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16 October, 22:35

The area of a parking lot is 805 square meters. A car requires 5 square meters and a bus requires 32 square meters of space. There can be at most 80 vehicles parked at one time. If the cost to park a car is $2.00 and a bus is $6.00, how many buses should be in the lot to maximize income?

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  1. 16 October, 23:31
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    B = number of buses parked

    C = number of cars parked

    We can write 3 formulas:

    Formula 1: Total number of vehicles cannot exceed 80.

    B+C<=80

    The maximum income will be when it is 80

    Formula 2: The space occupied cannot exceed 805

    5C+32B<=805

    The maximum income will be when it is 805

    So let’s try to get 805 m^2 with 80 vehicles at the same time

    B+C=80 - > C=80-B

    5C+32B=805 - > C = (805-32B) / 5

    Let’s do the equalization method:

    80-B = (805-32B) / 5

    5 * (80-B) = 805-32B

    400-5B=805-32B

    32B-5B=805-400

    27B=405

    B=405/27=15

    C=80-15=65

    Let’s verify:

    5*65+32*15=805 - > OK

    Formula 3: Income=6*15+2*65=220

    If we put more buses, it doesn’t fit. Let’s try with 1 more bus and 1 less car:

    5*34+32*16=832>805 - > NOT OK

    If we put less buses, it’s less income. Let’s try with 1 less bus and 1 more car:

    Income=6*14+2*66=216 - > Not so convenient.

    So the optimum is 15 buses and 65 cars.
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