Kathy drove from Dry Creek to Cactus, a distance of 250km. She increased her speed by 10km/h for the 360km trip from Cactus to spur. If the total trip took 1 hours, what was her speed from Dry creek to Cactus.
To solve this problem, we can use the formula for average speed:
Average Speed = Total Distance / Total Time
Let's break down the information given in the question:
- Kathy's total trip consisted of two parts: from Dry Creek to Cactus and from Cactus to Spur.
- The distance from Dry Creek to Cactus is 250 km.
- The distance from Cactus to Spur is 360 km.
- Kathy increased her speed by 10 km/h for the second part of her trip.
Let's calculate the total time it took for Kathy to complete her trip:
Total Time = Time from Dry Creek to Cactus + Time from Cactus to Spur
We know that the total trip took 1 hour, so:
1 hour = Time from Dry Creek to Cactus + Time from Cactus to Spur
Now, let's determine the time it took Kathy to travel from Cactus to Spur:
Time from Cactus to Spur = Distance from Cactus to Spur / Speed from Cactus to Spur
Since Kathy increased her speed by 10 km/h for this part of the trip, her speed from Cactus to Spur is: Speed from Dry Creek to Cactus + 10 km/h
Now we have everything we need to calculate the speed from Dry Creek to Cactus:
Average Speed = Total Distance / Total Time
But we know that Average Speed = Speed from Dry Creek to Cactus
So, we can rewrite the formula as:
Speed from Dry Creek to Cactus = Total Distance / Total Time
Speed from Dry Creek to Cactus = 250 km / (1 hour - Time from Cactus to Spur)
Now, let's substitute the value for Time from Cactus to Spur:
Speed from Dry Creek to Cactus = 250 km / (1 hour - Distance from Cactus to Spur / Speed from Cactus to Spur)
Since we already know the value for the distance from Cactus to Spur (360 km) and the speed from Cactus to Spur (Speed from Dry Creek to Cactus + 10 km/h), we can substitute those values:
Speed from Dry Creek to Cactus = 250 km / (1 hour - 360 km / (Speed from Dry Creek to Cactus + 10 km/h))
Now we have an equation with one unknown variable (Speed from Dry Creek to Cactus). To solve for it, we can use algebraic methods or approximation methods such as graphical or numerical methods.
To find Kathy's speed from Dry Creek to Cactus, we can use the formula for speed:
Speed = Distance / Time
Let's first calculate the time it took for Kathy to drive from Cactus to Spur. We know that the distance from Cactus to Spur is 360 km. Kathy increased her speed by 10 km/h for this leg of the trip. Let's assume her speed from Dry Creek to Cactus was "x" km/h. Therefore, her speed from Cactus to Spur would be "x + 10" km/h.
The time for the Cactus to Spur trip can be calculated using the equation:
Time = Distance / Speed
Plugging in the values, we get:
360 km / (x + 10) km/h = Time
Now, let's calculate the time it took for Kathy to drive from Dry Creek to Cactus. We know that the distance from Dry Creek to Cactus is 250 km. Her speed for this leg of the trip is "x" km/h. The time can be calculated using the same equation:
Time = Distance / Speed
Plugging in the values, we get:
250 km / x km/h = Time
We are given that the total trip took 1 hour. So, the sum of the individual times should be equal to 1 hour:
Time (Dry Creek to Cactus) + Time (Cactus to Spur) = 1
250 km / x km/h + 360 km / (x + 10) km/h = 1
To solve the equation, we can cross-multiply and solve for "x":
250(x + 10) + 360x = x(x + 10) [Cross-multiplication]
Now, expand and simplify the equation:
250x + 2500 + 360x = x^2 + 10x
Combine like terms:
610x + 2500 = x^2 + 10x
Rearrange the equation:
x^2 + 10x - 610x - 2500 = 0
Simplify:
x^2 - 600x - 2500 = 0
Next, we can use the quadratic formula to find the value of "x". The quadratic formula is:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation, the values are:
a = 1
b = -600
c = -2500
Substituting these values into the quadratic formula, we get:
x = (-(-600) ± √((-600)² - 4 * 1 * (-2500))) / (2 * 1)
Simplifying further:
x = (600 ± √(360000 + 10000)) / 2
x = (600 ± √(370000)) / 2
Finally, we can calculate the two possible values of "x" and determine Kathy's speed from Dry Creek to Cactus.
Assuming Kathy is a Formula-1 racer, driving 510km in 1 hour,
time = distance/speed
add up the times:
250/x + 360/(x+10) = 1
x = 604.13 km/hr - a new world land speed record!!!!
If 1 hour is not the correct time, then fix it and solve the new equation.