SOLUTION: Your group has been commissioned to design a new roller coaster for an amusement park. The roller coaster should be more spectacular than all existing roller coasters, and in parti

Question 1175757: Your group has been commissioned to design a new roller coaster for an amusement park. The roller coaster should be more spectacular than all existing roller coasters, and in particular it should include a vertical loop. In order to stay on the track through the loop, the cars must travel at a speed given (in miles per hour) by
v = 90r where r is the radius of the loop in feet.
1.a) The location of the amusement park is somewhere in Bukidnon and you are planning to have a total height of 160 feet. Find the speed of the car that it must reach on the roller coaster. (Write your answer in exact value form using the concept of simplifying radicals.) 2 points
Note: Because the vertical loops used in roller coasters are not perfect circles, the total height of the loop is about 2.5 times its radius.

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's break down this roller coaster design problem.
**1. Understand the Relationship Between Height and Radius**
* The total height of the loop is 2.5 times its radius (h = 2.5r).
* We're given the total height (h = 160 feet).
**2. Calculate the Radius (r)**
* 160 = 2.5r
* r = 160 / 2.5
* r = 64 feet
**3. Calculate the Required Speed (v)**
* We're given the formula: v = √(90r) (in miles per hour)
* Substitute the radius (r = 64 feet):
* v = √(90 * 64)
* v = √(5760)
**4. Simplify the Radical**
* To simplify √(5760), find the prime factorization of 5760:
* 5760 = 2^7 * 3^2 * 5
* Rewrite the radical:
* v = √(2^7 * 3^2 * 5)
* v = √(2^6 * 2 * 3^2 * 5)
* v = √(2^6 * 3^2) * √(2 * 5)
* v = 2^3 * 3 * √(10)
* v = 8 * 3 * √(10)
* v = 24√(10)
**Answer:**
1. a) The speed the car must reach is 24√(10) miles per hour.

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