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Zorluk: OrtaLinear Inequalities in One Variable

A catering company charges a flat setup fee of 500plus500 plus 35 per guest. A company has budgeted at most 2,400forabanquet.Ifthecompanyalsowantstopurchaseacelebrationcakefor2,400 for a banquet. If the company also wants to purchase a celebration cake for 150, what is the maximum number of guests they can invite to the banquet without exceeding their budget?

Cevap: 50 guests

Cevap

The maximum number of guests the company can invite without exceeding their budget is 50.
The total cost of the banquet consists of a flat setup fee (500),acelebrationcake(500), a celebration cake ( 150), and a variable cost of 35perguest(35 per guest ( 35g ,where, where g representsthenumberofguests).Sincethecompanyhasbudgetedatmost represents the number of guests). Since the company has budgeted at most 2,400, the sum of these expenses must be less than or equal to 2,400.Thisisrepresentedbytheinequality2,400. This is represented by the inequality 35g + 500 + 150 \le 2400 .Combiningtheconstantsyields. Combining the constants yields 35g + 650 \le 2400 .Subtracting650frombothsidesgives. Subtracting 650 from both sides gives 35g \le 1750 .Dividingbothsidesby35resultsin. Dividing both sides by 35 results in g \le 50$. Therefore, the maximum number of guests they can invite without exceeding their budget is 50.

Adım Adım Çözüm

1
Set up the inequality representing the total expenses.
35g+500+150≤240035g + 500 + 150 \le 2400
The sum of the fixed setup fee, the cake cost, and the variable per-guest cost must not exceed the maximum budget of $2,400.
2
Simplify the inequality by combining the constants.
35g+650≤240035g + 650 \le 2400
Combining the fixed expenses (500setupfeeand500 setup fee and 150 cake) simplifies the calculation.
3
Isolate the variable term by subtracting the fixed cost from both sides.
35g≤175035g \le 1750
This determines the portion of the budget that can be allocated specifically to guests.
4
Solve for the variable by dividing both sides by the cost per guest.
g≤50g \le 50
Dividing the remaining guest budget by the rate per guest gives the maximum number of guests allowed.

Anahtar Kavram

Formulating and solving multi-step linear inequalities in one variable to determine a maximum value in context.
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Linear Inequalities in One Variable

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