A cyclist bikes at a constant speed for 18 miles. He then returns home at the same speed, but takes a different route. His return trip takes one hour longer and is 23 miles. Find his speed
let the speed be x mph
time for first trip = 18/x
time for 2nd trip = 23/x
23/x - 18/x = 1
5/x = 1
x = 5
his speed is 5 mph
check:
time for 1st trip = 18/5 = 3.6 hrs
time for 2nd trip = 23/5 = 4.6 hrs, which is 1 hour more
Let X be distance per hour,
Ist trip=18miles =18/X
2nd trip=23miles =23/X,=1hr longer
speed,S=2nd trip-1st trip
X=23-18
X=5mph.
checkings
23/5 -18/5 =4.6-3.6=1hr
Let's assume the cyclist's speed is "x" miles per hour.
Step 1: Calculate the time taken for the first trip.
Distance = Speed * Time
Time = Distance / Speed
Given that the distance for the first trip is 18 miles and the speed is "x", the time for the first trip is 18/x hours.
Step 2: Calculate the time taken for the return trip.
Given that the distance for the return trip is 23 miles and the speed is also "x", the time for the return trip is 23/x hours.
Step 3: Determine the time difference.
The return trip took one hour longer than the first trip. So, we have the equation:
23/x = 18/x + 1
Step 4: Solve the equation.
To solve the equation, we need to cross multiply and then simplify:
23x = (18 + x) * (x)
23x = 18x + x^2
Step 5: Rearrange the equation.
Rearranging the equation, we have:
x^2 + 18x - 23x = 0
x^2 - 5x = 0
Step 6: Factor the equation.
Factoring the equation, we have:
x(x - 5) = 0
Step 7: Solve for "x".
Setting each factor equal to zero, we have two possible solutions for "x":
x = 0 or x - 5 = 0
Step 8: Determine the valid solution.
Since a speed of 0 miles per hour is not possible for a cyclist, we can discard the x = 0 solution. Therefore, the valid solution is x - 5 = 0.
Step 9: Calculate the speed.
To calculate the speed, substitute the valid solution back into the equation:
x - 5 = 0
x = 5
So, the cyclist's speed is 5 miles per hour.
To find the cyclist's speed, we need to first determine the time it took for the initial 18-mile trip and then use that information to find the speed.
Let's start by finding the time it took for the initial 18-mile trip. We can use the formula Time = Distance / Speed.
Let's call the speed of the cyclist "x" (in miles per hour).
For the initial 18-mile trip:
Time = 18 miles / x miles per hour
Now, let's find the time it took for the return trip, which was 23 miles. We're given that the return trip took one hour longer than the initial trip.
Time for the return trip = Time for the initial trip + 1 hour
Let's substitute the value of Time for the initial trip into this equation:
Time for the return trip = (18 miles / x miles per hour) + 1 hour
Since the speed for both trips was the same (x miles per hour), we can set up the equation:
23 miles / x miles per hour = (18 miles / x miles per hour) + 1 hour
Now, we can solve this equation to find the value of x, the cyclist's speed.
Let's multiply both sides of the equation by x to eliminate the denominator:
23 miles = 18 miles + x hours
Subtract 18 miles from both sides of the equation:
5 miles = x hours
Finally, divide both sides of the equation by x to solve for x:
x = 5 miles / 1 hour
Therefore, the cyclist's speed is 5 miles per hour.