Answer :
Let's address these questions one by one with detailed explanations and clear steps.
### Question 5:
The rear wheels of DeMarius' car complete 5 rotations for every full rotation of a front wheel. What is the radius, in feet, of the rear wheel on the car? Write your answer as a simplified fraction.
To solve this, let's go step-by-step:
1. Understanding the Relationship:
- One rotation of the front wheel covers the same distance as 5 rotations of the rear wheel.
2. Circumference Relation:
- The circumference of the front wheel is equal to 5 times the circumference of the rear wheel.
3. Formula for Circumference:
- The circumference [tex]\(C\)[/tex] of a wheel is given by [tex]\(C = 2\pi r\)[/tex], where [tex]\(r\)[/tex] is the radius of the wheel.
Let [tex]\( r_f \)[/tex] be the radius of the front wheel and [tex]\( r_r \)[/tex] be the radius of the rear wheel.
4. Equation Setup:
- Circumference of front wheel: [tex]\( 2\pi r_f \)[/tex]
- Circumference of rear wheel: [tex]\( 2\pi r_r \)[/tex]
- Since one rotation of the front wheel equals 5 rotations of the rear wheel:
[tex]\[ 2\pi r_f = 5 \times 2\pi r_r \][/tex]
5. Simplifying the Equation:
- Cancel out the common terms [tex]\( 2\pi \)[/tex] on both sides:
[tex]\[ r_f = 5r_r \][/tex]
6. Finding the Radius of the Rear Wheel:
- To find the radius of the rear wheel [tex]\( r_r \)[/tex] in terms of the front wheel's radius [tex]\( r_f \)[/tex]:
[tex]\[ r_r = \frac{r_f}{5} \][/tex]
- Since it’s not specified, we can only state the relationship: The radius of the rear wheel is [tex]\(\frac{1}{5}\)[/tex] of the radius of the front wheel.
Answer for Question 5:
The radius of the rear wheel is [tex]\(\frac{1}{5}\)[/tex] of the radius of the front wheel, expressed as the simplified fraction [tex]\( \frac{1}{5} \)[/tex].
---
### Question 6:
DeMarius' original design for his car used rear wheels with a radius of 12 inches. What is the measure of the central angle of this rear wheel such that the arc length is equivalent to that of a full rotation of the rear wheel that is actually used on DeMarius' car?
To find the measure of the central angle, let's break down the problem:
1. Given Data:
- Radius of the original rear wheel [tex]\( r_o = 12 \)[/tex] inches.
- Use the radius of the actual rear wheel [tex]\( r_r \)[/tex] from the previous part. Assuming the front wheel's radius [tex]\( r_f \)[/tex] is given or known:
[tex]\[ r_r = \frac{r_f}{5} \][/tex]
2. Arc Length Equivalence:
- Arc length in a full rotation of the actual rear wheel (circumference): [tex]\( 2\pi r_r \)[/tex]
- Let the central angle be [tex]\( \theta \)[/tex] (in radians) for the original rear wheel.
3. Relationship Between Arc Length and Central Angle:
- Arc length [tex]\( s \)[/tex] is given by the formula [tex]\( s = \theta \times r \)[/tex]
4. Set Up the Equation:
- Arc length of the original wheel with [tex]\( \theta \)[/tex] is equal to the circumference of the actual rear wheel:
[tex]\[ \theta \times 12 = 2\pi \times \frac{r_f}{5} \][/tex]
5. Solve for [tex]\( \theta \)[/tex]:
- Isolate [tex]\( \theta \)[/tex]:
[tex]\[ \theta = \frac{2\pi \times \frac{r_f}{5}}{12} \][/tex]
[tex]\[ \theta = \frac{2\pi r_f}{5 \times 12} \][/tex]
[tex]\[ \theta = \frac{2\pi r_f}{60} \][/tex]
[tex]\[ \theta = \frac{\pi r_f}{30} \][/tex]
Note: The exact value of [tex]\( \theta \)[/tex] depends on the specific radius of the front wheel [tex]\( r_f \)[/tex].
Answer for Question 6:
The measure of the central angle [tex]\( \theta \)[/tex] of the original rear wheel is [tex]\( \frac{\pi r_f}{30} \)[/tex] radians, where [tex]\( r_f \)[/tex] is the radius of the front wheel. Without the specific value of [tex]\( r_f \)[/tex], this expression represents the central angle.
