Maximum Minimum Problem: Vehicle Parking Lot

by Andrew
(New York)

Need Assistance

Up to 6000 square meters of land is available to build a parking lot for cars and small trucks. The local government has stipulated five conditions that must be met before the project can be put before the planning committee for approval.
a. There must be at least 50 spaces for trucks.
b. each car must be allocated a minimum space of 15 sq meters.
c. each truck must be allocated a minimum space of 30sq meters
d.the parking lot must have at least twice as many spaces for cars as for trucks
e. cars must be charged $2 per hr and trucks $5 per hr.

1. what is the largest number of vehicles the parking lot could hold?
2. What number of car spaces and truck spaces will actually be provided in order to generate maximum revenue when the lot is full?

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Sep 20, 2011

Maximum Minimum Problem - Vehicle Parking Lot
by: Staff

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Part II

2. What number of car spaces and truck spaces will actually be provided in order to generate maximum revenue when the lot is full?

Each Car requires 15 sq meters of space

Each Truck requires 30 sq meters of space

cars must be charged $2 per hr

trucks must be charged $5 per hr

Every truck will require the same space as two cars. However every truck will generate more income than 2 cars.

Every 2 cars generate an hourly revenue of $4 per hour

Every truck generates an hourly revenue of $5 per hour

Therefore maximum income will be achieved by maximizing the number of trucks and minimizing the number of cars.

T = number of trucks

T_max = The maximum number of trucks = unknown

C = number of cars

Since the parking lot must have at least twice as many spaces for cars as for trucks

C_min = 2 * T_max

C_min = minimum number of cars = unknown = 2 * T_max

6,000 sq meters = (T_ max) * 30 sq meters per truck + (C_ min) * 15 sq meters per car

6,000 sq meters = (T_ max) * 30 sq meters per truck + (2 * T_max) * 15 sq meters per car

6,000 = (T_ max) * 30 + (2 * T_max) * 15

6,000 = 30(T_ max) + 30(T_max)

6,000 = 60(T_ max)

6,000 / 60 = 60(T_ max) / 60

6,000 / 60 = (T_ max)* (60 / 60)

100 = (T_ max)* (60 / 60)

100 = (T_ max)* (1)

100 = (T_ max)

T_ max = 100 truck spaces

C_min = 2 * T_max

C_min = 2 * 100

C_min = 200 car spaces

the answer to question 2): 100 truck spaces, 200 car spaces

Thanks for writing.

Staff
www.solving-math-problems.com

Sep 20, 2011

Maximum Minimum Problem - Vehicle Parking Lot
by: Staff

Part I

The question:

by Andrew
(New York)

Need Assistance

Up to 6000 square meters of land is available to build a parking lot for cars and small trucks. The local government has stipulated five conditions that must be met before the project can be put before the planning committee for approval.

a. There must be at least 50 spaces for trucks.

b. each car must be allocated a minimum space of 15 sq meters.

c. each truck must be allocated a minimum space of 30 sq meters

d. the parking lot must have at least twice as many spaces for cars as for trucks

e. cars must be charged $2 per hr and trucks $5 per hr.

1. what is the largest number of vehicles the parking lot could hold?

2. What number of car spaces and truck spaces will actually be provided in order to generate maximum revenue when the lot is full?

The answer:

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1. what is the largest number of vehicles the parking lot could hold?

Available area = 6,000 sq meters

Each Truck requires 30 sq meters of space

Each Car requires 15 sq meters of space

Every truck will require the same space as two cars.

The maximum number of vehicles = minimum number of trucks + maximum number of cars.

T = number of trucks

T_min = The minimum number of trucks = 50

C = number of cars

C_max = maximum number of cars = unknown

6,000 sq meters = (T_min) * 30 sq meters per truck + (C_max) * 15 sq meters per car

6,000 sq meters = 50 * 30 sq meters per truck + (C_max) * 15 sq meters per car

6,000 = 50 * 30 + (C_max) * 15

6,000 = 1500 + (C_max) * 15

6,000 = 1500 + 15(C_max)

6,000 - 1500 = 1500 - 1500 + 15(C_max)

4500 = 1500 - 1500 + 15(C_max)

4500 = 0 + 15(C_max)

4500 = 15(C_max)

4500 / 15 = 15(C_max) / 15

4500 / 15 = (C_max) * (15 / 15)

300 = (C_max) * (15 / 15)

300 = (C_max) * (1)

300 = C_max

largest number of vehicles the parking lot could hold = minimum number of trucks + maximum number of cars.

largest number of vehicles the parking lot could hold = 50 trucks + 300 cars

largest number of vehicles the parking lot could hold = 350 vehicles (total

the answer to question 1): 350 vehicles

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Maximum Minimum Problem: Vehicle Parking Lot

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