Math  /  Data & Statistics

Question\begin{align*} \text{Table 1:} \\ \begin{array}{|c|c|} \hline \text{Input} & \text{Output} \\ \hline 3 & 18 \\ 8 & 43 \\ 9 & 48 \\ 2 & - \\ 5 & \\ \hline \end{array} \end{align*}
\begin{align*} \text{Table 2:} \\ \begin{array}{|c|c|} \hline \text{Input} & \text{Output} \\ \hline 6 & 8 \\ 8 & 9 \\ 2 & 6 \\ 10 & - \\ c & - \\ \hline \end{array} \end{align*}
\text{Find the missing numbers and then calculate the rule for each table.}

Studdy Solution

STEP 1

1. Each table represents a function with a specific rule that maps inputs to outputs.
2. The missing outputs in each table can be determined by identifying the pattern or rule from the given data.
3. The rule for each table can be expressed as a mathematical function or equation.

STEP 2

1. Analyze Table 1 to find the rule and determine missing outputs.
2. Analyze Table 2 to find the rule and determine missing outputs.

STEP 3

Identify the pattern in Table 1 by examining the given input-output pairs:
- For Input 33, Output is 1818. - For Input 88, Output is 4343. - For Input 99, Output is 4848.

STEP 4

Determine the rule for Table 1 by finding a consistent mathematical relationship:
Assume a linear relationship: Output=a×Input+b \text{Output} = a \times \text{Input} + b .
Using the points (3,18)(3, 18) and (8,43)(8, 43), solve for aa and bb.
\begin{align*} 18 &= 3a + b \\ 43 &= 8a + b \end{align*}
Subtract the first equation from the second:
\begin{align*} 43 - 18 &= (8a + b) - (3a + b) \\ 25 &= 5a \\ a &= 5 \end{align*}
Substitute a=5a = 5 into the first equation:
\begin{align*} 18 &= 3(5) + b \\ 18 &= 15 + b \\ b &= 3 \end{align*}
The rule is: Output=5×Input+3 \text{Output} = 5 \times \text{Input} + 3 .

STEP 5

Use the rule to find the missing outputs for Input 22 and 55:
- For Input 22: Output=5×2+3=13 \text{Output} = 5 \times 2 + 3 = 13 . - For Input 55: Output=5×5+3=28 \text{Output} = 5 \times 5 + 3 = 28 .

STEP 6

Identify the pattern in Table 2 by examining the given input-output pairs:
- For Input 66, Output is 88. - For Input 88, Output is 99. - For Input 22, Output is 66.

STEP 7

Determine the rule for Table 2 by finding a consistent mathematical relationship:
Assume a linear relationship: Output=a×Input+b \text{Output} = a \times \text{Input} + b .
Using the points (6,8)(6, 8) and (8,9)(8, 9), solve for aa and bb.
\begin{align*} 8 &= 6a + b \\ 9 &= 8a + b \end{align*}
Subtract the first equation from the second:
\begin{align*} 9 - 8 &= (8a + b) - (6a + b) \\ 1 &= 2a \\ a &= 0.5 \end{align*}
Substitute a=0.5a = 0.5 into the first equation:
\begin{align*} 8 &= 6(0.5) + b \\ 8 &= 3 + b \\ b &= 5 \end{align*}
The rule is: Output=0.5×Input+5 \text{Output} = 0.5 \times \text{Input} + 5 .

STEP 8

Use the rule to find the missing outputs for Input 1010 and cc:
- For Input 1010: Output=0.5×10+5=10 \text{Output} = 0.5 \times 10 + 5 = 10 . - For Input cc: Output=0.5×c+5 \text{Output} = 0.5 \times c + 5 .
The missing outputs and rules for the tables are:
- Table 1: Missing outputs are 1313 and 2828. Rule: Output=5×Input+3 \text{Output} = 5 \times \text{Input} + 3 . - Table 2: Missing outputs are 1010 and 0.5×c+50.5 \times c + 5. Rule: Output=0.5×Input+5 \text{Output} = 0.5 \times \text{Input} + 5 .

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