Question

a street light is at the top of a pole that has a height of 15 ft . a woman 5 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. how fast is the tip of her shadow moving away from the pole when she is 36 ft from the base of the pole? (leave your answer as an exact number.)

Answer 1

When the woman is 36 ft away from the base of the pole, the tip of her shadow is not moving away from the pole.

Step-by-step explanation:

To find how fast the tip of the woman's shadow is moving away from the pole, we can use similar triangles. The height of the pole is 15 ft and the height of the woman is 5 ft. When the woman is 36 ft from the base of the pole, the length of her shadow can be found using the ratios of the similar triangles: (15/5) = (x/36), where x is the length of the shadow. Solving this equation, we find that x = 108 ft.

To find the rate at which the tip of the shadow is moving away from the pole, we can differentiate the equation with respect to time. Let y be the distance between the woman and the pole. Then we have (15/5) = (x/y), where x is the length of the shadow and y is the distance between the woman and the pole. Differentiating this equation, we get (d/dt)(15/5) = (d/dt)(x/y). Since the woman's height is constant, the derivative of 15/5 is 0. So we have 0 = (d/dt)(x/y). Rearranging this equation, we get (dy/dt)(15/5) = (d/dt)(x). Substituting the given values, we have (dy/dt)(3) = (d/dt)(108).

To find the value of (dy/dt), we can use the equation from the previous step: (dy/dt)(3) = (d/dt)(108). Dividing both sides by 3, we get dy/dt = (d/dt)(108)/3. Since (d/dt)(108) is the rate at which the shadow length is changing, and we know that the shadow length is constant, the rate is 0. Therefore, dy/dt = 0. This means that the tip of the woman's shadow is not moving away from the pole.

Answer 2

The tip of the woman's shadow is moving away from the pole at a rate of 2/3 ft/s when she is 36 ft from the base of the pole.

Step-by-step explanation:

To find the speed at which the tip of the woman's shadow is moving away from the pole, we can use similar triangles. Let's call the distance between the woman and the pole x and the length of the shadow y. Since the height of the pole is 15 ft and the woman's height is 5 ft, we have the following proportion: x/y = (x+15)/(y+y). We can differentiate this equation with respect to time to find the rate of change of x with respect to y, or dx/dy. Then we substitute the given values to find the rate of change of y with respect to time, or dy/dt.

Let's differentiate the equation: y * d(x/y)/dt = (y+y) * d(x+15)/(y+y)/dt. We can simplify this to: y * dx/dy = 2 * dx/dt * (x+15)/(2y). Rearranging the equation and plugging in the given values, we get: dx/dt = (2/3) ft/s. Therefore, the tip of the woman's shadow is moving away from the pole at a rate of 2/3 ft/s when she is 36 ft from the base of the pole.