Question

a street light is at the top of a 19 foot tall pole. a 6 foot tall woman walks away from the pole with a speed of 5 ft/sec along a straight path. at what rate is the distance from the base of pole to the tip of the shadow changing when she is 45 ft from the base of the pole?

Answer 1

To find the rate at which the distance from the base of the pole to the tip of the shadow is changing, we can use similar triangles and differentiate the equation with respect to time. The rate of change can be calculated using the given values for dx/dt = (x + 45) * 5 / 45^2

Step-by-step explanation:

To find the rate at which the distance from the base of the pole to the tip of the shadow is changing, we need to find the rate at which the length of the shadow is changing.

Let's denote the length of the shadow as 'x' and the distance from the base of the pole to the woman as 'y'. We can use similar triangles to set up a ratio:

19/6 = (x + y) / y

Now, differentiate both sides of the equation with respect to time:

d(19/6) = (d(x+y)/dt) / y - (x+y) * (dy/dt) / y^2

Simplifying this equation gives us:

0 = (dx/dt)/y - (x+y) * (dy/dt) / y^2

We are given that dy/dt = 5 ft/sec when y = 45 ft. We can substitute these values into the equation and solve for dx/dt:

0 = (dx/dt)/45 - (x+45) * (5) / 45^2

dx/dt = (x + 45) * 5 / 45^2

Therefore, the rate at which the distance from the base of the pole to the tip of the shadow is changing is given by this equation.