer**: t=8t = 8t=8

STEP 8

The rotation time ttt is **8 minutes**.

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Question8. If the rotation angle $\theta=\frac{5 \pi}{2}$ radians and the angular speed $\omega=\frac{5 \pi}{16}$ radians per minute, find the rotation time $t$ in minut
Enter your next step here

What is this asking? How long does it take to rotate $\frac{5\pi}{2}$ radians if you're rotating at a speed of $\frac{5\pi}{16}$ radians per minute? Watch out! Make sure your units match up – we're dealing with radians and minutes here!

1. Set up the relationship between angle, angular speed, and time.
2. Solve for time.

We know $\theta = \frac{5\pi}{2}$ radians and $\omega = \frac{5\pi}{16}$ radians per minute.
Let's plug these values into our formula: $\frac{5\pi}{2} = \frac{5\pi}{16} \cdot t$

On the right side, $\frac{16}{5\pi} \cdot \frac{5\pi}{16}$ becomes 1, effectively isolating t.
On the left side, the $\pi$ and $5$ in the numerator and denominator divide to one, leaving us with: $\frac{16}{2} = t$

Finally, we divide 16 by 2 to get our final answer: $t = 8$

The rotation time $t$ is 8 minutes.