Question
A company offers four plans for customers who want to rent DVDs and video games. The costs of the plans can be modeled by a linear function that combines a one-time membership fee with a per disc rental rate. The table shows the total costs, including membership fees, for a person who rents 1, 2, and 5 discs in a month.
Cost of Repairs
Discs rented
Plan A
Plan B
Plan C
Plan D
1
$14
$12
$10
$12
2
$17
$16
$15
$21
5
$26
$28
$30
$48
Which plan has the smallest one-time membership fee?
plan A
plan B
plan C
plan D
Cost of Repairs
Discs rented
Plan A
Plan B
Plan C
Plan D
1
$14
$12
$10
$12
2
$17
$16
$15
$21
5
$26
$28
$30
$48
Which plan has the smallest one-time membership fee?
plan A
plan B
plan C
plan D
Answers
GPT-4o mini
To determine which plan has the smallest one-time membership fee, we can analyze the provided costs for different numbers of discs rented. The total cost can be expressed as:
\[ \text{Total Cost} = \text{Membership Fee} + (\text{Per Disc Rate} \times \text{Number of Discs}) \]
We can use the provided costs to compute the membership fee for each plan for the minimum number of discs rented, then validate it against other rental amounts.
Let's analyze the costs given for each plan:
1. For **1 disc rented**:
- Plan A: $14
- Plan B: $12
- Plan C: $10
- Plan D: $12
2. For **2 discs rented**:
- Plan A: $17
- Plan B: $16
- Plan C: $15
- Plan D: $21
3. For **5 discs rented**:
- Plan A: $26
- Plan B: $28
- Plan C: $30
- Plan D: $48
Now, we can calculate the membership fee for each plan by choosing one of the rental amounts (we will use all available to ensure our calculations are accurate).
### Calculate for each plan
#### Plan A
- For 1 disc:
\[
14 = M_A + (R_A \times 1) \rightarrow M_A + R_A = 14
\]
- For 2 discs:
\[
17 = M_A + (R_A \times 2) \rightarrow M_A + 2R_A = 17
\]
Let's subtract the first equation from the second:
\[
(M_A + 2R_A) - (M_A + R_A) = 17 - 14 \rightarrow R_A = 3
\]
Substituting \( R_A \) back to find \( M_A \):
\[
M_A + 3 = 14 \rightarrow M_A = 11
\]
#### Plan B
- For 1 disc:
\[
12 = M_B + (R_B \times 1) \rightarrow M_B + R_B = 12
\]
- For 2 discs:
\[
16 = M_B + (R_B \times 2) \rightarrow M_B + 2R_B = 16
\]
Subtract the equations:
\[
R_B = 4
\]
Substituting back:
\[
M_B + 4 = 12 \rightarrow M_B = 8
\]
#### Plan C
- For 1 disc:
\[
10 = M_C + (R_C \times 1) \rightarrow M_C + R_C = 10
\]
- For 2 discs:
\[
15 = M_C + (R_C \times 2) \rightarrow M_C + 2R_C = 15
\]
Subtract the equations:
\[
R_C = 5
\]
Substituting back:
\[
M_C + 5 = 10 \rightarrow M_C = 5
\]
#### Plan D
- For 1 disc:
\[
12 = M_D + (R_D \times 1) \rightarrow M_D + R_D = 12
\]
- For 2 discs:
\[
21 = M_D + (R_D \times 2) \rightarrow M_D + 2R_D = 21
\]
Subtract the equations:
\[
R_D = 9
\]
Substituting back:
\[
M_D + 9 = 12 \rightarrow M_D = 3
\]
### Summary of Membership Fees
- Plan A: $11
- Plan B: $8
- Plan C: $5
- Plan D: $3
The plan with the smallest one-time membership fee is **Plan D** with a membership fee of **$3**.
\[ \text{Total Cost} = \text{Membership Fee} + (\text{Per Disc Rate} \times \text{Number of Discs}) \]
We can use the provided costs to compute the membership fee for each plan for the minimum number of discs rented, then validate it against other rental amounts.
Let's analyze the costs given for each plan:
1. For **1 disc rented**:
- Plan A: $14
- Plan B: $12
- Plan C: $10
- Plan D: $12
2. For **2 discs rented**:
- Plan A: $17
- Plan B: $16
- Plan C: $15
- Plan D: $21
3. For **5 discs rented**:
- Plan A: $26
- Plan B: $28
- Plan C: $30
- Plan D: $48
Now, we can calculate the membership fee for each plan by choosing one of the rental amounts (we will use all available to ensure our calculations are accurate).
### Calculate for each plan
#### Plan A
- For 1 disc:
\[
14 = M_A + (R_A \times 1) \rightarrow M_A + R_A = 14
\]
- For 2 discs:
\[
17 = M_A + (R_A \times 2) \rightarrow M_A + 2R_A = 17
\]
Let's subtract the first equation from the second:
\[
(M_A + 2R_A) - (M_A + R_A) = 17 - 14 \rightarrow R_A = 3
\]
Substituting \( R_A \) back to find \( M_A \):
\[
M_A + 3 = 14 \rightarrow M_A = 11
\]
#### Plan B
- For 1 disc:
\[
12 = M_B + (R_B \times 1) \rightarrow M_B + R_B = 12
\]
- For 2 discs:
\[
16 = M_B + (R_B \times 2) \rightarrow M_B + 2R_B = 16
\]
Subtract the equations:
\[
R_B = 4
\]
Substituting back:
\[
M_B + 4 = 12 \rightarrow M_B = 8
\]
#### Plan C
- For 1 disc:
\[
10 = M_C + (R_C \times 1) \rightarrow M_C + R_C = 10
\]
- For 2 discs:
\[
15 = M_C + (R_C \times 2) \rightarrow M_C + 2R_C = 15
\]
Subtract the equations:
\[
R_C = 5
\]
Substituting back:
\[
M_C + 5 = 10 \rightarrow M_C = 5
\]
#### Plan D
- For 1 disc:
\[
12 = M_D + (R_D \times 1) \rightarrow M_D + R_D = 12
\]
- For 2 discs:
\[
21 = M_D + (R_D \times 2) \rightarrow M_D + 2R_D = 21
\]
Subtract the equations:
\[
R_D = 9
\]
Substituting back:
\[
M_D + 9 = 12 \rightarrow M_D = 3
\]
### Summary of Membership Fees
- Plan A: $11
- Plan B: $8
- Plan C: $5
- Plan D: $3
The plan with the smallest one-time membership fee is **Plan D** with a membership fee of **$3**.