A barista wants to use Arabica and Robusta beans to make a cup of coffee containing less than 280 milligrams of caffeine. Each ounce of Arabica beans contains 40 milligrams of caffeine and each ounce of Robusta beans contains 60 milligrams of caffeine. If the barista is required to use twice as many Arabica beans as Robusta beans, which inequality gives all possible values for the amount [tex]$a$[/tex], in ounces, of Arabica beans the barista could use?
(A) [tex]$0\ \textless \ a\ \textless \ 1 \frac{3}{4}$[/tex]
(B) [tex]$0\ \textless \ a\ \textless \ 2$[/tex]
(C) [tex]$0\ \textless \ a\ \textless \ 3 \frac{1}{2}$[/tex]
(D) [tex]$0\ \textless \ a\ \textless \ 4$[/tex]
Answer (2)
Express the total caffeine content as an inequality: 40 a + 60 r < 280 .
Use the relationship a = 2 r to substitute r = 2 a into the inequality.
Simplify the inequality to 70 a < 280 .
Solve for a to find the possible values: 0 < a < 4 .
The inequality that gives all possible values for a is 0 < a < 4 .
Explanation
Setting up the variables and constraints Let a be the amount of Arabica beans in ounces and r be the amount of Robusta beans in ounces. We know that the amount of caffeine from Arabica beans is 40 milligrams per ounce, and the amount of caffeine from Robusta beans is 60 milligrams per ounce. The total caffeine content must be less than 280 milligrams. Also, the barista uses twice as many Arabica beans as Robusta beans, which means a = 2 r .
Formulating the inequality We can write the inequality representing the total caffeine content as: 40 a + 60 r < 280
Substituting the relationship between a and r Since a = 2 r , we can express r in terms of a as r = 2 a . Substitute this into the inequality: 40 a + 60 ( 2 a ) < 280
Simplifying the inequality Simplify the inequality: 40 a + 30 a < 280
Combining like terms Combine the terms with a :
70 a < 280
Solving for a Divide both sides by 70 to solve for a :
a < 70 280 a < 4
Finding the possible values for a Since the amount of Arabica beans must be greater than 0, we have 0 < a < 4 .
Final Answer Therefore, the possible values for the amount a , in ounces, of Arabica beans the barista could use are given by the inequality 0 < a < 4 .
Examples
Imagine you're mixing ingredients for a special recipe, like our barista blending coffee beans. You need to control the amount of a key ingredient, similar to caffeine, to keep the recipe's flavor just right. By using inequalities, you can determine the range of ingredient amounts that will result in a perfect dish, ensuring it's neither too strong nor too weak. This is useful in baking, cooking, and even mixing drinks, where precise measurements are crucial for the best outcome.
The solution involves setting up an inequality based on caffeine content and the relationship between Arabica and Robusta beans. After substituting and simplifying, we find that the possible amounts of Arabica beans are given by the inequality 0 < a < 4 . The correct answer choice is (D) 0 < a < 4 .
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