A community skating rink is in the shape of a rectangle with semicircles attached at the ends The length of the rectangle is \( 20 \) feet less than twice the width. The thickness of the ice is \( 0.75 \) inch.
(a) Build a model that expresses the ice volume, \( V \) , as a function of the width, \( x \) .
(b) How much ice is in the rink if the width is \( 90 \) feet?
\(\left. \begin{array} { l l } { \text { (a) } V ( x ) = \frac { x ^ { 2 } } { 8 } - \frac { 5 x } { 4 } + \frac { \pi x ^ { 2 } } { 64 } } & { \text { (b) } 1297.61 ft ^ { 3 } } \end{array} \right.\)
Find the solution of the exponential equation \( 16 e ^ { x } = 12 \) in terms of logarithms, or correct to four decimal places. \( x = \)
Certain radioactive material decays in such a way that the mass remaining after \( t \) years is given by the function
\( m ( t ) = 100 e ^ { - 0.01 t } \)
where \( m ( t ) \) is measured in grams. (a) Find the mass at time \( t = 0 \) . Your answer is (b) How much of the mass remains after \( 20 \) years? Your answer is Round answers to \( 1 \) decimal place.
