Floating Ball The polynomial function \(f ( x ) = \frac { \pi } { 3 } x ^ { 3 } - 5 \pi x ^ { 2 } + \frac { 500 \pi d } { 3 }\)

\(\left. \begin{array} { l } { \text { 103. (a) about } 7.13 cm \text { ; The ball floats partly above the } } \\ { \text { surface. (b) The sphere is more dense than water and } } \\ { \text { sinks below the surface. } \quad \text { (c) } 10 cm \text { ; The balloon is } } \\ { \text { submerged with its top even with the surface. } } \end{array} \right.\)

Chris is going to rent a truck for one day. There are two companies he can choose from, and they have the following prices.

Company A charges an initial fee of \( \$ 97.50 \) and an additional \( 10 \) cents for every mile driven. Company B charges an initial fee of \( \$ 55 \) and an additional \( 60 \) cents for every mile driven.

For what mileages will Company A charge no more than Company B?

Write your answer as an inequality, using \( m \) for the number of miles driven.

A phone company offers two monthly charge plans. In Plan A, the customer pays a monthly fee of \( \$ 26.20 \) and then an additional \( 5 \) cents per minute of use. In Plan B, the customer pays a monthly fee of \( \$ 20 \) and then an additional \( 7 \) cents per minute of use.

For what amounts of monthly phone use will Plan A cost less than Plan B?

Use \( m \) for the number of minutes of phone use, and solve your inequality for \( m \) .

Consider the function \(y = g ( x ) = - x ^ { 2 } + 7 x + 5\) .

(a) Use the limit definition to compute a formula for \(y = g ^ { \prime } ( x ) \) .

\(y = \)

(b) Determine the slope of the tangent line to \(y = g ( x ) \) at the value \(x = 3\) .

slope \(= \)

(c) Compute \(g ( 3 ) \) .

\(g ( 3 ) = \)

(d) The equation for the tangent line to \(y = g ( x ) \) at the point \(( 3 , g ( 3 ) ) \) , written in point-slope form, is \(y - \square = \square ( x - 3\))

Fill in the blanks with the appropriate numbers.

Billy rides his bicycle on a straight race track. When he starts sprinting he increases \(2 m / s\) for every second, \(s\) , he continues to drive over \(9\) seconds in the race. At \(9\) seconds he is traveling \(10 m / s\) . He Writes the following functions to represent his speed after s seconds but he must combine them somehow:

\(a ( x ) = s - 9\)

\(s ( x ) = 2 s + 10\)

Write a composite function to represent his speed \(s\) seconds in the race \(( 3 pts ) \) .

Find out how fast he is going after \(14\) seconds ( \(3\) pts).

The sum of two numbers is \( 94 \) . The difference of the two numbers is \( 58 \) . What are the two numbers.

Let \( x \) be the larger number and \( y \) be the smaller number. Write an equation that expresses the information in the sentence "The sum of two numbers is \( 94 \) ."

\(\square \)

Write an equation that expresses the information in the sentence "The difference of the two numbers is

\( 58 . " \) \(\square \)

Solve the system you have written above.

The larger number, \( x \) is \(\square \). The smaller number, \(y\) is \(\square \).