14) A spot Light is on the ground \( (20 \mathrm{ft} \) away from a wall and 6 FT ) person is walking towards the wall at a rate of \( 2.5 \mathrm{tT} / \mathrm{sec} \). How fast is the height of the shadow changing when the person is 8 fT from the wall? Is the shadow increasing or decreasing in height at this Time?

To solve this problem, we can use similar triangles. Let \( h \) be the height of the shadow and \( x \) be the distance of the person from the wall. The light, the top of the person's head, and the tip of the shadow all form similar triangles. Using the distances, we can set up the relation: \[ \frac{h}{x} = \frac{6}{20} \Rightarrow h = \frac{3}{10}x \] As the person walks towards the wall, \( x \) decreases. The rate of change of \( x \) is given as \( -2.5 \text{ ft/sec} \). To find the rate of change of the shadow’s height, we differentiate \( h \) with respect to time \( t \): \[ \frac{dh}{dt} = \frac{3}{10} \frac{dx}{dt} \] Substituting \( \frac{dx}{dt} = -2.5 \): \[ \frac{dh}{dt} = \frac{3}{10}(-2.5) = -0.75 \text{ ft/sec} \] So, the height of the shadow is decreasing at a rate of \( 0.75 \text{ ft/sec} \) when the person is 8 ft from the wall.