Problem Solving (Word Problem)

Question

A plane, 50 miles away from its destination, is flying toward the airport at an altitude of 30,500 ft. Later, this same plane is 20 miles from the airport and has descended to an altitude of 18,000 ft. How far has the plane flown?

Recall that 1 mile= 5280 ft.

Answer 1

Since we are given a distance traveled horizontally (50 miles to 20 miles) and vertically (30,500 ft to 18,000 ft), in order to find our distance we need to create a right triangle. The vertical (y) and horizontal (x) sides that meet at the right angle are known as vectors; they are the value of our total distance traveled toward the destination x = 50 - 20 miles = 30 miles and total descent y = 30,500 - 18,000 ft = 12,500 ft.

We can see that we have two different units for distance and either one is acceptable given the question. We should always try to make units smaller so to make our lives easier so we should convert the descent to miles. in order to convert, we multiply y by the ratio of miles over feet;

y = 12,500 ft x (1 mile / 5280 ft) = 2.367 miles (rounded to three decimal places for accuracy)

Now we have two sides of a triangle in the same units (y = 2.367 miles ; x = 30 miles). If we picture a right triangle, we know the vertical and horizontal sides (x,y) are also known as a and b, as in the pythagorean theorem. Using that, we can now find the distance traveled in a straight line.

2.367² + 30² = c²

5.603 + 900 = c²

√905.63 = c

c ≈ 30.09 miles

For clarification: We can just as easily convert everything to feet and go that route.

30 miles x (5280 ft / 1 mile) = 158400 ft

But we can now see that those numbers, which we now need to square will be unnecessarily large and can lead to errors in computing.

12,500² + 158,400² = c²

156,250,000 + 25,090,560,000 = c²

√25,246,810,000 = c

c ≈ 158,892.45 ft

To check: 158,892.45 ft x (1mile / 5280 ft) = 30.09 miles

Both ways will get the same answer; but given the choice, its better to go with the smaller number.