Answer :
Let's approach this step-by-step to understand the question and find the correct solution.
### Step 1: Identify the Independent and Dependent Variables
In this situation, we need to identify the two main variables involved:
- Independent Variable: The depth of snow, [tex]\( d \)[/tex] (in inches), because it is the variable that we consider changing.
- Dependent Variable: The speed of the snowplow, [tex]\( s \)[/tex] (in miles per hour), because it depends on the depth of the snow.
Hence, the correct choice is:
OB. The independent variable is the depth of snow ([tex]\( d \)[/tex]), in inches, and the dependent variable is speed ([tex]\( s \)[/tex]), in miles per hour.
### Step 2: Formulate the Linear Function
The problem tells us that the snowplow has a maximum speed of 30 miles per hour on a dry highway and that this speed decreases by 1.7 miles per hour for every inch of snow on the highway.
We can express this relationship with the following linear equation:
[tex]\[ s = s_{\text{max}} - k \cdot d \][/tex]
where:
- [tex]\( s \)[/tex] is the speed of the snowplow (dependent variable)
- [tex]\( s_{\text{max}} \)[/tex] is the maximum speed of the snowplow on a dry highway (30 mph)
- [tex]\( k \)[/tex] is the rate at which the speed decreases per inch of snow (1.7 mph per inch)
- [tex]\( d \)[/tex] is the depth of snow (independent variable)
So, the linear equation becomes:
[tex]\[ s = 30 - 1.7 \cdot d \][/tex]
### Step 3: Determine When the Snowplow Will Be Unable to Move
The snowplow will be unable to move when its speed drops to 0 mph. We need to find the depth of snow [tex]\( d \)[/tex] at which [tex]\( s = 0 \)[/tex]. Therefore, we set the equation to 0:
[tex]\[ 0 = 30 - 1.7 \cdot d \][/tex]
Now, solve for [tex]\( d \)[/tex]:
[tex]\[ 1.7 \cdot d = 30 \][/tex]
[tex]\[ d = \frac{30}{1.7} \][/tex]
After performing the division:
[tex]\[ d \approx 17.647058823529413 \][/tex]
When rounded to the nearest inch:
[tex]\[ d \approx 18 \][/tex]
Thus, the snowplow will be unable to move when the snow is approximately 18 inches deep.
### Conclusion
So, the correct choice to complete the statement is:
OB. The independent variable is the depth of snow ([tex]\( d \)[/tex]), in inches, and the dependent variable is speed ([tex]\( s \)[/tex]), in miles per hour. The linear function that models this situation is:
[tex]\[ \boxed{s = 30 - 1.7 \cdot d} \][/tex]
The snowplow will be unable to move when the snow is [tex]\( \boxed{18} \)[/tex] inch(es) deep.
### Reasonableness of the Linear Model
As for the reasonableness of using a linear model for this situation:
OD. The linear model is reasonable because the plowing rate is always constant for any snowplow. Given the constraints described, it is logical to assume that the reduction in speed per inch of snow is consistent up to the point where the snow depth is too much for the plow to handle.
Therefore, the most reasonable answer for the linear model is:
[tex]\[ \boxed{\text{OD}} \][/tex]
### Step 1: Identify the Independent and Dependent Variables
In this situation, we need to identify the two main variables involved:
- Independent Variable: The depth of snow, [tex]\( d \)[/tex] (in inches), because it is the variable that we consider changing.
- Dependent Variable: The speed of the snowplow, [tex]\( s \)[/tex] (in miles per hour), because it depends on the depth of the snow.
Hence, the correct choice is:
OB. The independent variable is the depth of snow ([tex]\( d \)[/tex]), in inches, and the dependent variable is speed ([tex]\( s \)[/tex]), in miles per hour.
### Step 2: Formulate the Linear Function
The problem tells us that the snowplow has a maximum speed of 30 miles per hour on a dry highway and that this speed decreases by 1.7 miles per hour for every inch of snow on the highway.
We can express this relationship with the following linear equation:
[tex]\[ s = s_{\text{max}} - k \cdot d \][/tex]
where:
- [tex]\( s \)[/tex] is the speed of the snowplow (dependent variable)
- [tex]\( s_{\text{max}} \)[/tex] is the maximum speed of the snowplow on a dry highway (30 mph)
- [tex]\( k \)[/tex] is the rate at which the speed decreases per inch of snow (1.7 mph per inch)
- [tex]\( d \)[/tex] is the depth of snow (independent variable)
So, the linear equation becomes:
[tex]\[ s = 30 - 1.7 \cdot d \][/tex]
### Step 3: Determine When the Snowplow Will Be Unable to Move
The snowplow will be unable to move when its speed drops to 0 mph. We need to find the depth of snow [tex]\( d \)[/tex] at which [tex]\( s = 0 \)[/tex]. Therefore, we set the equation to 0:
[tex]\[ 0 = 30 - 1.7 \cdot d \][/tex]
Now, solve for [tex]\( d \)[/tex]:
[tex]\[ 1.7 \cdot d = 30 \][/tex]
[tex]\[ d = \frac{30}{1.7} \][/tex]
After performing the division:
[tex]\[ d \approx 17.647058823529413 \][/tex]
When rounded to the nearest inch:
[tex]\[ d \approx 18 \][/tex]
Thus, the snowplow will be unable to move when the snow is approximately 18 inches deep.
### Conclusion
So, the correct choice to complete the statement is:
OB. The independent variable is the depth of snow ([tex]\( d \)[/tex]), in inches, and the dependent variable is speed ([tex]\( s \)[/tex]), in miles per hour. The linear function that models this situation is:
[tex]\[ \boxed{s = 30 - 1.7 \cdot d} \][/tex]
The snowplow will be unable to move when the snow is [tex]\( \boxed{18} \)[/tex] inch(es) deep.
### Reasonableness of the Linear Model
As for the reasonableness of using a linear model for this situation:
OD. The linear model is reasonable because the plowing rate is always constant for any snowplow. Given the constraints described, it is logical to assume that the reduction in speed per inch of snow is consistent up to the point where the snow depth is too much for the plow to handle.
Therefore, the most reasonable answer for the linear model is:
[tex]\[ \boxed{\text{OD}} \][/tex]
