Hard SAT Math question involving rates, distance, and time

Question

A car is traveling at x feet per second. The driver sees a red light ahead, and after 1.5 seconds reaction time, the driver applies the brake. After the brake is applied, the car takes x/24 seconds to stop, during which time the average speed of the car is x/2 feet per second. If the car travels 165 feet from the time the driver saw the red light to the time it comes to a complete stop, which of the following equations can be used to find the value of x?

Answer 1

We use the distance formula:

Distance = rate * time

Here is a visual representation of the problem:

               D = rt                              D = rt

              D = x(1.5)                    D = (x/2)*(x/24)

   I----------------------------------I---------------------------------I

Driver                              Driver                                   Car

Sees                                applies                                 Stops

Red                                  brake

Light

   I<---------------------- 165 feet ------------------------------->I

Since they told us the total distance is 165 feet, we add to get this equation:

x(1.5) + (x/2)*(x/24) = 165

Now we can rewrite this using algebraic methods:

1.5x + x²/48 = 165

Subtract 165 from both sides:

1.5x + x²/48 = 165

              -165     -165

-------------------------------

1.5x + x²/48 – 165 =0



Multiply both sides by 48:

72x + x² – 7920 = 0

Reorder:

x² + 72x − 7920 = 0   Answer

Answer 2

After seeing the red light, the driver drives a total of 1.5 seconds (before applying the brake) + x/24 seconds (after applying the brake). Then the car stops. Distance = velocity * time. Before applying the brakes, the car travels (x feet/sec) times 1.5 seconds, or 1.5x. The distance after applying the brakes is (x/2 feet/sec) times (x/24 seconds), or x/2 * x/24 = x²/48. Adding these two distances (before and after applying the brakes) is a total of 165 feet. So, the equation is 1.5x + x²/48 = 165. Subtracting 165 from each side and rearranging terms, the quadratic equation would be x²/48 + 1.5x - 165 = 0. This would answer the problem stated Going a step further and solving for x, use the quadratic formula x = [-b +/- square root of (b²-4ac)] / 2a. First multiply all terms by 48 to make the solution simpler: x²+72x-7920=0. Solve for x by plugging in 1 for a, 72 for b, and 7920 for c. The answer is 60 feet per second. To double check and ensure this answer makes sense, before braking the car travels 60 * 1.5 = 90 feet. After braking the car travels x/2 or 60/2 = 30 feet per second for x/24 or 60/24 = 2.5 seconds, or a total of 30*2.5 = 75 feet. This confirms the total distance travelled of 90 + 75 = 165 feet.

Answer 3

orig speed=x'/sec

reaction=1.5sec

stopping time=x/24sec

auto avg speed=x/2ftsec

165' total

x/2*t=165