A hang glider is standing at the top of a 3,000 foot cliff. The hang glider jumps off and begins to descend at a constant rate of 50 feet per second, how fast is the area of the triangle formed by the cliff, the hang glider, and the ground changing at the instant when the hang glider is 2,000 feet above the ground?
You need to know the slope of the glider's descent. If he drops straight down, there is no triangle. That is, you need to know his horizontal speed. The vertical drop of 50 ft/s is no good by itself.
To find the rate at which the area of the triangle is changing, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, the base of the triangle remains constant at 3,000 feet because the cliff is not moving. However, the height of the triangle is changing as the hang glider descends.
Let's define the height of the triangle as h, and the rate at which the height is changing as dh/dt (where t represents time). We need to find dh/dt when the hang glider is 2,000 feet above the ground.
To do this, we can use similar triangles. The large triangle formed by the cliff, the hang glider, and the ground is similar to the smaller triangle formed by the cliff, the hang glider (2,000 feet above the ground), and the ground. This means that the ratio of their heights is equal to the ratio of their bases:
h / 3000 = (h - 2000) / 3000
To solve for h, we can cross-multiply:
h * 3000 = (h - 2000) * 3000
3000h = 3000h - 6000000
Simplifying:
6000000 = 3000h
h = 2000
Now that we know h = 2000, we can differentiate both sides of the equation with respect to time (t) to find dh/dt:
d(h) / dt = d(2000) / dt
dh/dt = 0
The rate of change of the height is 0 at the instant when the hang glider is 2,000 feet above the ground. Therefore, the area of the triangle is not changing at that instant.