Chapter 3.4, Problem 54PPS

(a)

To determine

Create a table to show the amount D would earn for 20, 50 , and 100 cars for both options.

Explanation of Solution

Given:

Ms. S has given D two payment options :

1. He can choose to be paid $3 per car
2. He can choose to be paid $75 fee plus $1.50 per car.

Calculation :

Let

y = payment of D

x = number of cars

So, the equations for the payment that D gets ( y ) in x cars in both options :

1. y = 3x
2. y = 75 + 1. 5x

Now, create a table for the payment of D for 20 , 50 , and 100 cars :

Number of cars x	Payment of D in choice 1. y = 3x	Payment of D in choice 2. y = 75 + 1. 5x
20	$y=3\cdot 20=\$60$	$y=75+1.5\times 20=75+30=\$105$
50	$y=3\cdot 50=\$150$	$y=75+1.5\times 50=75+75=\$150$
100	$y=3\cdot 100=\$300$	$y=75+1.5\times 100=75+150=\$225$

(b)

To determine

Write an equation to represent D’s earnings for each payment option.

Answer to Problem 54PPS

The equations for the payment that D gets ( y ) in x cars in both options :

1. y = 3x
2. y = 75 + 1. 5x

Explanation of Solution

Given:

Ms. S has given D two payment options :

1.He can choose to be paid $3 per car

2.He can choose to be paid $75 fee plus $1.50 per car.

Calculation :

Let

y = payment of D

x = number of cars

So, the equations for the payment that D gets ( y ) in x cars in both options :

1. Since D gets $3 each car , so payment he gets is 3x for x cars . Hence the equation is :

y = 3x

2. Since D gets $1.50 each car and $75 fee , so payment he gets is 75+ 1. 5x for x cars . Hence the equation is :

y = 75 + 1. 5x

(c)

To determine

Graph the equations for both options.

Explanation of Solution

Calculation :

From part (b),

The equations for the payment that D gets ( y ) in x cars in both options :

1. y = 3x
2. y = 75 + 1. 5x

From part (a),

x	Option 1: y = 3x	Option 2: y = 75 + 1. 5x
20	60	105
50	150	150

Plot the points (20,60) and (50,150) and join to get the graph for option 1.

Plot the points (20,105) and (50,150) and join to get the graph for option 2.

Graphs for both the options :

  

(d)

To determine

Find the more profitable option if:

1. 35 people attend the party
2. 75 people attend the party

Answer to Problem 54PPS

When there are 35 people , option 2. is more profitable.

When there are 75 people, option 1. is more profitable.

Explanation of Solution

Calculation :

From part (b),

The equations for the payment that D gets ( y ) in x cars in both options :

1. y = 3x
2. y = 75 + 1. 5x

Now, create a table for the payment of D for :

1. x = 35
2. x = 75

Number of cars/people x	Payment of D in choice 1. y = 3x	Payment of D in choice 2.y = 75 + 1. 5x
35	$y=3\cdot 35=\$105$	$y=75+1.5\times 35=75+52.5=\$127.5$
75	$y=3\cdot 75=\$225$	$y=75+1.5\times 75=75+112.5=\$187.5$

So, when there are 35 people , option 2. is more profitable.

When there are 75 people, option 1. is more profitable.

(e)

To determine

Write a statement to determine the best payment option based of the number of cars D parks.

Answer to Problem 54PPS

If the number of cars is less than 50 , then option 2 of payment is more profitable and if the number of cars is more than 50 , the option 1 of payment is more profitable and for number of cars equal to 50 , both the options of payment are equally profitable.

Explanation of Solution

Calculation :

From part (c) , we have the graph of both the payment options :

  

Clearly , the lines of both the equations meet at (50,150) .

Below that point , the graph of equation y = 75 + 1. 5x is higher , whereas , above the point the graph of equation y = 3x is higher.

y -axis represents the payment of D and x- axis represents the number of cars D parks.

So, we can conclude that :

If the number of cars is less than 50 , then option 2 of payment is more profitable and if the number of cars is more than 50 , the option 1 of payment is more profitable and for number of cars equal to 50 , both the options of payment are equally profitable.

(f)

To determine

If Ms.D send 50 invites , find the option that D should choose.

Answer to Problem 54PPS

Second option

Explanation of Solution

Calculation :

From part (e) , we have :

If the number of cars is less than 50 , then option 2 of payment is more profitable and if the number of cars is more than 50 , the option 1 of payment is more profitable and for number of cars equal to 50 , both the options of payment are equally profitable.

Since Ms. D has send 50 invites , it is not sure that all 50 people will come to the party . 50 is the maximum number of cars/people that will come.

So, D should choose option 2 , since it is more profitable for payment of D , if the number of cars is less than 50 and for 50 cars , both the options are equally profitable.

Hence , D must choose second option .