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Sagot :
Certainly! Let's break down each part of the problem step-by-step to derive our results.
### Part 1: Olivia's Money After Buying Bagels
1. Initial Amount of Money:
Olivia starts with [tex]$23. 2. Cost of One Bagel: Each bagel costs $[/tex]3.
3. Number of Bagels Bought:
Olivia buys 5 bagels.
4. Calculate the Total Money Spent:
Total money spent = Number of bagels * Cost per bagel
[tex]\[ 5 \text{ bagels} \times \$3/\text{bagel} = \$15 \][/tex]
5. Calculate the Money Left:
Money left = Initial amount of money - Total money spent
[tex]\[ \$23 - \$15 = \$8 \][/tex]
So, Olivia spends \[tex]$15 and she has \$[/tex]8 left.
### Part 2: Total Number of Computers in the Server Room
1. Initial Number of Computers:
There are 9 computers initially.
2. Number of Computers Added Per Day:
5 computers are added each day.
3. Number of Days:
The computers are added for 4 days.
4. Calculate Total Number of Computers Added:
Total computers added = Number of computers added per day * Number of days
[tex]\[ 5 \text{ computers/day} \times 4 \text{ days} = 20 \text{ computers} \][/tex]
5. Calculate the Total Number of Computers:
Total number of computers = Initial number of computers + Total computers added
[tex]\[ 9 + 20 = 29 \text{ computers} \][/tex]
So, a total of 20 computers are added, making the total 29 computers in the server room.
### Part 3: Solving the Equation [tex]\(4(18 - 3k) = 9(k + 1)\)[/tex]
1. Distribute Both Sides of the Equation:
[tex]\[ 4(18 - 3k) = 72 - 12k \][/tex]
[tex]\[ 9(k + 1) = 9k + 9 \][/tex]
2. Set the Equation:
[tex]\[ 72 - 12k = 9k + 9 \][/tex]
3. Isolate the Variable [tex]\(k\)[/tex]:
[tex]\[ 72 - 12k = 9k + 9 \implies 72 - 9 = 12k + 9k \implies 63 = 21k \implies k = \frac{63}{21} = 3 \][/tex]
So, the solution to the equation is [tex]\(k = 3\)[/tex].
### Part 4: Probability Calculation (Between 19 and 23 in a Population)
1. Population Mean ([tex]\(\mu\)[/tex]):
The mean is 22.
2. Population Standard Deviation ([tex]\(\sigma\)[/tex]):
The standard deviation is 13.
3. Sample Size (n):
The sample size is 85.
4. Lower Bound for [tex]\(x\)[/tex]:
The lower bound is 19.
5. Upper Bound for [tex]\(x\)[/tex]:
The upper bound is 23.
6. Calculate Z-Scores:
Z-Score formula: [tex]\(\frac{x - \mu}{\frac{\sigma}{\sqrt{n}}}\)[/tex]
- Lower Bound Z-Score (19):
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{\frac{13}{\sqrt{85}}} \approx -2.1275871824522046 \][/tex]
- Upper Bound Z-Score (23):
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{\frac{13}{\sqrt{85}}} \approx 0.7091957274840682 \][/tex]
7. Calculate the Probability:
Using cumulative distribution function (CDF) of normal distribution:
[tex]\[ \text{Probability} = \Phi(z_{\text{upper}}) - \Phi(z_{\text{lower}}) \approx 0.7442128248197002 \][/tex]
So, the probability that [tex]\(x\)[/tex] will be between 19 and 23 is approximately 0.7442.
### Summary of Results:
1. Olivia spends \[tex]$15 and has \$[/tex]8 left.
2. A total of 20 computers are added, making the total 29 computers in the server room.
3. The solution to the equation [tex]\(4(18 - 3k) = 9(k + 1)\)[/tex] is [tex]\(k = 3\)[/tex].
4. The probability that [tex]\(x\)[/tex] will be between 19 and 23 is approximately 0.7442.
### Part 1: Olivia's Money After Buying Bagels
1. Initial Amount of Money:
Olivia starts with [tex]$23. 2. Cost of One Bagel: Each bagel costs $[/tex]3.
