Question

Tommy and Zach are starting out at the same position. Tommy runs north at 5 miles per hour and Zach starts to run east 2 hours later at the rate of 8 miles per hour. How long until Tommy and Zach are 16 miles apart? Round to the nearest tenth if necessary.

Answer 1

Alright, so that means sqrt(x^2+y^2) is the distance apart using the Pythagorean Theorem, using x=Tommy's distance from the start and y=Zach's distance. Since sqrt(x^2+y^2)=16, that means that x^2+y^2=256 by squaring both sides. Tommy's length can be used as 5z (z=hours), and Zach's length is 8z-16 (since 16 miles is what he missed by starting 2 hours later). We could then write it as (5z)^2+(8z-16)^2=256, and then by multiplying it out we get
25z^2+64z^2−256z+256=256. Subtracting 256 from both sides, we get
25z^2+64z^2−256z=0 and dividing by z we get 89z-256=0. Adding 256 to both sides, we get 89z=256 and z=256/89 and rounded to 2.9 hours

Answer 2

After 2.90 hours Tommy and Zach will be 16 mils apart

Explanation:

Let Tommy and Zach run for t hours until they are 16 miles apart.

Tommy runs North with the speed of 5 miles per hour.

So, distance covered by Tommy in t hours = 5t miles

Zach runs East with the speed of 8 miles per hour.

Since Zach started 2 hours late so he will run for (t - 2) hours.

Distance covered in (t - 2) hours with the speed of 8 miles per hour = 8(t - 2) miles

Now both are 16 miles apart so we will apply Pythagoras theorem in the given triangle.

[8(t - 2)]²+ (5t)² = 16²

64(t - 2)² + 25t² = 256

64(t² + 4 - 4t) + 25t² = 256

64t² + 256 - 256t + 25t² = 256

89t² -256t + 256 - 256 = 0

89t² - 256t = 0

t( 89t - 256) = 0

(89t - 256) = 0

89t = 256

t =
`(256)/(89)=2.876`

t = 2.90 hours

After 2.90 hours Tommy and Zach will be 16 mils apart.