Two landscapers must mow a rectangular lawn that measures 100 feet by 200 feet. Each wants to mow no more than half of the lawn.
The first starts by mowing around the outside of the lawn. How wide a strip must the first landscaper mow on each of the four sides in order to mow no more than half of the lawn? The mower has a 24-inch cut. Approximate the required number of trips around the lawn.
The total area of the complete lawn is (100-ft x 200-ft) = 20,000 ft². One half of the lawn is 10,000 ft². That's the limit that the first man must be careful not to exceed, lest he blindly mow a couple of blades more than his partner does, and become the laughing stock of the whole company when the word gets around. 10,000 ft² ... no mas !
When you think about it ... massage it and roll it around in your mind's eye, and then soon give up and make yourself a sketch ... you realize that if he starts along the length of the field, then with a 2-ft cut, the lengths of the strips he cuts will line up like this:
These are the lengths of each strip. They're 2-ft wide, so the area of each one is (2 x the length).
I expected to be able to see a pattern developing, but my brain cells are too fatigued and I don't see it. So I'll just keep going for another lap, then add up all the areas and see how close he is:
So far, after four laps around the yard, the 16 lengths add up to 2,272-ft, for a total area of 4,544-ft². If I kept this up, I'd need to do at least four more laps ... probably more, because they're getting smaller all the time, so each lap contributes less area than the last one did.
Hey ! Maybe that's the key to the approximate pattern !
Each lap around the yard mows a 2-ft strip along the length ... twice ... and a 2-ft strip along the width ... twice. (Approximately.) So the area that gets mowed around each lap is (2-ft) x (the perimeter of the rectangle), (approximately), and then the NEXT lap is a rectangle with 4-ft less length and 4-ft less width.
So now we have rectangles measuring
(200 x 100), (196 x 96), (192 x 92), (188 x 88), (184 x 84) ... etc.
and the areas of their rectangular strips are 1200-ft², 1168-ft², 1136-ft², 1104-ft², 1072-ft² ... etc.
==> I see that the areas are decreasing by 32-ft² each lap. So the next few laps are 1040-ft², 1008-ft², 976-ft², 944-ft², 912-ft² ... etc.
How much area do we have now:
After 9 laps, Area = 9,648-ft² After 10 laps, Area = 10,560-ft².
And there you are ... Somewhere during the 10th lap, he'll need to stop and call the company surveyor, to come out, measure up, walk in front of the mower, and put down a yellow chalk-line exactly where the total becomes 10,000-ft².
There must still be an easier way to do it. For now, however, I'll leave it there, and go with my answer of: During the 10th lap.
This problem can be solved by using different methods. I will use vectors since it's the simplest way in which we can solve it. This can be solved by using related rates of change though.
First, we start by drawing a diagram with the velocity vectors.
A= velocity of the first person
B= velocity of the second person
C= velocity in which they are moving away from each other.
Since there is no acceleration in the problem, we can suppose we are talking about constant speeds, so the velocity at which they are moving away from each other will always remain constant. (It doesn't matter what time it is, the velocity will always be the same)
Having said this we can solve this problem by using the components, by using law of cosines or graphically. I will use law of cosines. The idea is to find the length of side c.
Law of cosines:
so we can solve the formula for C so we get:
and now we can substitute the values we know:
if we want an exact answer, then that will be the exact answer, which approximates to:
