Answer :
To find the total parking cost, [tex]\( p(n) \)[/tex], for [tex]\( n \)[/tex] days at the airport parking garage, we need to consider the initial cost for the first day and the incremental costs for subsequent days.
Given the problem:
- The cost is [tex]$\$[/tex] 15[tex]$ for the first day. - The cost is $[/tex]\[tex]$ 10$[/tex] for each subsequent day.
### Analysis of Each Function
1. Function 1: [tex]\( p(1) = 10; p(n) = p(n-1) + 15, \, \text{for} \, n \geq 2 \)[/tex]
For the first day, this function states the cost is [tex]\(10\)[/tex]. This does not match the given condition, as the first day cost should be [tex]$15$[/tex]. Therefore, this function is not correct.
2. Function 2: [tex]\( p(1) = 5; p(n) = p(n-1) + 10, \, \text{for} \, n \geq 2 \)[/tex]
For the first day, this function states the cost is [tex]$5. This also does not match the given condition where the first day cost should be \$[/tex]15. Therefore, this function is not correct.
3. Function 3: [tex]\( p(1) = 15; p(n) = p(n-1) + 10, \, \text{for} \, n \geq 2 \)[/tex]
For the first day, this function gives the correct cost of [tex]$15. For subsequent days, each additional day adds $[/tex]10, which matches the problem statement:
- First day: [tex]$15 - Second day: $[/tex]15 + [tex]$10 = $[/tex]25
- Third day: [tex]$25 + $[/tex]10 = [tex]$35 Thus, this function is correct. 4. Function 4: \( p(n) = 5n + 10 \) This function states a direct formula for any number of days. However, substituting \(n = 1\): \[ p(1) = 5(1) + 10 = 5 + 10 = 15 \] This matches the first day's cost but does not adhere to the incremental cost structure for subsequent days. Thus, checking for \(n = 2\): \[ p(2) = 5(2) + 10 = 10 + 10 = 20 \] But according to the problem, the second day should be $[/tex]25. So, this function is not correct.
5. Function 5: [tex]\( p(n) = 10n + 5 \)[/tex]
This function states a direct formula for any number of days. Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ p(1) = 10(1) + 5 = 10 + 5 = 15 \][/tex]
This matches the first day's cost. For subsequent days:
[tex]\[ p(2) = 10(2) + 5 = 20 + 5 = 25 \][/tex]
[tex]\[ p(3) = 10(3) + 5 = 30 + 5 = 35 \][/tex]
This follows the given cost structure perfectly. Thus, this function is correct.
6. Function 6: [tex]\( p(n) = 10n + 15 \)[/tex]
This function states a direct formula for any number of days. Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ p(1) = 10(1) + 15 = 10 + 15 = 25 \][/tex]
This does not match the first day's cost and thus does not adhere to the given cost structure. Therefore, this function is not correct.
### Summary of Correct Functions
The correct functions are:
- [tex]\( p(1) = 15; p(n) = p(n-1) + 10, \, \text{for} \, n \geq 2 \)[/tex]
- [tex]\( p(n) = 10n + 5 \)[/tex]
Given the problem:
- The cost is [tex]$\$[/tex] 15[tex]$ for the first day. - The cost is $[/tex]\[tex]$ 10$[/tex] for each subsequent day.
### Analysis of Each Function
1. Function 1: [tex]\( p(1) = 10; p(n) = p(n-1) + 15, \, \text{for} \, n \geq 2 \)[/tex]
For the first day, this function states the cost is [tex]\(10\)[/tex]. This does not match the given condition, as the first day cost should be [tex]$15$[/tex]. Therefore, this function is not correct.
2. Function 2: [tex]\( p(1) = 5; p(n) = p(n-1) + 10, \, \text{for} \, n \geq 2 \)[/tex]
For the first day, this function states the cost is [tex]$5. This also does not match the given condition where the first day cost should be \$[/tex]15. Therefore, this function is not correct.
3. Function 3: [tex]\( p(1) = 15; p(n) = p(n-1) + 10, \, \text{for} \, n \geq 2 \)[/tex]
For the first day, this function gives the correct cost of [tex]$15. For subsequent days, each additional day adds $[/tex]10, which matches the problem statement:
- First day: [tex]$15 - Second day: $[/tex]15 + [tex]$10 = $[/tex]25
- Third day: [tex]$25 + $[/tex]10 = [tex]$35 Thus, this function is correct. 4. Function 4: \( p(n) = 5n + 10 \) This function states a direct formula for any number of days. However, substituting \(n = 1\): \[ p(1) = 5(1) + 10 = 5 + 10 = 15 \] This matches the first day's cost but does not adhere to the incremental cost structure for subsequent days. Thus, checking for \(n = 2\): \[ p(2) = 5(2) + 10 = 10 + 10 = 20 \] But according to the problem, the second day should be $[/tex]25. So, this function is not correct.
5. Function 5: [tex]\( p(n) = 10n + 5 \)[/tex]
This function states a direct formula for any number of days. Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ p(1) = 10(1) + 5 = 10 + 5 = 15 \][/tex]
This matches the first day's cost. For subsequent days:
[tex]\[ p(2) = 10(2) + 5 = 20 + 5 = 25 \][/tex]
[tex]\[ p(3) = 10(3) + 5 = 30 + 5 = 35 \][/tex]
This follows the given cost structure perfectly. Thus, this function is correct.
6. Function 6: [tex]\( p(n) = 10n + 15 \)[/tex]
This function states a direct formula for any number of days. Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ p(1) = 10(1) + 15 = 10 + 15 = 25 \][/tex]
This does not match the first day's cost and thus does not adhere to the given cost structure. Therefore, this function is not correct.
### Summary of Correct Functions
The correct functions are:
- [tex]\( p(1) = 15; p(n) = p(n-1) + 10, \, \text{for} \, n \geq 2 \)[/tex]
- [tex]\( p(n) = 10n + 5 \)[/tex]
