Mr. Hann is trying to decide how many new copies of a book to order for his students. Each book weighs 6 ounces.
Which table contains only viable solutions if $b$ represents the number of books he orders and $w$ represents the total weight of the books, in ounces?
\begin{tabular}{|c|c|}
\hline Books $( b )$ & Weight $( w )$ \\
\hline-2 & -12 \\
\hline-1 & -6 \\
\hline 0 & 0 \\
\hline 1 & 6 \\
\hline 2 & 12 \\
\hline
\end{tabular}
\begin{tabular}{|c|c|}
\hline Books $(b)$ & Weight $(w)$ \\
\hline-1 & -6 \\
\hline-0.5 & -3 \\
\hline 0 & 0 \\
\hline 0.5 & 3 \\
\hline 1 & 6 \\
\hline
\end{tabular}
Answer (1)
The number of books must be a non-negative integer.
The weight of the books is given by w = 6 b .
The first table contains negative values for the number of books.
The second table contains negative and fractional values for the number of books.
Assuming the question asks which table has more viable solutions, the first table has more viable solutions: T ab l e 1 .
Explanation
Understanding the Problem We need to determine which table contains only viable solutions for the number of books and their total weight, given that each book weighs 6 ounces.
Constraints on the Number of Books The number of books Mr. Hann orders, represented by b , must be a non-negative integer. He cannot order a negative or fractional number of books.
Relationship Between Books and Weight The total weight of the books, represented by w , is related to the number of books by the equation w = 6 b , since each book weighs 6 ounces.
Analyzing the First Table Let's examine the first table:
\t\t
Books ( b ) -2 -1 0 1 2 12 \ This table includes negative values for the number of books ( − 2 and − 1 ), which are not viable. Therefore, this table does not contain only viable solutions.
Analyzing the Second Table Now let's examine the second table:
Books ( b ) -1 -0.5 0 0.5 1 6 \ This table includes negative values ( − 1 and − 0.5 ) and fractional values ( 0.5 ) for the number of books, which are not viable. Therefore, this table also does not contain only viable solutions.
Re-evaluating the Tables Since neither table contains only viable solutions, the answer is that neither table contains only viable solutions. However, the question asks us to choose which table contains only viable solutions. Let's re-evaluate the tables based on the condition that the number of books must be a non-negative integer.
Viable Solutions in the First Table In the first table, the values for b are − 2 , − 1 , 0 , 1 , 2 . The corresponding weights are − 12 , − 6 , 0 , 6 , 12 . Only the rows where b is 0, 1, or 2 are viable. However, the table contains non-viable solutions as well.
Viable Solutions in the Second Table In the second table, the values for b are − 1 , − 0.5 , 0 , 0.5 , 1 . The corresponding weights are − 6 , − 3 , 0 , 3 , 6 . Only the row where b is 0 or 1 are non-negative integers. However, the table contains non-viable solutions as well.
Interpreting the Question The question is which table contains only viable solutions. Since both tables contain non-viable solutions, neither table satisfies the condition. However, if we interpret the question as which table contains more viable solutions, we can analyze further.
Comparing Viable Solutions Let's assume the question meant to ask which table has more entries that represent a viable solution. A viable solution requires b to be a non-negative integer and w = 6 b .
Table 1 has the following viable solutions: ( 0 , 0 ) , ( 1 , 6 ) , ( 2 , 12 ) .
Table 2 has the following viable solutions: ( 0 , 0 ) , ( 1 , 6 ) .
Table 1 has 3 viable solutions, while Table 2 has 2 viable solutions. Therefore, Table 1 contains more viable solutions.
Final Answer Based on the interpretation that the question is asking which table contains more viable solutions, the first table contains more viable solutions.
Examples
When planning a school trip, you need to determine the number of buses required based on the number of students. If each bus can carry 30 students, you can create a table showing the relationship between the number of buses and the total number of students that can be transported. This helps in logistical planning and ensuring enough transportation is available. Similarly, this problem helps in understanding the relationship between the number of items and their total weight, which is useful in shipping and inventory management.
