Step 1

Please have a look at the picture below to understand what's going on:

For the sake of clarity,

s = length of the shadow = AD

and x = distance of the lady from the wall = BD

\(\displaystyle\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}={0.75}\frac{{m}}{{s}}\)

Step 2

Triangle ADE is similar to triangle ABC (AAA criterion of similarity)

Hence, \(\displaystyle{D}\frac{{E}}{{B}}{C}={A}\frac{{D}}{{A}}{B}\) (Corresponding sides of similar triangles are proportional)

Hence, \(\displaystyle\frac{{1.75}}{{8}}=\frac{{s}}{{{s}+{x}}}\)

Hence, \(\displaystyle{1.75}{\left({s}+{x}\right)}={8}{s}\)

Or, \(\displaystyle{1.75}{x}={\left({8}-{1.75}\right)}{s}={6.25}{s}\)

Step 3

Part (a)

Differentiate both sides w.r.t time t to get:

\(\displaystyle\frac{{{1.75}{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}=\frac{{{6.25}{d}{s}}}{{{\left.{d}{t}\right.}}}\)

Hence, the rate at which length of her shadow is increasing = \(\displaystyle\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}=\frac{{{\left(\frac{{1.75}}{{6.25}}\right)}{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}={\left(\frac{{1.75}}{{6.25}}\right)}{x}{0.75}={0.21}\frac{{m}}{{s}}\)

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 = \(\displaystyle\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}+\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}={0.75}+{0.21}={0.96}\frac{{m}}{{s}}\)

Step 5

Final answers:

Part (a) \(\displaystyle{0.21}\frac{{m}}{{s}}\)

Part (b) \(\displaystyle{0.96}\frac{{m}}{{s}}\)

Please have a look at the picture below to understand what's going on:

For the sake of clarity,

s = length of the shadow = AD

and x = distance of the lady from the wall = BD

\(\displaystyle\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}={0.75}\frac{{m}}{{s}}\)

Step 2

Triangle ADE is similar to triangle ABC (AAA criterion of similarity)

Hence, \(\displaystyle{D}\frac{{E}}{{B}}{C}={A}\frac{{D}}{{A}}{B}\) (Corresponding sides of similar triangles are proportional)

Hence, \(\displaystyle\frac{{1.75}}{{8}}=\frac{{s}}{{{s}+{x}}}\)

Hence, \(\displaystyle{1.75}{\left({s}+{x}\right)}={8}{s}\)

Or, \(\displaystyle{1.75}{x}={\left({8}-{1.75}\right)}{s}={6.25}{s}\)

Step 3

Part (a)

Differentiate both sides w.r.t time t to get:

\(\displaystyle\frac{{{1.75}{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}=\frac{{{6.25}{d}{s}}}{{{\left.{d}{t}\right.}}}\)

Hence, the rate at which length of her shadow is increasing = \(\displaystyle\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}=\frac{{{\left(\frac{{1.75}}{{6.25}}\right)}{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}={\left(\frac{{1.75}}{{6.25}}\right)}{x}{0.75}={0.21}\frac{{m}}{{s}}\)

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 = \(\displaystyle\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}+\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}={0.75}+{0.21}={0.96}\frac{{m}}{{s}}\)

Step 5

Final answers:

Part (a) \(\displaystyle{0.21}\frac{{m}}{{s}}\)

Part (b) \(\displaystyle{0.96}\frac{{m}}{{s}}\)