The cities of Toronto and Vancouver are \( 5415 km \) apart. A train leaves Toronto headed towards Vancouver travelling at \( 160 km / hr \) . At the same time, a train leaves Vancouver headed towards Toronto travelling at \( 125 km / hr \) . How many hours is it before the trains meet? [Full algebraic solution required including the "Let" statements. A table may be helpful].

let the time they take = t

let the distance the train from Vancover covers =x

Toronto to vancover: 160 = (5415-x)/t

therefore t=(5415-x)/160

Vancover to Toronto: 125=x/t

therefore t=x/125

equate both t's to get

(5415-x)/160 = x/125

125(5415-x)= 160x

676875-125x=160x

676875 = 285x

x= 2375 km

t= 2375/125 = 19 hours

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 \(\square \).