Quadratic Word Problem?

Question

Bill and Wilma start from the same point and travel along straight roads that are at right angles to each other. Wilma travels 3 miles an hour faster than Bill. At the end of the 2 hours, they are 30 miles apart. How fast is each traveling?

I know this represents a right triangle but I can't figure out how to set up the equation and solve it. Can someone please show me step by step on how to solve this? Thanks.

Answer 1

Bill's speed is x mph.

After 2 hours, Bill has gone 2x, and Wilma has gone 2(x + 3). So you have a right triangle with sides 2x, 2x + 6, and 30.

900 = 4x^2 + 4x^2 + 24x + 36
0 = 8x^2 + 24x - 864 = x^2 + 3x - 108 = (x - 9)(x + 12)

So x = 9 or -12, in this case you want the positive answer, Bill's speed is 9, Wilma's 12. And 324 + 576 = 900, so this checks out.

Answer 2

Bills rate = x
wilmas rate = x+3
after 2 hours bill has gone 2x and
wilma has gone 2(x+3)=2x + 6
that creates the length of the legs of the right triangle
bills distance squared + wilmas distance squared = 30^2
(2x)^2+(2x+6)^2 =900
4x^2 + 4x^2+24x +36 =900

8x^2 +24x +36 =900
8x^2 + 24x - 864 =0
divide by 4 to make the numbers easier to work with
2x^2 +6x - 216=0
(2x+24)(x-9)=0
2x+24=0 and x-9=0
so x =-12 or 9
-12 doesn't make any sense because speed cannot be neg so x = 9

so bill is traveling 9 mph and wilma is going 12 mph
checking so bill went 18 miles and wilma 24
18^2 +24^2 = 324+576=900 and sqrt 900 = 30

Answer 3

Let

x = speed Bill is traveling
x + 3 = speed Wilma is traveling
d = distance they are apart
t = time

Remember distance = rate * time

d = t√{x² + (x + 3)²}
30 = 2√{x² + (x + 3)²}
15 = √{x² + x² +6x + 9} = √{2x² + 6x + 9}
225 = 2x² + 6x + 9
2x² + 6x - 216 = 0
x² + 3x - 108 = 0
(x + 12)(x - 9) = 0
x = 9
The negative solution is rejected.

x + 3 = 12

Bill travels 9 miles/hour
Wilma travels 12 miles/hour

Answer 4

after 2 hrs, bill has gone x miles & wilma x+6
from the puthagorean theorem:
30^2=x^2+(x+6)^2
900=x^2+x^2+12x+36
2x^2+12x-864=0
x^2+6x-432=0
(x+24)(x-18)=0 since x must be >0
x-18=0
x=18
x+6=24
Bill traveled 18 miles in 2 hrs for 9 mph
wilma travelled 24 mi in 2 hrs for 12 mph.