John drove for two hours at the speed of 50 miles per hour (mph) and another x hours at the speed of 55 mph. If the average speed of the entire journey is 53 mph, which of the following could be used to find x?

Question

Eva Abramson · Accepted Answer

#x = "3 hours"#

The idea here is that you need to work backwards from the definition of the average speed to determine how much time did John spend driving at 55 mph.

The average speed can be thought of as being the ratio between the total distance travelled and the total time needed to travel it.

#"average speed" = "total distance"/"total time"#

At the same time, distance can be expressed as the product between velocity (in this case, speed) and time.

So, if John drove for 2 hours at 50 mph, then he covered a distance of

#d_1 = 50 "miles"/color(red)(cancel(color(black)("h"))) * 2 color(red)(cancel(color(black)("h"))) = "100 miles"#

The second part of the total distance was travelled at 55 mph for x hours, so you can say that

#d_2 = 55"miles"/color(red)(cancel(color(black)("h"))) * x color(red)(cancel(color(black)("h"))) = 55*x " miles"#

The total distance travelled is equal to

#d_"total" = d_1 + d_2#

#d_"total" = 100 + 55x" miles"#

The total time needed was

#t_"total" = 2 + x" hours"#

This means that the average speed is

#bar(v) = color(blue)((100 + 55x)/(2 + x) = 53)# #-># the equation that will lead you to #x#.

Solve this equation for #x# to get

#53 * (2 + x) = 100 + 55x#

#106 + 53x = 100 + 55x#

#2x = 6 => x = 6/2 = color(green)("3 hours")#

Jameson Allen · Answer

undefined To find the value of x, you can use the formula for average speed:

Average speed = Total distance / Total time

In this case, the total time is the sum of the time John drove at 50 mph and the time he drove at 55 mph.

Let's denote the time John drove at 50 mph as 2 hours (since he drove for two hours at that speed).

So, the total time is 2 hours + x hours.

The total distance traveled is the sum of the distances traveled at each speed. Distance = Speed × Time.

Distance traveled at 50 mph = 50 mph × 2 hours = 100 miles

Distance traveled at 55 mph = 55 mph × x hours = 55x miles

Therefore, the total distance is 100 miles + 55x miles.

We know that the average speed for the entire journey is 53 mph.

So, we can set up the equation:

53 mph = (100 miles + 55x miles) / (2 hours + x hours)

To solve for x, you can cross multiply and solve the resulting equation. Once you have the value of x, you can use it to determine the correct option from the given choices.