Every morning, a bakery makes wheat bread. The staff bakes 20 loaves before the bakery opens at 7 am and then bakes 10 more loaves every half-hour for the next three hours. If the number of loaves they have finished baking depends on the number of hours that have passed since the bakery opened, what is the range of this relation?
* {0,0.5,1,1.5,2,2.5,3}
* {20,30,40,50,60,70,80}
* {20,25,30,35,40,45,50}
* {0,1,2,3,4,5,6}
Answer (2)
The bakery starts with 20 loaves.
They bake 10 loaves every half-hour for 3 hours, which is 6 intervals.
Calculate the number of loaves baked at each half-hour interval using the formula: L o a v es = 20 + 20 h .
The range of the relation is 20 , 30 , 40 , 50 , 60 , 70 , 80 .
Explanation
Understanding the Problem We are given that a bakery makes wheat bread every morning. They start by baking 20 loaves before the bakery opens at 7am. Then, they bake 10 more loaves every half-hour for the next three hours. We need to find the range of the relation between the number of hours passed since opening and the number of loaves baked.
Defining the Variable Let's denote the number of hours passed since the bakery opened as h . The bakery bakes for 3 hours after opening, and they bake 10 loaves every half-hour. This means they bake 10 loaves every 0.5 hours. The possible values for h are 0, 0.5, 1, 1.5, 2, 2.5, and 3.
Calculating the Number of Loaves We can calculate the number of loaves baked for each value of h . The number of loaves baked before opening is 20. For each half-hour, 10 loaves are baked. So, the total number of loaves baked after h hours is given by the formula: L o a v es = 20 + 10 × ( 0.5 h ) = 20 + 20 h
Finding the Range Now, let's calculate the number of loaves for each value of h :
When h = 0 , L o a v es = 20 + 20 ( 0 ) = 20
When h = 0.5 , L o a v es = 20 + 20 ( 0.5 ) = 20 + 10 = 30
When h = 1 , L o a v es = 20 + 20 ( 1 ) = 20 + 20 = 40
When h = 1.5 , L o a v es = 20 + 20 ( 1.5 ) = 20 + 30 = 50
When h = 2 , L o a v es = 20 + 20 ( 2 ) = 20 + 40 = 60
When h = 2.5 , L o a v es = 20 + 20 ( 2.5 ) = 20 + 50 = 70
When h = 3 , $Loaves = 20 + 20(3) = 20 + 60 = 80
Determining the Range The range of the relation is the set of all possible numbers of loaves baked. From our calculations, the range is {20, 30, 40, 50, 60, 70, 80}.
Final Answer Therefore, the correct answer is {20, 30, 40, 50, 60, 70, 80}.
Examples
Understanding the range of a function is crucial in many real-world scenarios. For instance, if you're managing a store, knowing the range of possible sales based on different marketing efforts helps you plan inventory and staffing. Similarly, in manufacturing, understanding the range of production output based on available resources allows for efficient resource allocation and meeting customer demand. This concept is also applicable in financial planning, where understanding the range of possible investment returns helps in making informed decisions and managing risk.
The bakery bakes a total of 20 loaves initially, and then 10 loaves for each of the 6 half-hour intervals over three hours. The calculated total number of loaves baked ranges from 20 to 80. Therefore, the correct answer is {20, 30, 40, 50, 60, 70, 80}.
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