### Question 5:
The rear wheels of DeMarius' car complete 5 rotations for every full rotation of a front wheel. What is the radius, in feet, of the rear wheel on the car? Write your answer as a simplified fraction.
To solve this, let's go step-by-step:
1. Understanding the Relationship:
- One rotation of the front wheel covers the same distance as 5 rotations of the rear wheel.
2. Circumference Relation:
- The circumference of the front wheel is equal to 5 times the circumference of the rear wheel.
3. Formula for Circumference:
- The circumference [tex]\(C\)[/tex] of a wheel is given by [tex]\(C = 2\pi r\)[/tex], where [tex]\(r\)[/tex] is the radius of the wheel.
Let [tex]\( r_f \)[/tex] be the radius of the front wheel and [tex]\( r_r \)[/tex] be the radius of the rear wheel.
4. Equation Setup:
- Circumference of front wheel: [tex]\( 2\pi r_f \)[/tex]
- Circumference of rear wheel: [tex]\( 2\pi r_r \)[/tex]
- Since one rotation of the front wheel equals 5 rotations of the rear wheel:
[tex]\[ 2\pi r_f = 5 \times 2\pi r_r \][/tex]
5. Simplifying the Equation:
- Cancel out the common terms [tex]\( 2\pi \)[/tex] on both sides:
[tex]\[ r_f = 5r_r \][/tex]
6. Finding the Radius of the Rear Wheel:
- To find the radius of the rear wheel [tex]\( r_r \)[/tex] in terms of the front wheel's radius [tex]\( r_f \)[/tex]:
[tex]\[ r_r = \frac{r_f}{5} \][/tex]
- Since it’s not specified, we can only state the relationship: The radius of the rear wheel is [tex]\(\frac{1}{5}\)[/tex] of the radius of the front wheel.
Answer for Question 5:
The radius of the rear wheel is [tex]\(\frac{1}{5}\)[/tex] of the radius of the front wheel, expressed as the simplified fraction [tex]\( \frac{1}{5} \)[/tex].
---
### Question 6:
DeMarius' original design for his car used rear wheels with a radius of 12 inches. What is the measure of the central angle of this rear wheel such that the arc length is equivalent to that of a full rotation of the rear wheel that is actually used on DeMarius' car?
To find the measure of the central angle, let's break down the problem:
1. Given Data:
- Radius of the original rear wheel [tex]\( r_o = 12 \)[/tex] inches.
- Use the radius of the actual rear wheel [tex]\( r_r \)[/tex] from the previous part. Assuming the front wheel's radius [tex]\( r_f \)[/tex] is given or known:
[tex]\[ r_r = \frac{r_f}{5} \][/tex]
2. Arc Length Equivalence:
- Arc length in a full rotation of the actual rear wheel (circumference): [tex]\( 2\pi r_r \)[/tex]
- Let the central angle be [tex]\( \theta \)[/tex] (in radians) for the original rear wheel.
3. Relationship Between Arc Length and Central Angle:
- Arc length [tex]\( s \)[/tex] is given by the formula [tex]\( s = \theta \times r \)[/tex]
4. Set Up the Equation:
- Arc length of the original wheel with [tex]\( \theta \)[/tex] is equal to the circumference of the actual rear wheel:
[tex]\[ \theta \times 12 = 2\pi \times \frac{r_f}{5} \][/tex]
5. Solve for [tex]\( \theta \)[/tex]:
- Isolate [tex]\( \theta \)[/tex]:
[tex]\[ \theta = \frac{2\pi \times \frac{r_f}{5}}{12} \][/tex]
[tex]\[ \theta = \frac{2\pi r_f}{5 \times 12} \][/tex]
[tex]\[ \theta = \frac{2\pi r_f}{60} \][/tex]
[tex]\[ \theta = \frac{\pi r_f}{30} \][/tex]
Note: The exact value of [tex]\( \theta \)[/tex] depends on the specific radius of the front wheel [tex]\( r_f \)[/tex].
Answer for Question 6:
The measure of the central angle [tex]\( \theta \)[/tex] of the original rear wheel is [tex]\( \frac{\pi r_f}{30} \)[/tex] radians, where [tex]\( r_f \)[/tex] is the radius of the front wheel. Without the specific value of [tex]\( r_f \)[/tex], this expression represents the central angle.
