Question
Multiple Representations The equation yequalsStartFraction 6 Over 20 EndFraction
x describes the number of kilometers y that a van travels in x minutes. What is the constant speed of the van in terms of hours? Use pencil and paper. Express the speed in three different ways.
Question content area bottom
Part 1
The constant speed of the van is
enter your response here kilometers per hour.
x describes the number of kilometers y that a van travels in x minutes. What is the constant speed of the van in terms of hours? Use pencil and paper. Express the speed in three different ways.
Question content area bottom
Part 1
The constant speed of the van is
enter your response here kilometers per hour.
Answers
GPT-4o mini
To find the constant speed of the van in terms of hours from the equation \( y = \frac{6}{20} x \), we can follow these steps:
### Step 1: Simplify the Equation
The equation tells us the distance \( y \) traveled in kilometers as a function of time \( x \) in minutes.
We can simplify the fraction:
\[
\frac{6}{20} = \frac{3}{10}
\]
So, the equation can be rewritten as:
\[
y = \frac{3}{10} x
\]
### Step 2: Convert Minutes to Hours
Since the speed needs to be expressed in kilometers per hour (km/h), we need to convert the time from minutes to hours.
There are 60 minutes in an hour, so:
\[
x \text{ minutes} = \frac{x}{60} \text{ hours}
\]
### Step 3: Substitute and Calculate Speed
Substituting in terms of hours, we have:
\[
y = \frac{3}{10} \left(\frac{x}{60}\right)
\]
This simplifies to:
\[
y = \frac{3x}{600} = \frac{x}{200}
\]
### Step 4: Calculate Speed in km/h
The equation \( y = \frac{x}{200} \) tells us that for every hour (when \( x = 60 \) minutes), the distance \( y \) is:
\[
y = \frac{60}{200} = \frac{3}{10} \text{ kilometers}
\]
To find the speed in km/h, we can express this as follows for 60 minutes:
To find out how far the van travels in 1 hour (60 minutes):
\[
\text{Speed} = \frac{y \text{ (kilometers)}}{1 \text{ hour}} = \frac{3}{10} \text{ km/min} \times 60 \text{ min/hr} = \frac{3 \times 60}{10} = 18 \text{ km/h}
\]
### Different Representations
1. **Standard form**: **18 km/h**
2. **As a fraction**: **\(\frac{18}{1}\) km/h**
3. **In terms of m/s**:
- Convert km/h to m/s:
- \(\text{Speed in m/s} = \frac{18 \text{ km/h} \times 1000 \text{ m/km}}{3600 \text{ s/h}} = \frac{18000}{3600} = 5 \text{ m/s}\).
In conclusion, the constant speed of the van is:
- **18 kilometers per hour**
- **\(\frac{18}{1}\) kilometers per hour**
- **5 meters per second**
### Step 1: Simplify the Equation
The equation tells us the distance \( y \) traveled in kilometers as a function of time \( x \) in minutes.
We can simplify the fraction:
\[
\frac{6}{20} = \frac{3}{10}
\]
So, the equation can be rewritten as:
\[
y = \frac{3}{10} x
\]
### Step 2: Convert Minutes to Hours
Since the speed needs to be expressed in kilometers per hour (km/h), we need to convert the time from minutes to hours.
There are 60 minutes in an hour, so:
\[
x \text{ minutes} = \frac{x}{60} \text{ hours}
\]
### Step 3: Substitute and Calculate Speed
Substituting in terms of hours, we have:
\[
y = \frac{3}{10} \left(\frac{x}{60}\right)
\]
This simplifies to:
\[
y = \frac{3x}{600} = \frac{x}{200}
\]
### Step 4: Calculate Speed in km/h
The equation \( y = \frac{x}{200} \) tells us that for every hour (when \( x = 60 \) minutes), the distance \( y \) is:
\[
y = \frac{60}{200} = \frac{3}{10} \text{ kilometers}
\]
To find the speed in km/h, we can express this as follows for 60 minutes:
To find out how far the van travels in 1 hour (60 minutes):
\[
\text{Speed} = \frac{y \text{ (kilometers)}}{1 \text{ hour}} = \frac{3}{10} \text{ km/min} \times 60 \text{ min/hr} = \frac{3 \times 60}{10} = 18 \text{ km/h}
\]
### Different Representations
1. **Standard form**: **18 km/h**
2. **As a fraction**: **\(\frac{18}{1}\) km/h**
3. **In terms of m/s**:
- Convert km/h to m/s:
- \(\text{Speed in m/s} = \frac{18 \text{ km/h} \times 1000 \text{ m/km}}{3600 \text{ s/h}} = \frac{18000}{3600} = 5 \text{ m/s}\).
In conclusion, the constant speed of the van is:
- **18 kilometers per hour**
- **\(\frac{18}{1}\) kilometers per hour**
- **5 meters per second**