3. Number of Bagels Bought:
Olivia buys 5 bagels.
4. Calculate the Total Money Spent:
Total money spent = Number of bagels * Cost per bagel
[tex]\[ 5 \text{ bagels} \times \$3/\text{bagel} = \$15 \][/tex]
5. Calculate the Money Left:
Money left = Initial amount of money - Total money spent
[tex]\[ \$23 - \$15 = \$8 \][/tex]
So, Olivia spends \[tex]$15 and she has \$[/tex]8 left.
### Part 2: Total Number of Computers in the Server Room
1. Initial Number of Computers:
There are 9 computers initially.
2. Number of Computers Added Per Day:
5 computers are added each day.
3. Number of Days:
The computers are added for 4 days.
4. Calculate Total Number of Computers Added:
Total computers added = Number of computers added per day * Number of days
[tex]\[ 5 \text{ computers/day} \times 4 \text{ days} = 20 \text{ computers} \][/tex]
5. Calculate the Total Number of Computers:
Total number of computers = Initial number of computers + Total computers added
[tex]\[ 9 + 20 = 29 \text{ computers} \][/tex]
So, a total of 20 computers are added, making the total 29 computers in the server room.
### Part 3: Solving the Equation [tex]\(4(18 - 3k) = 9(k + 1)\)[/tex]
1. Distribute Both Sides of the Equation:
[tex]\[ 4(18 - 3k) = 72 - 12k \][/tex]
[tex]\[ 9(k + 1) = 9k + 9 \][/tex]
2. Set the Equation:
[tex]\[ 72 - 12k = 9k + 9 \][/tex]
3. Isolate the Variable [tex]\(k\)[/tex]:
[tex]\[ 72 - 12k = 9k + 9 \implies 72 - 9 = 12k + 9k \implies 63 = 21k \implies k = \frac{63}{21} = 3 \][/tex]
So, the solution to the equation is [tex]\(k = 3\)[/tex].
### Part 4: Probability Calculation (Between 19 and 23 in a Population)
1. Population Mean ([tex]\(\mu\)[/tex]):
The mean is 22.
2. Population Standard Deviation ([tex]\(\sigma\)[/tex]):
The standard deviation is 13.
3. Sample Size (n):
The sample size is 85.
4. Lower Bound for [tex]\(x\)[/tex]:
The lower bound is 19.
5. Upper Bound for [tex]\(x\)[/tex]:
The upper bound is 23.
6. Calculate Z-Scores:
Z-Score formula: [tex]\(\frac{x - \mu}{\frac{\sigma}{\sqrt{n}}}\)[/tex]
- Lower Bound Z-Score (19):
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{\frac{13}{\sqrt{85}}} \approx -2.1275871824522046 \][/tex]
- Upper Bound Z-Score (23):
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{\frac{13}{\sqrt{85}}} \approx 0.7091957274840682 \][/tex]
7. Calculate the Probability:
Using cumulative distribution function (CDF) of normal distribution:
[tex]\[ \text{Probability} = \Phi(z_{\text{upper}}) - \Phi(z_{\text{lower}}) \approx 0.7442128248197002 \][/tex]
So, the probability that [tex]\(x\)[/tex] will be between 19 and 23 is approximately 0.7442.
### Summary of Results:
1. Olivia spends \[tex]$15 and has \$[/tex]8 left.
2. A total of 20 computers are added, making the total 29 computers in the server room.
3. The solution to the equation [tex]\(4(18 - 3k) = 9(k + 1)\)[/tex] is [tex]\(k = 3\)[/tex].
4. The probability that [tex]\(x\)[/tex] will be between 19 and 23 is approximately 0.7442.
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