A spotlight at a building corridor is fastened to a wall 8m above the floor. A lady 1.75m tall moves away from the wall at a speed of 0.75ms. a.)at what rate is the lenght of her shadow increasing? b.)at what speed is the tip of her shadow moving?

Question

A spotlight at a building corridor is fastened to a wall 8m above the floor. A lady 1.75m tall moves away from the wall at a speed of 0.75ms.  a.)at what rate is the lenght of her shadow increasing?  b.)at what speed is the tip of her shadow moving?

Theodore Schwartz · Accepted Answer

Step 1 Please have a look at the picture below to understand what&#039;s going on:  For the sake of clarity, s = length of the shadow = AD and x = distance of the lady from the wall = BD dxdt=0.75ms Step 2 Triangle ADE is similar to triangle ABC (AAA criterion of similarity) Hence, DEBC=ADAB (Corresponding sides of similar triangles are proportional) Hence, 1.758=ss+x Hence, 1.75(s+x)=8s Or, 1.75x=(8−1.75)s=6.25s Step 3 Part (a) Differentiate both sides w.r.t time t to get: 1.75dxdt=6.25dsdt Hence, the rate at which length of her shadow is increasing = dsdt=(1.756.25)dxdt=(1.756.25)x0.75=0.21ms Step 4 Part (b) the rate at which the tip of her shadow is moving = rate t which she is moving + rate at which the length of the shadow is increasing = dxdt+dsdt=0.75+0.21=0.96ms Step 5 Final answers: Part (a) 0.21ms Part (b) 0.96ms

A spotlight at a building corridor is fastened to a wall 8m above the floor. A lady 1.75m tall moves away from the wall at a speed of 0.75m/s. a.)at what rate is the lenght of her shadow increasing? b.)at what speed is the tip of her shadow moving?

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