A chef is preparing portions of roasted vegetables and mashed potatoes for a catered event… — SAT

A chef is preparing portions of roasted vegetables and mashed potatoes for a catered event. Each portion of roasted vegetables requires $3$ ounces of potatoes, and each portion of mashed potatoes requires $6$ ounces of potatoes. The chef has a total of at most $90$ ounces of potatoes available. The chef must prepare at least $8$ portions of roasted vegetables and at least $5$ portions of mashed potatoes. If $v$ represents the number of portions of roasted vegetables that the chef prepares, what is the maximum possible value of $v$ ?

Cevap: 20

Cevap

The maximum possible value of

v

is 20.

The maximum number of portions of roasted vegetables the chef can prepare is 20 because minimizing the number of mashed potato portions to its boundary constraint of 5 maximizes the remaining resources, yielding

3v \leq 90 - 6(5) \implies 3v \leq 60 \implies v \leq 20

Adım Adım Çözüm

Formulate the inequality representing the potato weight constraint.

3v + 6p \leq 90

, where

v

is the number of portions of roasted vegetables and

p

is the number of portions of mashed potatoes.

Each portion of roasted vegetables uses

3

ounces of potatoes, each portion of mashed potatoes uses

6

ounces of potatoes, and the total amount used cannot exceed

90

ounces.

Formulate the inequalities representing the minimum quantity constraints.

v \geq 8

and

p \geq 5

The chef must make at least

8

portions of roasted vegetables and at least

5

portions of mashed potatoes.

Isolate the variable

v

in the potato limit inequality.

v \leq 30 - 2p

Subtracting

6p

from both sides of

3v + 6p \leq 90

yields

3v \leq 90 - 6p

, and dividing the entire inequality by

3

gives

v \leq 30 - 2p

Determine the maximum value of

v

by substituting the minimum possible value of

p

v \leq 30 - 2(5) \implies v \leq 20

To maximize

v

, we must minimize the subtracted term

2p

. The minimum allowed value of

p

5

Anahtar Kavram

Optimization in systems of linear inequalities

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