Rate of decreasing shadow problem solved with derivative and similar triangles

Boat Moving Rate Solution
Known values given: y = 5 z = 13 $\frac{dz}{dt}$ = -2; x = $\sqrt{z² - y²}$ = $\sqrt{13² - 5²}$ = 12 Use Pythagorean Theorem Solve for $\frac{dx}{dt}$ x² = z² - y² Substitute: y = 5 x² = z² - 25 Take derivative 2x $\frac{dx}{dt}$ = 2z $\frac{dz}{dt}$ Substitute: x = 12 z = 13 $\frac{dz}{dt}$ = -2 2 * 12 $\frac{dx}{dt}$ = 2 * 13 * -2 $\frac{dx}{dt}$ = $\frac{-52}{24}$ = -2 $\frac{1}{6}$ The boat is moving to shore at 2 $\frac{1}{6}$ feet per second.
156A

Boat Moving Rate
Solution

Known values given:
y = 5 z = 13 $\frac{dz}{dt}$ = -2;
x = $\sqrt{z² - y²}$ = $\sqrt{13² - 5²}$ = 12
Drawing of man pulling rope attached to boat and x, y and z coordinated.

Use Pythagorean Theorem
Solve for $\frac{dx}{dt}$
x² = z² - y²

Substitute: y = 5
x² = z² - 25

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

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

$\frac{dx}{dt}$ = $\frac{-52}{24}$ = -2 $\frac{1}{6}$

The boat is moving to shore
at 2

\frac{1}{6}

feet per second.

156A

Boat Moving RateSolution

Boat Moving Rate
Solution