Rate of decreasing shadow problem solved with derivative and similar triangles

Rope Pulling Rate Solution
Known values given: y = 5 z = 13 $\frac{dx}{dt}$ = -2 x = $\sqrt{z² - y²}$ = $\sqrt{13² - 5²}$ = 12 Use Pythagorean Theorem Solve for $\frac{dz}{dt}$ z² = x² + y² Substitute: y = 5 z² = x² + 25 Take derivative 2z $\frac{dz}{dt}$ = 2x $\frac{dx}{dt}$ Substitute: z = 13 x = 12 $\frac{dx}{dt}$ = -2 2 * 13 $\frac{dz}{dt}$ = 2 * 12 * -2 26 $\frac{dz}{dt}$ = -48 $\frac{dz}{dt}$ = $\frac{-48}{26}$ = $\frac{-24}{13}$ = -1 $\frac{11}{13}$ The rope is being pulled at 1 $\frac{11}{13}$ feet per second.
156B

Rope Pulling Rate
Solution

Known values given:
y = 5 z = 13 $\frac{dx}{dt}$ = -2
x = $\sqrt{z² - y²}$ = $\sqrt{13² - 5²}$ = 12
Drawing of water filling a cone with r and h coordinates

Use Pythagorean Theorem
Solve for $\frac{dz}{dt}$
z² = x² + y²

Substitute: y = 5
z² = x² + 25

Take derivative
2z $\frac{dz}{dt}$ = 2x $\frac{dx}{dt}$

Substitute:
z = 13 x = 12 $\frac{dx}{dt}$ = -2
2 * 13 $\frac{dz}{dt}$ = 2 * 12 * -2
26 $\frac{dz}{dt}$ = -48

$\frac{dz}{dt}$ = $\frac{-48}{26}$ = $\frac{-24}{13}$ = -1 $\frac{11}{13}$

The rope is being pulled at
1

\frac{11}{13}

feet per second.

156B

Rope Pulling RateSolution

Rope Pulling Rate
Solution