Question
Donald fills his gasoline tank before leaving on a trip. His car uses gasoline at a constant rate and the amount of gasoline, g, remaining after m miles can be
modeled by m + 20g = 640.
Select the correct values to complete the range of the function in context.
modeled by m + 20g = 640.
Select the correct values to complete the range of the function in context.
Answers
GPT-4o mini
To analyze the situation described in the equation \( m + 20g = 640 \), we can rewrite this to express \( g \) in terms of \( m \):
\[
20g = 640 - m
\]
\[
g = \frac{640 - m}{20}
\]
This function indicates that the remaining amount of gasoline \( g \) is dependent on the number of miles \( m \) traveled.
1. **Find the Constraints:**
- Since \( g \) represents the gallons of gasoline remaining, it cannot be negative. Therefore, we need \( g \geq 0 \).
- Setting \( g = 0 \) in our equation gives us:
\[
0 = \frac{640 - m}{20}
\]
Multiplying both sides by 20 results in:
\[
640 - m = 0 \implies m = 640
\]
So the maximum distance \( m \) that can be traveled with a full tank without running out of gas is 640 miles.
2. **Minimum Value of \( m \):**
- The minimum distance that can be traveled is \( m = 0 \).
3. **Determine the Values of \( g \):**
- At \( m = 0 \):
\[
g = \frac{640 - 0}{20} = \frac{640}{20} = 32
\]
- At \( m = 640 \):
\[
g = \frac{640 - 640}{20} = 0
\]
4. **Range of Function:**
- Therefore, as \( m \) varies from 0 to 640, \( g \) will vary from 32 gallons (when \( m = 0 \)) to 0 gallons (when \( m = 640 \)).
- Thus, the range of the function in this context is:
\[
g \in [0, 32]
\]
In conclusion, the completed range of values for gasoline remaining \( g \) as a function of miles traveled \( m \) in this context is \( g \in [0, 32] \).
\[
20g = 640 - m
\]
\[
g = \frac{640 - m}{20}
\]
This function indicates that the remaining amount of gasoline \( g \) is dependent on the number of miles \( m \) traveled.
1. **Find the Constraints:**
- Since \( g \) represents the gallons of gasoline remaining, it cannot be negative. Therefore, we need \( g \geq 0 \).
- Setting \( g = 0 \) in our equation gives us:
\[
0 = \frac{640 - m}{20}
\]
Multiplying both sides by 20 results in:
\[
640 - m = 0 \implies m = 640
\]
So the maximum distance \( m \) that can be traveled with a full tank without running out of gas is 640 miles.
2. **Minimum Value of \( m \):**
- The minimum distance that can be traveled is \( m = 0 \).
3. **Determine the Values of \( g \):**
- At \( m = 0 \):
\[
g = \frac{640 - 0}{20} = \frac{640}{20} = 32
\]
- At \( m = 640 \):
\[
g = \frac{640 - 640}{20} = 0
\]
4. **Range of Function:**
- Therefore, as \( m \) varies from 0 to 640, \( g \) will vary from 32 gallons (when \( m = 0 \)) to 0 gallons (when \( m = 640 \)).
- Thus, the range of the function in this context is:
\[
g \in [0, 32]
\]
In conclusion, the completed range of values for gasoline remaining \( g \) as a function of miles traveled \( m \) in this context is \( g \in [0, 32] \).