Answer :
Let's determine the unknown values [tex]\( x \)[/tex] in each row step by step.
### Row 1: Find the Angular Speed
We are given:
- Radius = 8 cm
- Angular Speed = [tex]\( x \)[/tex] rpm
- Linear Speed = 12 cm/min
The relationship between linear speed ([tex]\( V \)[/tex]), angular speed ([tex]\( \omega \)[/tex]), and radius ([tex]\( r \)[/tex]) is given by:
[tex]\[ V = \omega \cdot r \][/tex]
From the given information, the linear speed [tex]\( V = 12 \)[/tex] cm/min and the radius [tex]\( r = 8 \)[/tex] cm.
Therefore, we need to find [tex]\( \omega \)[/tex]:
[tex]\[ 12 = \omega \cdot 8 \][/tex]
Solving for [tex]\( \omega \)[/tex]:
[tex]\[ \omega = \frac{12}{8} = 1.5 \, \text{cm/min}\][/tex]
To convert the angular speed from cm/min to rpm, recall that angular speed in rpm is:
[tex]\[ \text{revolutions per minute} = \frac{1.5}{2\pi} \, \text{radians per minute} = \frac{0.75}{\pi} \, \text{rpm} \][/tex]
Thus, the angular speed [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{0.75}{\pi} \, \text{rpm} \][/tex]
### Row 2: Find the Linear Speed
We are given:
- Radius = 18 feet
- Angular Speed = 4 rpm
- Linear Speed = [tex]\( x \)[/tex] mi/hr
The relationship between the linear speed [tex]\( V \)[/tex], angular speed [tex]\( \omega \)[/tex], and radius [tex]\( r \)[/tex] is:
[tex]\[ V = \omega \cdot r \cdot 2\pi \][/tex]
We first convert the angular speed to radians per minute:
[tex]\[ \omega = 4 \text{ rpm} = 4 \cdot 2\pi \, \text{radians per minute} \][/tex]
Next, calculate the linear speed in feet per minute:
[tex]\[ V = 4 \cdot 18 \cdot 2\pi \, \text{feet per minute} = 144\pi \, \text{feet per minute} \][/tex]
Converting feet per minute to miles per hour:
[tex]\[ V = 144\pi \cdot \frac{1 \, \text{mile}}{5280 \, \text{feet}} \cdot 60 \, \text{minutes per hour} = \frac{144\pi}{88} \, \text{miles per hour} = \frac{3\pi}{110} \, \text{mi/hr} \][/tex]
Thus, the linear speed [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{3\pi}{110} \, \text{mi/hr} \][/tex]
### Row 3: Find the Radius
We are given:
- Radius = [tex]\( x \)[/tex] feet
- Angular Speed = 800 rpm
- Linear Speed = 60 mi/hr
Convert 60 mi/hr to feet per minute:
[tex]\[ V = 60 \cdot 5280 \, \text{feet per hour} = 60 \cdot 5280 \cdot \frac{1}{60} \, \text{feet per minute} = 5280 \, \text{feet per minute} \][/tex]
The relationship between linear speed ([tex]\( V \)[/tex]), angular speed ([tex]\( \omega \)[/tex]), and radius ([tex]\( r \)[/tex]) is:
[tex]\[ 5280 = 800 \cdot 2\pi \cdot r \cdot \frac{1}{60} \][/tex]
Solving for [tex]\( r \)[/tex]:
[tex]\[ 5280 = \frac{800 \cdot 2\pi \cdot r}{60} \][/tex]
[tex]\[ 5280 = \frac{800 \cdot 2\pi \cdot r}{60} \][/tex]
[tex]\[ r \approx 63.0253574643906 \, \text{feet} \][/tex]
Thus, the radius [tex]\( x \)[/tex] is:
[tex]\[ x \approx 63.0253574643906 \, \text{feet} \][/tex]
### Row 4: Find the Angular Speed
We are given:
- Radius = 15 inches
- Angular Speed = [tex]\( x \)[/tex] rpm
- Linear Speed = 60 mi/hr
First, convert radius from inches to centimeters:
[tex]\[ r = 15 \times 2.54 \, \text{cm} = 38.1 \, \text{cm} \][/tex]
Convert the linear speed from miles/hr to cm/min:
[tex]\[ V = 60 \, \text{mi/hr} \cdot 5280 \, \text{feet/mile} \cdot 30.48 \, \text{cm/foot} \cdot \frac{1}{60} \, \text{min/hr} = 160934.4 \, \text{cm/min} \][/tex]
Using the relationship [tex]\( V = \omega \cdot r \)[/tex]:
[tex]\[ 160934.4 = \omega \cdot 38.1 \][/tex]
Solving for [tex]\( \omega \)[/tex]:
[tex]\[ \omega = \frac{160934.4}{38.1} = 4225.43 \text{cm/min} \][/tex]
Convert to rpm:
[tex]\[ x = \frac{4225.43}{2\pi} \approx 10560.0 \][/tex]
Thus, the angular speed [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{10560.0}{\pi} \, \text{rpm} \][/tex]
So the final answers are:
[tex]\[ x = \left( \frac{0.75}{\pi}, \frac{3\pi}{110}, 63.0253574643906, \frac{10560.0}{\pi} \right) \][/tex]
### Row 1: Find the Angular Speed
We are given:
- Radius = 8 cm
- Angular Speed = [tex]\( x \)[/tex] rpm
- Linear Speed = 12 cm/min
The relationship between linear speed ([tex]\( V \)[/tex]), angular speed ([tex]\( \omega \)[/tex]), and radius ([tex]\( r \)[/tex]) is given by:
[tex]\[ V = \omega \cdot r \][/tex]
From the given information, the linear speed [tex]\( V = 12 \)[/tex] cm/min and the radius [tex]\( r = 8 \)[/tex] cm.
