A man of height 1.7 meters walks away from a 5- meter lamppost at a speed of 1.1 m/s. Find the rate at which his shadow is increasing in length.
An illustration shows a man walking past a lamp post. The lamp is 5 units above the ground. The man is horizontally x units away from lamp. A line is drawn from the lamp to the man’s head and is extended to the ground. The distance between the man and the point on the ground where the line ends is labeled y.
(Use decimal notation. Give your answer to three decimal places.)
when the man is x meters from the pole, if the shadow's length is s, then
(s+x)/5 = s/1.7
s = 1.7/3.5 x
so, with dx/dt = 1.1,
ds/dt = 1.7/3.5 * dx/dt = 0.534 m/s
To find the rate at which the man's shadow is increasing in length, we need to use the concept of similar triangles.
Let's first define our variables:
- x: the horizontal distance between the man and the lamp post
- y: the vertical distance between the point on the ground and the end of the line drawn from the lamp post to the man's head
- h: the man's height
- s: the length of the man's shadow
We are given:
- h = 1.7 meters
- x = ?
- y = ?
- s = ?
In the given scenario, we have a right triangle formed by the man, the lamppost, and the point on the ground where the line extends. This right triangle is similar to another right triangle formed by the man's shadow, the lamppost, and the point on the ground where the shadow ends.
The ratio of the corresponding sides of similar triangles is always constant. This means that the ratio of the sides of the first triangle is equal to the ratio of the sides of the second triangle. Therefore, we can set up the following proportion:
s / x = (s + h) / y
Now we can substitute the values we know:
s / x = (s + 1.7) / y
To solve for the rate at which the shadow is increasing (ds / dt), we need to find the derivative of both sides of the equation with respect to time (t):
ds / dt * (1 / x) = d(s + h) / dt * (1 / y)
Since the man is moving horizontally and the length of the shadow depends on this horizontal distance, dx / dt = -1.1 m/s (negative because the man is moving away from the lamppost).
ds / dt * 1 / x = dh / dt * 1 / y
Now we can substitute the values we know and solve for ds / dt:
ds / dt * 1 / x = 0 * 1 / y
ds / dt = 0
Therefore, the rate at which the man's shadow is increasing in length is 0.