Question

A light is fixed on a wall 10 meters above the floor. twelve meters away from the wall a pole is being raised straight up at a rate of 45 cm/sec. when the pole is 6 meters tall at what rate is the tip of the shadow moving away from the pole?

Answer 1

To find the rate at which the tip of the shadow is moving away from the pole when the pole is 6 meters tall, we can differentiate the equation for the length of the shadow and substitute t = 6.

Step-by-step explanation:

To determine the rate at which the tip of the shadow is moving away from the pole, we need to find the rate at which the length of the shadow is changing. Let's assume that the length of the shadow is represented by the variable 's'.

From the given information, we know that the height of the light on the wall is 10 meters and the pole is being raised at a rate of 45 cm/sec. Therefore, the length of the shadow at any given time 't' is given by: s = sqrt((10+t)^2 + 12^2).

To find the rate at which the length of the shadow is changing, we differentiate the equation with respect to time 't': ds/dt = (10+t)/(sqrt((10+t)^2 + 12^2)) * dt/dt + 0. Since dt/dt = 1, we can simplify the equation to: ds/dt = (10+t)/(sqrt((10+t)^2 + 12^2)).

Now, substitute t = 6 into the equation to find the rate at which the tip of the shadow is moving away from the pole when the pole is 6 meters tall.