a 6 - foot - tall woman walks at 9 ft/s toward a street light that is 30 ft above the ground. what is the…

a 6 - foot - tall woman walks at 9 ft/s toward a street light that is 30 ft above the ground. what is the rate of change of the length of her shadow when she is 14 ft from the street light? at what rate is the tip of her shadow moving? let l be the length of the womans shadow and let x be the womans distance from the street light. write an equation that relates l and x.

a 6 - foot - tall woman walks at 9 ft/s toward a street light that is 30 ft above the ground. what is the rate of change of the length of her shadow when she is 14 ft from the street light? at what rate is the tip of her shadow moving? let l be the length of the womans shadow and let x be the womans distance from the street light. write an equation that relates l and x.

Answer

Explanation:

Step1: Set up similar - triangles proportion

We have two similar right - triangles. One triangle is formed by the street light from the ground to the top and from the base of the street light to the end of the shadow. The other is formed by the woman from the ground to the top and from the woman to the end of the shadow. The ratio of their heights and bases are equal. The height of the street light is (h_1 = 30) ft, the height of the woman is (h_2=6) ft. The base of the larger triangle is (x + L) and the base of the smaller triangle is (L). So, (\frac{30}{x + L}=\frac{6}{L}). Cross - multiply to get (30L=6(x + L)).

Step2: Simplify the equation

Expand the right - hand side: (30L = 6x+6L). Subtract (6L) from both sides: (30L-6L=6x), which simplifies to (24L = 6x), and then (L=\frac{1}{4}x).

Answer:

(L=\frac{1}{4}x)