The tour de France is a bicycle race that begins in Ireland but is primarily a race through France. On each day, racers cover a certain distance depending on the steepness of the terrain. Suppose that one particulAr day the racers must comp,tee 210 mi. One cyclist traveling 10 mph faster than the second cyclist, covers this distance in 2.4 h less time than the second cyclist. Find the rate of the first cyclist.

Rate of 2nd cyclist = X mi/h.

Rate of 1st cyclist = (x+10) mi/h.

Eq1: d2 = Xt = 210mi.
X = 210 / t.

Eq2: d1 = (X+10)(t-2.4) = 210mi.

In Eq2, substitute 210/t for X:
(210/t+10)(t-2.4) = 210,
210-504/t+10t-24 = 210,
-504/t+10t-24 = 210-210,
-504/t+10t-24 = 0,
Multiply both sides by t:
10t^2-24t-504 = 0,
Divide both sides by 2:
5t^2-12t-252 = 0.
Solve using Quadratic Formula and get:
t = 8.4, and -6. Select positive t.

t = 8.4h.

X = 210/t = 210 / 8.4 = 25mi/h.

X+10 = 25+10 = 35mi/h = Rate of 1st cyclist.

To solve this problem, we need to set up a system of equations based on the given information.

Let's say the speed of the second cyclist is x mph. According to the problem, the first cyclist is traveling 10 mph faster, so his speed can be represented as (x + 10) mph.

Now, we can calculate the time it takes for each cyclist to cover the distance of 210 miles.

For the second cyclist, who is traveling at a speed of x mph, the time taken can be calculated using the formula: time = distance / speed.
So for the second cyclist, the time taken is 210 / x hours.

For the first cyclist, who is traveling at a speed of (x + 10) mph, the time taken is 210 / (x + 10) hours.

According to the problem, the first cyclist completes 2.4 hours less than the second cyclist, so we can write the equation:

210 / x - 210 / (x + 10) = 2.4

To solve this equation, we can multiply both sides by x(x + 10) to eliminate the denominators:

210(x + 10) - 210x = 2.4x(x + 10)

Simplifying this equation gives:

210x + 2100 - 210x = 2.4x^2 + 24x

Now, rearrange the equation:

2.4x^2 + 24x - 210x - 2100 = 0

Combine like terms:

2.4x^2 - 186x - 2100 = 0

Now we have a quadratic equation. To solve it, we can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, a = 2.4, b = -186, and c = -2100.

After solving this quadratic equation, we find two possible values for x (the speed of the second cyclist). However, we know that the first cyclist is traveling 10 mph faster than the second cyclist, so we need to choose the positive solution for x.

Once we have the value for x, we can calculate the speed of the first cyclist by adding 10 to x.

So, by solving the quadratic equation and finding the positive root for x, we can determine the rate of the first cyclist.