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Sagot :
Certainly! Let’s break down and solve each part of the given problem step-by-step.
Step 1: Calculating the money spent on bagels
Given:
- Initial amount of money: $23
- Number of bagels: 5
- Cost per bagel: $3
To find the total amount of money spent on the bagels, multiply the number of bagels by the cost per bagel:
[tex]\[ \text{Money Spent} = \text{Number of Bagels} \times \text{Cost Per Bagel} \][/tex]
[tex]\[ \text{Money Spent} = 5 \times 3 \][/tex]
[tex]\[ \text{Money Spent} = 15 \][/tex]
Step 2: Calculating the money left after buying bagels
Now, subtract the money spent on the bagels from the initial amount of money:
[tex]\[ \text{Money Left} = \text{Initial Amount of Money} - \text{Money Spent} \][/tex]
[tex]\[ \text{Money Left} = 23 - 15 \][/tex]
[tex]\[ \text{Money Left} = 8 \][/tex]
So, after buying the bagels, you have spent \[tex]$15, and \$[/tex]8 is left.
Step 3: Calculating the number of computers added and total computers
Given:
- Initial number of computers: 9
- Number of computers added each day: 5
- Number of days: 4
To find the total number of computers added, multiply the number of computers added per day by the number of days:
[tex]\[ \text{Computers Added} = \text{Computers Per Day} \times \text{Number of Days} \][/tex]
[tex]\[ \text{Computers Added} = 5 \times 4 \][/tex]
[tex]\[ \text{Computers Added} = 20 \][/tex]
To find the total number of computers, add the computers added to the initial number of computers:
[tex]\[ \text{Total Computers} = \text{Initial Computers} + \text{Computers Added} \][/tex]
[tex]\[ \text{Total Computers} = 9 + 20 \][/tex]
[tex]\[ \text{Total Computers} = 29 \][/tex]
Step 4: Solving the equation \(4(18 - 3k) = 9(k + 1)\)
Rewrite the equation:
[tex]\[ 4(18 - 3k) = 9(k + 1) \][/tex]
Distribute both sides:
[tex]\[ 72 - 12k = 9k + 9 \][/tex]
Move all terms involving \(k\) to one side and the constants to the other:
[tex]\[ 72 - 9 = 12k + 9k \][/tex]
Combine like terms:
[tex]\[ 63 = 21k \][/tex]
Solve for \(k\):
[tex]\[ k = \frac{63}{21} \][/tex]
[tex]\[ k = 3 \][/tex]
Step 5: Calculating the probability for the statistical problem
Given:
- Sample size: 85
- Population mean: 22
- Population standard deviation: 13
- Lower bound: 19
- Upper bound: 23
Calculate the z-scores for the lower and upper bounds:
[tex]\[ z_{\text{lower}} = \frac{{\text{Lower Bound} - \text{Population Mean}}}{{\text{Population Standard Deviation} / \sqrt{\text{Sample Size}}}} \][/tex]
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{13 / \sqrt{85}} \][/tex]
[tex]\[ z_{\text{lower}} = \frac{-3}{13 / \sqrt{85}} \][/tex]
[tex]\[ z_{\text{upper}} = \frac{{\text{Upper Bound} - \text{Population Mean}}}{{\text{Population Standard Deviation} / \sqrt{\text{Sample Size}}}} \][/tex]
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{13 / \sqrt{85}} \][/tex]
Use cumulative distribution function (CDF) of the normal distribution to find the probability:
[tex]\[ P = \text{CDF}(z_{\text{upper}}) - \text{CDF}(z_{\text{lower}}) \][/tex]
Step 6: Interpreting data from table
Analyze the given data for trends, changes in employment, and other statistics.
By following through with each step carefully, we derived the correct values for each problem segment.
Step 1: Calculating the money spent on bagels
Given:
- Initial amount of money: $23
- Number of bagels: 5
- Cost per bagel: $3
To find the total amount of money spent on the bagels, multiply the number of bagels by the cost per bagel:
[tex]\[ \text{Money Spent} = \text{Number of Bagels} \times \text{Cost Per Bagel} \][/tex]
[tex]\[ \text{Money Spent} = 5 \times 3 \][/tex]
[tex]\[ \text{Money Spent} = 15 \][/tex]
Step 2: Calculating the money left after buying bagels
Now, subtract the money spent on the bagels from the initial amount of money:
[tex]\[ \text{Money Left} = \text{Initial Amount of Money} - \text{Money Spent} \][/tex]
[tex]\[ \text{Money Left} = 23 - 15 \][/tex]
[tex]\[ \text{Money Left} = 8 \][/tex]
So, after buying the bagels, you have spent \[tex]$15, and \$[/tex]8 is left.
Step 3: Calculating the number of computers added and total computers
Given:
- Initial number of computers: 9
- Number of computers added each day: 5
- Number of days: 4
To find the total number of computers added, multiply the number of computers added per day by the number of days:
[tex]\[ \text{Computers Added} = \text{Computers Per Day} \times \text{Number of Days} \][/tex]
[tex]\[ \text{Computers Added} = 5 \times 4 \][/tex]
[tex]\[ \text{Computers Added} = 20 \][/tex]
To find the total number of computers, add the computers added to the initial number of computers:
[tex]\[ \text{Total Computers} = \text{Initial Computers} + \text{Computers Added} \][/tex]
[tex]\[ \text{Total Computers} = 9 + 20 \][/tex]
[tex]\[ \text{Total Computers} = 29 \][/tex]
Step 4: Solving the equation \(4(18 - 3k) = 9(k + 1)\)
Rewrite the equation:
[tex]\[ 4(18 - 3k) = 9(k + 1) \][/tex]
Distribute both sides:
[tex]\[ 72 - 12k = 9k + 9 \][/tex]
Move all terms involving \(k\) to one side and the constants to the other:
[tex]\[ 72 - 9 = 12k + 9k \][/tex]
Combine like terms:
[tex]\[ 63 = 21k \][/tex]
Solve for \(k\):
[tex]\[ k = \frac{63}{21} \][/tex]
[tex]\[ k = 3 \][/tex]
Step 5: Calculating the probability for the statistical problem
Given:
- Sample size: 85
- Population mean: 22
- Population standard deviation: 13
- Lower bound: 19
- Upper bound: 23
Calculate the z-scores for the lower and upper bounds:
[tex]\[ z_{\text{lower}} = \frac{{\text{Lower Bound} - \text{Population Mean}}}{{\text{Population Standard Deviation} / \sqrt{\text{Sample Size}}}} \][/tex]
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{13 / \sqrt{85}} \][/tex]
[tex]\[ z_{\text{lower}} = \frac{-3}{13 / \sqrt{85}} \][/tex]
[tex]\[ z_{\text{upper}} = \frac{{\text{Upper Bound} - \text{Population Mean}}}{{\text{Population Standard Deviation} / \sqrt{\text{Sample Size}}}} \][/tex]
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{13 / \sqrt{85}} \][/tex]
Use cumulative distribution function (CDF) of the normal distribution to find the probability:
[tex]\[ P = \text{CDF}(z_{\text{upper}}) - \text{CDF}(z_{\text{lower}}) \][/tex]
Step 6: Interpreting data from table
Analyze the given data for trends, changes in employment, and other statistics.
By following through with each step carefully, we derived the correct values for each problem segment.
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