Problem Solving Using Algebraic Models

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## Problem Solving Using Algebraic Models

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Problem Solving Using

Algebraic Models

Corporate Average Fuel Economy (CAFE) standards require auto manufacturers to produce cars so that the average fuel efficiency between all of its car models is at least 27.5 miles per gallon. Your new car travels about 385 miles on 11 gallons of gas. Is your car above or below the CAFE standard?

Verbal

Model

Fuel Efficiency

Total miles

÷

=

# Gallons

Verbal

Model

Fuel Efficiency

Total miles

÷

=

# Gallons

Fuel efficiency = x

Total miles = 385

Amount of gas = 11

Labels

(miles/gallon)

(miles)

(gallons)

Alg. Model

x = 385

11

x = 35 miles/gallon

Above CAFE standards.

You used all of a \$50 gift certificate watching movies at the local theater. You have watched a total of 10 movies.

How much did you spent watching evening showings if afternoon showings are \$4 and evening showings are \$6.

# evening shows

=

Verbal

Model

Total \$

\$ evening

# afternoon shows

+

\$ afternoon

# evening showings

=

Verbal

Model

Total \$

\$ evening

# afternoon showings

+

\$ afternoon

Total cost = 50

Movie cost (evening) = \$6

# Movies (evening) = x

Movie cost (afternoon) = \$4

# Movies (afternoon) = 10 - x

Labels

Total cost = 50

Movie cost (evening) = \$6

# Movies (evening) = x

Movie cost (afternoon) = \$4

# Movies (afternoon) = 10 - x

Labels

Alg. Model

50 = 6x + 4(10 – x)

50 = 6x + 40 – 4x

50 = 2x + 40

10 = 2x

5 = x

You watched 5 evening showings.

The cost was 5(6) = \$30.

Driving to a concert, you average 24 miles per hour in the city and 65 miles per hour on the highway. If the 142 mile trip takes 2.5 hours, how much of the time were you driving on the highway?

Miles/hr hwy

=

Verbal

Model

Distance

time (hwy)

Miles/hr city

+

time (city)

Miles/hr hwy

=

Verbal

Model

Distance

time (hwy)

Miles/hr city

+

time (city)

Distance traveled = 142

Miles/hr (hwy) = 65

Time (hwy) = x

Miles/hr (city) = 24

Time (city) = 2.5 - x

Labels

Distance traveled = 142

Miles/hr (hwy) = 65

Time (hwy) = x

Miles/hr (city) = 24

Time (city) = 2.5 - x

Labels

Alg. Model

142 = 65x + 24(2.5 – x)

142 = 65x + 60 – 24x

142 = 41x + 60

82 = 41x

2 = x

2.5 – 2 = ½ hour.

A car and a truck are 245 miles apart traveling towards each other until they meet. The car averages 65 miles per hour and the truck averages 55 miles per hour. The truck started one hour after the car. After how long do the two automobiles meet? How many miles did each automobile travel?

rate of car

=

Verbal

Model

Distance

time (car)

rate of truck

+

time (truck)

rate of car

=

Verbal

Model

Distance

time (car)

rate of truck

+

time (truck)

Distance traveled = 245

Rate (car) = 65 mph

Time (car) = x hr.

Rate (truck) = 55 mph

Time (truck) = (x – 1) hr.

Labels

Distance traveled = 245

Rate (car) = 65 mph

Time (car) = x hr.

rate (truck) = 55 mph

Time (truck) = (x – 1) hr.

Labels

Alg. Model

245 = 65x + 55(x - 1)

245 = 65x + 55x - 55

245 = 120x - 55

300 = 120x

2.5 = x

The car traveled for 2.5 hours

and the truck traveled for

2.5 – 1 = 1.5 hours

before they met.

The car traveled 65(2.5) = 162.5 miles.

The truck traveled 55(1.5) = 82.5 miles.

McDougal Littell/Houghton Mifflin Algebra 2 Notetaking Guide