Question

21 mark for review the cost of renting a carpet cleaner is $52 for the first day and $26 for each additional day. which of the following functions gives the cost c(d), in dollars, of renting the carpet cleaner for d days, where d is a positive integer? a ( c(d) = 26d + 26 ) b ( c(d) = 26d + 52 ) c ( c(d) = 52d - 26 ) d ( c(d) = 52d + 78 )

Explanation:

Step1: Analyze the cost structure

The first day costs $52, and each additional day (so for LXI0 days after the first) costs $26. So the total cost \(C(d)\) is the first - day cost plus the cost of the additional days.
The cost of the additional days: For \(d\) days, the number of additional days is \(d - 1\), and each additional day costs $26, so the cost for additional days is \(26(d - 1)\). The first - day cost is $52.

Step2: Derive the cost function

$$
LATEXBLOCK0
$$

Wait, there is a mistake above. Wait, maybe the first - day cost is $52, and the additional days: if the first day is $52, and each subsequent day is $26. Wait, let's re - examine. Suppose the first day is $52, and then for each of the remaining \(d - 1\) days, we pay $26. So:

$$
LATEXBLOCK1
$$

But let's check with \(d = 1\): \(C(1)=26\times1 + 26=52\), which is correct. For \(d = 2\): \(C(2)=26\times2+26 = 52 + 26=78\), and the cost should be \(52 + 26=78\), which is correct.
Wait, but let's check the options. Option A is \(C(d)=26d + 26\), option C is \(C(d)=52d-26\). Let's check \(d = 1\) for option C: \(52\times1-26 = 26
eq52\), so C is wrong. Option B: \(26d + 52\), for \(d = 1\), \(26 + 52=78
eq52\), wrong. Option D: \(52d+78\), for \(d = 1\), \(52 + 78 = 130
eq52\), wrong. Wait, maybe my initial analysis of the first - day cost is wrong. Wait, maybe the first day is $52, and the rate per day is $26, but the first day is included in the "per - day" rate? No, the problem says "the cost of renting a carpet cleaner is $52 for the first day and $26 for each additional day". So for \(d = 1\), cost is $52; for \(d = 2\), cost is \(52 + 26=78\); for \(d = 3\), cost is \(52+26\times2 = 104\).
Now let's check the options:

• Option A: \(C(1)=26\times1+26 = 52\), \(C(2)=26\times2 + 26=78\), \(C(3)=26\times3+26 = 104\). This matches.
• Option B: \(C(1)=26\times1+52 = 78

eq52\), so B is wrong.

• Option C: \(C(1)=52\times1-26 = 26

eq52\), so C is wrong.

• Option D: \(C(1)=52\times1+78 = 130

eq52\), so D is wrong.

Answer:

A. \(C(d)=26d + 26\)