A tool manufacturer wants to produce a wrench set that consists of a toolbox and 7 wrenches. The toolbox costs $t$ dollars to produce and each wrench costs $w$ dollars to produce. The total cost of producing the wrench set is at least 4 times but no more than 6 times the cost of producing the toolbox. Which inequality represents all possible values of [tex]$w$[/tex]?
(A) [tex]$\frac{3}{7} t \leq w \leq \frac{5}{7} t$[/tex]
(B) [tex]$3 t \leq w \leq 5 t$[/tex]
(C) [tex]$\frac{4}{7} t \leq w \leq \frac{6}{7} t$[/tex]
(D) [tex]$4 t \leq w \leq 6 t$[/tex]
Answer (2)
Express the total cost of the wrench set: t + 7 w .
Set up the inequality based on the given conditions: 4 t ≤ t + 7 w ≤ 6 t .
Isolate w by subtracting t and dividing by 7: 7 3 t ≤ w ≤ 7 5 t .
The possible values of w are represented by: 7 3 t ≤ w ≤ 7 5 t .
Explanation
Understanding the Problem We are given that the total cost of producing the wrench set is at least 4 times but no more than 6 times the cost of producing the toolbox. The toolbox costs t dollars to produce and each of the 7 wrenches costs w dollars to produce. We need to find the inequality that represents all possible values of w .
Setting up the Inequality The total cost of the wrench set is the cost of the toolbox plus the cost of the 7 wrenches, which is t + 7 w . The problem states that this total cost is at least 4 t and no more than 6 t . This can be written as the compound inequality: 4 t ≤ t + 7 w ≤ 6 t
Isolating the Term with w To isolate w , we first subtract t from all parts of the inequality: 4 t − t ≤ t + 7 w − t ≤ 6 t − t 3 t ≤ 7 w ≤ 5 t
Solving for w Now, we divide all parts of the inequality by 7: 7 3 t ≤ 7 7 w ≤ 7 5 t 7 3 t ≤ w ≤ 7 5 t
Final Answer The inequality that represents all possible values of w is 7 3 t ≤ w ≤ 7 5 t . This corresponds to option (A).
Examples
Imagine you're running a small business that produces tool sets. This problem helps you determine the acceptable cost range for each wrench to ensure your total production costs stay within a desired budget, relative to the cost of the toolbox. By understanding these inequalities, you can make informed decisions about pricing and production efficiency. For example, if the toolbox costs 14 , t h e n t h ecos t o f e a c h w re n c h s h o u l d b e b e tw ee n \frac{3}{7} \times 14 = 6 an d \frac{5}{7} \times 14 = $10 to keep the total cost within the specified range.
The inequality representing the cost of each wrench, w , in relation to the toolbox cost t is 7 3 t ≤ w ≤ 7 5 t . This means the cost of each wrench should be between three-sevenths and five-sevenths of the toolbox cost. The correct multiple-choice option is (A).
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