Answer :
Sure, let's work through this step-by-step to make sense of the problem. However, before jumping into the solution, it's crucial to ensure we understand each part of the question and how to approach it.
Based on the provided results:
1. Olivia's bagel purchase problem.
2. Installation of additional computers.
3. Solving an equation involving distributive property.
4. Finding the probability for a given range in a normally distributed population.
Let's summarize and explain each part:
1. Olivia's Bagel Purchase Problem:
- Initial money: [tex]$23 - Cost per bagel: $[/tex]3
- Number of bagels bought: 5
Money spent on bagels:
[tex]\[ \text{Money spent} = 5 \text{ bagels} \times 3 \text{ dollars/bagel} = 15 \text{ dollars} \][/tex]
Remaining money:
[tex]\[ \text{Remaining money} = 23 \text{ dollars} - 15 \text{ dollars} = 8 \text{ dollars} \][/tex]
Result: Olivia spent [tex]$15 and has $[/tex]8 left.
2. Installation of Additional Computers:
- Initial computers: 9
- Additional computers per day: 5
- Number of days (Monday to Thursday): 4
Total additional computers installed:
[tex]\[ \text{Total additional computers} = 5 \text{ computers/day} \times 4 \text{ days} = 20 \text{ computers} \][/tex]
Total computers after installation:
[tex]\[ \text{Total computers} = 9 \text{ initial computers} + 20 \text{ additional computers} = 29 \text{ computers} \][/tex]
Result: 20 computers were added, making a total of 29 computers.
3. Solving the Equation using the Distributive Property:
- Equation: [tex]\( 4(18 - 3k) = 9(k + 1) \)[/tex]
Step-by-step solution:
[tex]\[ 4 \times (18 - 3k) = 72 - 12k \][/tex]
[tex]\[ 9 \times (k + 1) = 9k + 9 \][/tex]
Set the equations equal to each other:
[tex]\[ 72 - 12k = 9k + 9 \][/tex]
Combine like terms:
[tex]\[ 72 - 9 = 9k + 12k \][/tex]
[tex]\[ 63 = 21k \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{63}{21} = 3 \][/tex]
Result: The solution for [tex]\( k \)[/tex] is 3.
4. Probability Calculation:
- Sample Size: 85
- Population Mean: 22
- Population Standard Deviation: 13
- Lower bound (x): 19
- Upper bound (x): 23
Z-scores for the lower and upper bound:
Z-score for lower bound:
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{13 / \sqrt{85}} \approx -2.128 \][/tex]
Z-score for upper bound:
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{13 / \sqrt{85}} \approx 0.709 \][/tex]
Probability (using cumulative distribution function):
[tex]\[ \text{Probability} = \text{CDF}(0.709) - \text{CDF}(-2.128) \approx 0.744 \][/tex]
Result: The probability that [tex]\( x \)[/tex] will be between 19 and 23 is approximately 0.744.
By going through these structured steps, each part of the problem is methodically solved, and we end up with the following results:
- [tex]$(15, 8)$[/tex] for Olivia's purchase,
- [tex]$(20, 29)$[/tex] for the computers,
- [tex]$3$[/tex] for the equation solution,
- and [tex]$( -2.1275871824522046, 0.7091957274840682, 0.7442128248197002)$[/tex] for the probability problem.
Based on the provided results:
1. Olivia's bagel purchase problem.
2. Installation of additional computers.
3. Solving an equation involving distributive property.
4. Finding the probability for a given range in a normally distributed population.
Let's summarize and explain each part:
1. Olivia's Bagel Purchase Problem:
- Initial money: [tex]$23 - Cost per bagel: $[/tex]3
- Number of bagels bought: 5
Money spent on bagels:
[tex]\[ \text{Money spent} = 5 \text{ bagels} \times 3 \text{ dollars/bagel} = 15 \text{ dollars} \][/tex]
Remaining money:
[tex]\[ \text{Remaining money} = 23 \text{ dollars} - 15 \text{ dollars} = 8 \text{ dollars} \][/tex]
Result: Olivia spent [tex]$15 and has $[/tex]8 left.
2. Installation of Additional Computers:
- Initial computers: 9
- Additional computers per day: 5
- Number of days (Monday to Thursday): 4
Total additional computers installed:
[tex]\[ \text{Total additional computers} = 5 \text{ computers/day} \times 4 \text{ days} = 20 \text{ computers} \][/tex]
Total computers after installation:
[tex]\[ \text{Total computers} = 9 \text{ initial computers} + 20 \text{ additional computers} = 29 \text{ computers} \][/tex]
Result: 20 computers were added, making a total of 29 computers.
3. Solving the Equation using the Distributive Property:
- Equation: [tex]\( 4(18 - 3k) = 9(k + 1) \)[/tex]
Step-by-step solution:
[tex]\[ 4 \times (18 - 3k) = 72 - 12k \][/tex]
[tex]\[ 9 \times (k + 1) = 9k + 9 \][/tex]
Set the equations equal to each other:
[tex]\[ 72 - 12k = 9k + 9 \][/tex]
Combine like terms:
[tex]\[ 72 - 9 = 9k + 12k \][/tex]
[tex]\[ 63 = 21k \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{63}{21} = 3 \][/tex]
Result: The solution for [tex]\( k \)[/tex] is 3.
4. Probability Calculation:
- Sample Size: 85
- Population Mean: 22
- Population Standard Deviation: 13
- Lower bound (x): 19
- Upper bound (x): 23
Z-scores for the lower and upper bound:
Z-score for lower bound:
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{13 / \sqrt{85}} \approx -2.128 \][/tex]
Z-score for upper bound:
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{13 / \sqrt{85}} \approx 0.709 \][/tex]
Probability (using cumulative distribution function):
[tex]\[ \text{Probability} = \text{CDF}(0.709) - \text{CDF}(-2.128) \approx 0.744 \][/tex]
Result: The probability that [tex]\( x \)[/tex] will be between 19 and 23 is approximately 0.744.
By going through these structured steps, each part of the problem is methodically solved, and we end up with the following results:
- [tex]$(15, 8)$[/tex] for Olivia's purchase,
- [tex]$(20, 29)$[/tex] for the computers,
- [tex]$3$[/tex] for the equation solution,
- and [tex]$( -2.1275871824522046, 0.7091957274840682, 0.7442128248197002)$[/tex] for the probability problem.
