quadratic equations

Question

The formula d=.054x^2+.058x describes the distance in feet D that it takes to stop a vehicletoraveling x miles per hour on dry pavement. how fast can you drive if you wish to be able to stop within 65 ft?

Answer 1

d(x) = .054x² + .058x

 

We want to find x when the distance it takes to stop is 65 ft

 

.054x² + .058x = 65

.054x² + .058x - 65 = 0

 

This is a quadratic equation with a = .054, b=.058, c = -65

 

x = -.058/2(.054) ±√(.058² -4(.054)(-65))/2(.054)

   = -.058/.108 ±√14.043364/.108

   = -.058/.108 +√14.043364/.108    or  -.058/.108 - √14.043364/.108

   = 34.16 mph  or  -35.24 mph

   Disregard the negative answer..

   x ≅ 34.16 miles/hour

 

 

 

 

 

Answer 2

Plug in d = 65 into the formula. subtract 65 from both sides to obtain 0 = 0.054x^2+0.058x-65. Now solve using the quadratic formula with a=0.054 b=0.058 and c = -65. This yields x =34.2 and x=-35.2.  -35.2 must be discarded because it is a negative speed. Therefore, 34.2 is the proper solution.

Answer 3

 

Substitute 65 ft for d. Then subtract 65 from both sides:

 

This becomes a quadratic function of the form ax2 + bx + c = 0 with   a=0.054    b=0.058    c= -65       

The roots are found using the quadratic formula. Solve by hand if your teacher requires, but there are plenty of websites to solve these by inputting a,b, and c:  search for "quadratic equation solver".

 

 

The roots of the equation are +34.2 miles per hour (and -35.2, which is thrown out since it is a speed < 0 )

 

Always review the answer to see if it sounds reasonable. Stopping in 65 feet is a quick stop, so a speed of around 34 miles/hr, similar to a speed limit in a residential area, makes perfect sense here.