13. On a camping trip, you decide to hang your backpack from a rope stretched
between two trees, so bears can't steal your food while you sleep. The trees are 20
feet apart, and the rope is tied to each tree at a spot 12 feet above the ground. When
you hang your backpack on the rope, the weight of the pack pulls the rope down a
little each side of the rope now makes an angle of 15 degrees with the horizontal.
That night, a bear comes into your camp and tries to get your backpack with the
food in it. Standing on its back legs, the bear can reach 8 feet into the air. Is your
stuff safe? (Your backpack is two feet long.)

The sag in the rope can be found using the tangent relation. The side opposite the angle is the amount of sag in the middle of the rope. The side adjacent to the angle is half the distance between the trees, (10 ft). So the amount of sag is found from ...

Tan = Opposite / Adjacent

tan(15°) = sag/(10 ft)

sag = (10 ft)tan(15°) ≈ 2.68 feet

The 2 ft backpack will have its lowest extent a total of ...

12 ft -2.68 ft -2 ft = 7.32 ft

off the ground. Since the bear can reach up to 8 feet high, it can easily take a swipe at your backpack.

Your stuff is not safe.