If any body can solve this I would be greatly appreciative. I spent an hour in the tutoring center and no one could solve. Here goes:
A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?
When he is 4 m from the building, he is 8 m from the spotlight. The length of his shadow on the building at that time is 3 m.
The length of his shadow on the building when he is x meters from the spotlight is
Y = 2*(12/x) = 24/x
dY/dt = (dY/dx)*(dx/dt) = -(24/x^2)*1.6
= -(24/64)*1.6 = -0.6 m/s
Joseph spent 3/4 of an hour working out in a gym. He spent 1/2 of that time lifting weights. What fraction of a hour did he spend lifting weights?
To solve this problem, we can use related rates, which involves finding the rate at which one quantity is changing with respect to another quantity. In this case, we want to find the rate at which the length of the man's shadow on the building is changing (ds/dt) when he is 4 m from the building.
Let's break down the problem into different components:
1. Let's define the variables:
- l: length of the man's shadow on the building (we want to find dl/dt)
- x: distance between the man and the spotlight
- y: distance between the man and the building
2. We are given the following information:
- The distance between the spotlight and the wall (building) is 12 m.
- The man's height is 2 m.
- The man is walking towards the building at a speed of 1.6 m/s.
3. We need to find an equation that relates the variables:
- From similar triangles, we know that x/l = (x + y)/(l + 2)
(The ratio of corresponding sides in similar triangles is equal)
- Rearranging the equation, we get l = x(l + 2) / (x + y)
4. We can differentiate the equation implicitly with respect to time t:
- d(l)/dt = (d(x)/dt * (l + 2) + x * (dl/dt) - (dx/dt) * (l + 2)) / (x + y)^2
5. Now, let's substitute the known values to find the rate at which the length of the shadow is changing, dl/dt, when the man is 4 m from the building:
- x = 4 m (distance between the man and the spotlight)
- y = 12 - x = 8 m (distance between the man and the building)
- dx/dt = 1.6 m/s (rate at which the man is walking towards the building)
- l = ? and dl/dt = ? (what we want to find)
6. Substitute the known values and solve for dl/dt:
- dl/dt = (d(x)/dt * (l + 2) + x * (dl/dt) - (dx/dt) * (l + 2)) / (x + y)^2
- dl/dt = (1.6 * (l + 2) + 4 * (dl/dt) - 1.6 * (l + 2)) / (4 + 8)^2
- dl/dt = (1.6 * (l + 2) - 1.6 * (l + 2)) / 12^2
- dl/dt = 0 / 144
- dl/dt = 0
Therefore, when the man is 4 m from the building, the rate at which the length of his shadow on the building is decreasing is 0 m/s.