Rate of decreasing shadow problem solved with derivative and similar triangles

Decreasing Shadow Solution
Rate Bill is walking is: $\frac{dy}{dt}$ = -4 Rate shadow is decreasing: $\frac{dx}{dt}$ Compare overlapping similar triangle. $\frac{x}{6}$ = $\frac{x + y}{30}$ Cross multiply 30x = 6x + 6y 24x = 6y (See note below) Take derivative 24 $\frac{dx}{dt}$ = 6 $\frac{dy}{dt}$ Substitute $\frac{dy}{dt}$ = -4 24 $\frac{dx}{dt}$ = 6 * (-4) 24 $\frac{dx}{dt}$ = -24 $\frac{dx}{dt}$ = -1 When Bill walks towards the light at 4 feet per second his shadow decreases by 1 foot per second. Note: At this point there are two non- overlapping similar triangles. $\frac{x}{6}$ = $\frac{y}{24}$ Substituting -4 for y and solving for x gives the same results as applying the derivative. 24x = 6y 24x = 6 * (-4) 24x = -24 x = -1
154

Decreasing Shadow
Solution

Rate Bill is walking is:

\frac{dy}{dt}

= -4
Rate shadow is decreasing:

\frac{dx}{dt}

Drawing of water filling a cone with r and h coordinates

Compare overlapping similar triangle.

\frac{x}{6}

\frac{x + y}{30}

Cross multiply
30x = 6x + 6y
24x = 6y (See note below)

Take derivative
24 $\frac{dx}{dt}$ = 6 $\frac{dy}{dt}$

Substitute $\frac{dy}{dt}$ = -4
24 $\frac{dx}{dt}$ = 6 * (-4)
24 $\frac{dx}{dt}$ = -24
$\frac{dx}{dt}$ = -1

When Bill walks towards the light at 4
feet per second his shadow decreases
by 1 foot per second.

Note: At this point there are two non-
overlapping similar triangles.

\frac{x}{6}

\frac{y}{24}

Substituting -4 for y and solving
for x gives the same results as
applying the derivative.

24x = 6y
24x = 6 * (-4)
24x = -24
x = -1

154

Decreasing ShadowSolution

Decreasing Shadow
Solution