Join the growing community of curious minds on IDNLearn.com and get the answers you need. Join our community to access reliable and comprehensive responses to your questions from experienced professionals.
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.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for trusting IDNLearn.com with your questions. Visit us again for clear, concise, and accurate answers.
