Jane and David are preparing for their annual trip to Yosemite. After careful analysis of route options, and based on traffic patterns, David determines the following:
while driving at a constant pace, they will travel 40 miles in the first hour, 255 miles in the next 3 hours, and 120 miles in the next 2 hours.
I need an equation that will show David’s traveling time vs. distance when i graph it.
c. What does domain mean in the context of this situation? Identify the domain of the function.
d. What does range mean in the context of this situation? Identify the range of the function.
To create an equation that represents David's traveling time vs. distance, we need to understand the pattern in his travel.
From the information provided, David drives 40 miles in the first hour, 255 miles in the next 3 hours, and 120 miles in the following 2 hours. We can consider each segment of his travel as a separate rate of travel.
Let's break down David's travel into three segments:
Segment 1: 40 miles in 1 hour
Segment 2: 255 miles in 3 hours
Segment 3: 120 miles in 2 hours
To find the equation for David's traveling time vs. distance, we will use the formula:
Distance = Rate x Time
For each segment, we can calculate the rate using the given distance and the time.
Segment 1:
Rate = Distance / Time = 40 miles / 1 hour = 40 miles/hour
Segment 2:
Rate = Distance / Time = 255 miles / 3 hours = 85 miles/hour
Segment 3:
Rate = Distance / Time = 120 miles / 2 hours = 60 miles/hour
Now, we can create the equation for David's traveling time vs. distance:
Distance = Rate1 * Time1 + Rate2 * Time2 + Rate3 * Time3
Distance = 40 * Time1 + 85 * Time2 + 60 * Time3
Now, let's address the other questions:
c. The domain represents the set of input values or independent variables of a function. In this situation, the domain would be the possible values for the traveling time. Since David can travel for any positive amount of time, the domain of the function representing David's traveling time vs. distance would be all positive real numbers.
d. The range represents the set of output values or dependent variables of a function. In this situation, the range would be the possible values for the distance covered. As David's travel distance can be any positive number, the range of the function would also be all positive real numbers.
what you really need is to understand that speed = distance/time
they will travel 40 miles in the first hour, 255 miles in the next 3 hours, and 120 miles in the next 2 hours.
So, the speed changes for each trip segment. It is, in sequence,
40/1 = 40 mi/hr
255/3 = 85 mi/hr
120/2 = 60 mi/hr
Now, what do you need to provide?
traveling time vs. distance
That is, let x = the distance
and let y = the time
and recall that time = distance/speed.
So, now we know that y =
x/40 for 0 <= x < 40
1+(x-40)/85 for 40 <= x < 295
4+(x-295)/60 for 295 <= x <= 415
The rest should be doable now.