Therefore, we need to find [tex]\( \omega \)[/tex]:
[tex]\[ 12 = \omega \cdot 8 \][/tex]
Solving for [tex]\( \omega \)[/tex]:
[tex]\[ \omega = \frac{12}{8} = 1.5 \, \text{cm/min}\][/tex]
To convert the angular speed from cm/min to rpm, recall that angular speed in rpm is:
[tex]\[ \text{revolutions per minute} = \frac{1.5}{2\pi} \, \text{radians per minute} = \frac{0.75}{\pi} \, \text{rpm} \][/tex]
Thus, the angular speed [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{0.75}{\pi} \, \text{rpm} \][/tex]
### Row 2: Find the Linear Speed
We are given:
- Radius = 18 feet
- Angular Speed = 4 rpm
- Linear Speed = [tex]\( x \)[/tex] mi/hr
The relationship between the linear speed [tex]\( V \)[/tex], angular speed [tex]\( \omega \)[/tex], and radius [tex]\( r \)[/tex] is:
[tex]\[ V = \omega \cdot r \cdot 2\pi \][/tex]
We first convert the angular speed to radians per minute:
[tex]\[ \omega = 4 \text{ rpm} = 4 \cdot 2\pi \, \text{radians per minute} \][/tex]
Next, calculate the linear speed in feet per minute:
[tex]\[ V = 4 \cdot 18 \cdot 2\pi \, \text{feet per minute} = 144\pi \, \text{feet per minute} \][/tex]
Converting feet per minute to miles per hour:
[tex]\[ V = 144\pi \cdot \frac{1 \, \text{mile}}{5280 \, \text{feet}} \cdot 60 \, \text{minutes per hour} = \frac{144\pi}{88} \, \text{miles per hour} = \frac{3\pi}{110} \, \text{mi/hr} \][/tex]
Thus, the linear speed [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{3\pi}{110} \, \text{mi/hr} \][/tex]
### Row 3: Find the Radius
We are given:
- Radius = [tex]\( x \)[/tex] feet
- Angular Speed = 800 rpm
- Linear Speed = 60 mi/hr
Convert 60 mi/hr to feet per minute:
[tex]\[ V = 60 \cdot 5280 \, \text{feet per hour} = 60 \cdot 5280 \cdot \frac{1}{60} \, \text{feet per minute} = 5280 \, \text{feet per minute} \][/tex]
The relationship between linear speed ([tex]\( V \)[/tex]), angular speed ([tex]\( \omega \)[/tex]), and radius ([tex]\( r \)[/tex]) is:
[tex]\[ 5280 = 800 \cdot 2\pi \cdot r \cdot \frac{1}{60} \][/tex]
Solving for [tex]\( r \)[/tex]:
[tex]\[ 5280 = \frac{800 \cdot 2\pi \cdot r}{60} \][/tex]
[tex]\[ 5280 = \frac{800 \cdot 2\pi \cdot r}{60} \][/tex]
[tex]\[ r \approx 63.0253574643906 \, \text{feet} \][/tex]
Thus, the radius [tex]\( x \)[/tex] is:
[tex]\[ x \approx 63.0253574643906 \, \text{feet} \][/tex]
### Row 4: Find the Angular Speed
We are given:
- Radius = 15 inches
- Angular Speed = [tex]\( x \)[/tex] rpm
- Linear Speed = 60 mi/hr
First, convert radius from inches to centimeters:
[tex]\[ r = 15 \times 2.54 \, \text{cm} = 38.1 \, \text{cm} \][/tex]
Convert the linear speed from miles/hr to cm/min:
[tex]\[ V = 60 \, \text{mi/hr} \cdot 5280 \, \text{feet/mile} \cdot 30.48 \, \text{cm/foot} \cdot \frac{1}{60} \, \text{min/hr} = 160934.4 \, \text{cm/min} \][/tex]
Using the relationship [tex]\( V = \omega \cdot r \)[/tex]:
[tex]\[ 160934.4 = \omega \cdot 38.1 \][/tex]
Solving for [tex]\( \omega \)[/tex]:
[tex]\[ \omega = \frac{160934.4}{38.1} = 4225.43 \text{cm/min} \][/tex]
Convert to rpm:
[tex]\[ x = \frac{4225.43}{2\pi} \approx 10560.0 \][/tex]
Thus, the angular speed [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{10560.0}{\pi} \, \text{rpm} \][/tex]
So the final answers are:
[tex]\[ x = \left( \frac{0.75}{\pi}, \frac{3\pi}{110}, 63.0253574643906, \frac{10560.0}{\pi} \right) \][/tex]
