Question 1175757
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.