These are a couple of rate related problems. One of them I do not know how to do. What is the diagram for the first one?
1.
A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft away from the pole?
Answer Attempt:
I'm a bit stuck on how to do this. I'm a bit thrown off on how to handle the parts that the man is 6 feet tall and 40 ft away but we're trying to find the rate at which the tip of his shadow is moving. Any hints?
2.
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?
Answer Attempt:
So let's say w = distance west, s = distance south, and d = the distance between the two points.
$$w^2 + s^2 = d^2$$ $$2w\frac{dw}{dt} + 2s\frac{ds}{dt} = 2d\frac{dd}{dt}$$ $$2 \cdot 50mi \cdot 25\frac{mi}{h} + 240mi \cdot 60\frac{mi}{hr} = 260mi \frac{dd}{dt}$$ $$\frac{16900mi}{260mi} \cdot \frac{mi}{h} = \frac{dd}{dt}$$ $$65\frac{mi}{hr} = \frac{dd}{dt}$$