1. Total distance traveled varies directly with the amount of time spent traveling. If traveling 4.5 hours results in 180 miles traveled, write the direct linear variation equation. Then determine how long it would take you to travel 320 miles, assuming you traveled at the same rate.
a. What are the variables (the input and output)?
b. Solve for k.
c. Write the direct variation equation.
d. How long will it take you to travel 320 miles? Please write your answer in a complete sentence.
a) depends on how you define your variables
e.g. let the time be t hours, and the distance be d miles
then :
d = kt
plug in d= 180 and t = 4.5 to solve for k
rewrite the equation , now that you know k
sub in d = 320 and solve for t
a. The input variable is the time spent traveling, and the output variable is the total distance traveled.
b. To solve for k, we'll use the given information that traveling 4.5 hours results in 180 miles traveled. We can set up the equation:
distance = k * time
Given that distance = 180 miles and time = 4.5 hours, we substitute these values into the equation:
180 = k * 4.5
To solve for k, divide both sides of the equation by 4.5:
k = 180 / 4.5
Simplifying the right side:
k = 40
So, k equals 40.
c. The direct variation equation is:
distance = k * time
Now that we know k is 40, the equation becomes:
distance = 40 * time
d. To calculate how long it would take to travel 320 miles, we can use the direct variation equation we found in part c. We'll substitute distance = 320 miles into the equation:
320 = 40 * time
To solve for time, divide both sides of the equation by 40:
time = 320 / 40
Simplifying the right side:
time = 8
Therefore, it would take 8 hours to travel 320 miles.