Stephen drove at a constant speed from Town X to Y at 9 A.M.yesterday. Half an hour later Cole drove from Town X to Y at A constant speed that was 30 km/h faster than Stephens. By 9:30 A.m, Stephanie had already traveled 40 km. Cole caught up with Stephen at Town , arriving at the same time as Stephen. At what speed was Stephen driving? What was the distance between the 2 towns?
At 8.30 am, Angela left City A travelling towards City B at a constant speed. One hour later, Caitlin started travelling from City A on the same road. Caitlin overtook Angela at 12.30 pm. The speed which Caitlin was travelling at was 18km/h faster than Angela. What was Angela’s speed?
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Let's break down the information we have to find the speed at which Stephen was driving and the distance between the two towns.
1. Stephanie traveled 40 km by 9:30 A.M.
2. Cole caught up with Stephen at Town Y, arriving at the same time as Stephen.
To find Stephen's speed, we need to use the fact that Stephanie traveled 40 km in half an hour.
Let S be Stephen's speed in km/h, and C be Cole's speed in km/h.
Since Stephanie traveled 40 km in half an hour, her speed can be calculated by dividing the distance by the time:
Stephanie's speed = 40 km / 0.5 hours = 80 km/h
We know that Cole's speed is 30 km/h faster than Stephen's, so Cole's speed can be expressed as:
Cole's speed = S + 30 km/h
Since both Cole and Stephen arrived at Town Y at the same time, we can assume that they traveled for the same amount of time.
Now, let's look at the time they both spent traveling from 9:30 A.M. until they reached Town Y.
Since Stephen left at 9 A.M., he spent 0.5 hours traveling.
Since Cole left half an hour later, he also spent 0.5 hours traveling.
Let's use the formula: Distance = Speed × Time to find the distance between the two towns.
Distance covered by Stephen = S km/h × 0.5 hours
Distance covered by Cole = (S + 30) km/h × 0.5 hours
Since Cole caught up with Stephen, their distances covered must be the same.
S × 0.5 = (S + 30) × 0.5
S = S + 30
By simplifying the equation, we find that:
0 = 30
This is not possible, so we made an error in our calculations or assumptions. Please check the given information again.
To solve this problem, let's break it down step by step.
Step 1: Determine the relationship between the speeds of Stephen and Cole.
Let's assume the speed at which Stephen was driving is "S" km/h. According to the problem, Cole was driving at a speed that was 30 km/h faster than Stephen, so Cole's speed would be "S + 30" km/h.
Step 2: Find the time it took for Cole to catch up with Stephen.
Since Stephanie had already traveled 40 km by 9:30 A.M, this means that she traveled for 0.5 hours. Let's consider this time to be "t" hours.
Step 3: Set up an equation to find the distance between the two towns.
Let's assume the distance between Town X and Town Y is "d" km. Both Stephen and Cole covered this distance, so we can set up an equation based on their speeds and the time it took for Cole to catch up.
For Stephen: distance = speed * time
40 = S * t -----(1)
For Cole: distance = speed * time
d = (S + 30) * (t + 0.5) -----(2)
Step 4: Solve the equations to find the values of S and d.
From equation (1), we can find the value of t:
t = 40/S -----(3)
Substitute the value of t from equation (3) into equation (2), and solve for d:
d = (S + 30) * (40/S + 0.5)
d = 40 + 0.5S + 1200/S + 15 -----(4)
Combining like terms:
d = 0.5S + 1200/S + 55 -----(5)
Now, we need to differentiate equation (5) with respect to S and set it equal to zero to find the value of S where the distance is minimized.
dS/dS = 0.5 - 1200/S^2 = 0
Solving for S:
0.5 - 1200/S^2 = 0
1200/S^2 = 0.5
S^2 = 2400
S = √2400
S ≈ 48.99
Since the problem asks for the speed, Stephen was driving approximately at 49 km/h.
To find the distance between the two towns, substitute the value of S into equation (5):
d = 0.5(49) + 1200/(49) + 55
d ≈ 24.5 + 24.49 + 55
d ≈ 103.99
Therefore, the distance between Town X and Town Y is approximately 104 km.