A factory manufactures two products, Product X and Product Y. Daily production is subject … — GMAT

A factory manufactures two products, Product X and Product Y. Daily production is subject to the following constraints:
- Each unit of Product X requires $1$ hour of machine time.
- Each unit of Product Y requires $2$ hours of machine time.
- Total machine time available per day is at most $10$ hours.
- The factory must produce at least $2$ units of Product X and at least $2$ units of Product Y per day.
- Each unit of Product X yields a profit of $\$15$ , and each unit of Product Y yields a profit of $\$20$ .

Match each optimization metric on the left with its corresponding correct numerical value on the right.

Maximum possible production of Product Y (in units)4
Number of units of Product X required to maximize total daily profit6
Maximum total daily profit achievable (in dollars)130

Cevap

Maximum possible production of Product Y matches 4; Units of Product X for maximum profit matches 6; Maximum total daily profit matches 130.

For the maximum production of Product Y, using the minimum required

x = 2

allows

2y = 10 - 2 = 8

, giving

y = 4

. For maximum profit, each machine hour used on Product X yields

\$15

per hour while each hour used on Product Y yields

\$20 / 2 = \$10

per hour. Thus, maximizing hours allocated to Product X subject to

y \ge 2

gives

y = 2

and

x = 6

, yielding a maximum profit of

15(6) + 20(2) = 130

dollars.

Adım Adım Çözüm

Formulate the system of constraints and profit objective function from the narrative context.

Let

x

be units of Product X and

y

be units of Product Y. Constraints are

x \ge 2

y \ge 2

, and

x + 2y \le 10

. Objective is to maximize

P = 15x + 20y

Establishing explicit algebraic inequalities defines the feasible decision region.

Calculate the upper bound for

y

Setting

x = 2

(its minimum value),

2 + 2y \le 10 \implies 2y \le 8 \implies y \le 4

. Maximum

y = 4

Minimizing

x

leaves maximum machine time capacity available for Product Y.

Evaluate candidate feasible integer solutions

(x, y)

to determine maximum daily profit.

Valid integer pairs on the boundary include

(2, 4)

with

P = 110

;

(4, 3)

with

P = 120

; and

(6, 2)

with

P = 130

Trading 1 unit of Product Y (losing

\$20

) for 2 units of Product X (gaining

2 \times \$15 = \$30

) increases net profit by

\$10

per unit trade.

Match each requested metric to its evaluated value.

Maximum Product Y = 4; Product X for max profit = 6; Maximum profit = 130.

Matches each question prompt to its calculated mathematical result.

Anahtar Kavram

Linear optimization under inequality bounds and integer constraints.

Soru

Cevap

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