5. A ball is thrown straight upward from the ground and attains a height of \( s(t)=-2 t^{2}+28 t+68 \mathrm{ft} \) above the ground after \( t \mathrm{sec} \). When does the ball reach the maximum height? What is the maximum height?

A rancher has 800 feet of fencing to put around a rectangular field and then subdivide the field into 3 identical smaller rectangular plots by placing two fences parallel to one of the field's shorter sides. Find the dimensions that maximize the enclosed area. Write your answers as fractions reduced to lowest terms.

\( \int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 14 - y ^ { 2 } } } \int _ { - \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } y ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d z d x d y \)

\( f ( x ) = ( x - 2 ) ^ { 4 } - 2 , x \geq 2 \)

Resuelve la ecuación diferencial \( \frac{dy}{dx} = 3y^{2} \) dado que \( y(0) = 1 \).

Calcular el área de las siguientes regiones que están delimitadas por las siguientes curvas cuyas ecuaciones son: A) \( \mathrm{y}^{2}=-\mathrm{x} \quad ; \mathrm{x}-\mathrm{y}=4 \quad ; \mathrm{y}=-1 \quad ; \mathrm{y}=2 \)