You've dropped a lucky penny into the middle of a hemispherical water basin of radius 3m. There is a light on the upper edge of the basin. The penny falls vertically along the axis of symmetry of the basin, and when it is 1m from the bottom of the basin it is falling at 2m/s. How fast is the shadow of the penny moving along the surface of the basin at this instant?
To find the speed of the shadow of the penny along the surface of the basin, we need to determine the rate of change of the shadow's position with respect to time.
Let's consider the situation at the given instant when the penny is 1m from the bottom of the basin and falling at 2m/s. We can use similar triangles to relate the distance the penny has fallen to the distance its shadow has moved.
Let h be the distance from the bottom of the basin to the penny, and x be the distance from the base of the shadow to the center of the basin. Then, we have the following relationship between h and x:
h : 1m = x : 3m
We can rearrange this equation to express h in terms of x:
h = (x * 1m) / 3m
Now, we need to differentiate both sides of this equation with respect to time. We'll denote dh/dt as the rate at which h is changing with respect to time, and dx/dt as the rate at which x is changing with respect to time.
Differentiating the equation, we get:
dh/dt = (d/dt)((x * 1m) / 3m)
dh/dt = (1/3) * (dx/dt)
From the given information, we know that dh/dt = -2m/s (since the penny is falling downward at 2m/s).
Plugging in the values, we have:
-2m/s = (1/3) * (dx/dt)
Simplifying, we find:
dx/dt = -6m/s
Therefore, the shadow of the penny is moving along the surface of the basin at a speed of 6m/s in the opposite direction at the given instant.
To find the speed of the shadow of the penny along the surface of the basin, you can use similar triangles.
Let's assume that the shadow of the penny forms a right triangle with the radius of the basin and the vertical distance between the penny and the bottom of the basin.
1. First, find the height of the triangle formed by the shadow of the penny. This height is the vertical distance between the penny and the bottom of the basin. We are given that it is 1m.
2. Next, find the base of the triangle formed by the shadow of the penny. This base is the horizontal distance between the penny and the vertical line passing through the center of the basin.
To find the base, we need to calculate the horizontal distance from the center of the basin to the vertical line passing through the penny. This distance is the radius of the basin minus the height of the triangle formed by the shadow of the penny.
Given that the radius of the basin is 3m and the height of the triangle is 1m, the base is 3 - 1 = 2m.
3. Now, we can calculate the ratio of the height to the base of the triangle formed by the shadow of the penny. This ratio will be the same as the ratio of the speed of the penny falling to the speed of the shadow moving along the surface of the basin.
Given that the speed of the penny falling is 2m/s, the ratio is 1/2.
4. Finally, using the ratio, we can calculate the speed of the shadow along the surface of the basin. Multiply the ratio by the speed of the penny.
1/2 * 2m/s = 1m/s
Therefore, the shadow of the penny is moving at a speed of 1m/s along the surface of the basin at this instant.