Jim lives three miles east of State College. At noon, he leaves his house and begins to walk due east at a constant speed of 2 miles per hour. Annie lives four miles north of State College. At noon, she leaves her house and begins to bicycle due north at a constant speed of 8 miles per hour. Calculate the rate at which the distance between the two people is changing when it is 1 p. m.

The rate is 13 miles per hour

Step-by-step explanation:

* Lets explain how to solve the problem

- Jim lives three miles east of State College

- At noon, he leaves his house and begins to walk due east at a

constant speed of 2 miles per hour

- Annie lives four miles north of State College

- At noon, she leaves her house and begins to bicycle due north at a

constant speed of 8 miles per hour

- The east is perpendicular to the north

* Lets solve the problem

∵ At noon means 12 p. m

∵ They moved till 1 p. m

∵ Jim walked for 1 hour and Annie bicycled for 1 hour

∵ The rate of Jim is 2 miles per hour

∵ The rate of Annie is 8 miles per hour

- The distance = rate * time

∴ Jim walked = 2 * 1 = 2 miles

∴ Annie bicycled = 8 * 1 = 8 miles

- Lets calculate the distance of Jim from the State College till his

position at 1 p. m

∵ Jim lives three miles east of State College

∴ His distance at 1 p. m = 3 + 2 = 5 miles east

- Lats calculate the distance of Annie from the State College till her

position at 1 p. m

∵ Annie lives four miles north of State College

∴ Her distance at 1 p. m = 4 + 8 = 12 miles North

- Lets find the distance between them at 1 p. m

∵ The north ⊥ east

- Use Pythagoras Theorem to find the distance

∴ The distance = √ (5² + 12) ² = √ (25 + 144) = √169 = 13 miles

- The rate = distance/time

∵ The distance between them is 13 miles in 1 hour

∴ The rate = 13/1 = 13 miles per hour

* The rate is 13 miles per hour