The monthly cost in dollars for using a cell phone is given by the function
𝐶
(
𝑛
)
=
1
4
𝑛
+
25
, where
𝑛
is the number of minutes used each month. Which of the following is (are) true?

I. The monthly cost of a cell phone increases by $1 every 4 minutes of use time.

II. Each minute of use time increases the monthly cost by a factor of
1
4
.

III. The monthly cost of a cell phone increases by
1
4
dollar every minute of use time.

Question

The monthly cost in dollars for using a cell phone is given by the function 
𝐶
(
𝑛
)
=
1
4
𝑛
+
25
, where 
𝑛
 is the number of minutes used each month.  Which of the following is (are) true?

I.  The monthly cost of a cell phone increases by $1 every 4 minutes of use time.

II. Each minute of use time increases the monthly cost by a factor of 
1
4
.

III.  The monthly cost of a cell phone increases by 
1
4
 dollar every minute of use time.

BlackTom AI · Accepted Answer

We start by restating the given information and the statements to evaluate.
The cost function is C(n) = (1/4) n + 25, where n is the number of minutes.
Now we analyze each statement:

Option I: The monthly cost increases by $1 every 4 minutes of use time.
- If n increases by 4 (n -> n+4), the change in cost is C(n+4) - C(n) = [(1/4)(n+4) + 25] - [(1/4)n + 25] = (1/4)n + 1 + 25 - (1/4)n - 25 = 1. 
- This shows a $1 increase for every additional 4 minutes, so this statement is true.

Option II: Each minute of use time increases the monthly cost by a factor of 1/4.
- The phrase “by a factor of 1/4” implies a multiplicative change: the cost would be multiplied by 1/4 for each additional minute, which would give an exponential, not linear, growth. 
- In our linear model, the per-minute change is additive, not a constant multiplicative factor. Specifically, C(n+1) - C(n) = (1/4), not a multiplication by 1/4. Therefore, this statement is false.

Option III: The monthly cost increases by 1/4 dollar every minute of use time.
- The derivative (or per-minute increase) is the coefficient of n in C(n) = (1/4) n + 25, which is 1/4. This means each additional minute adds 0.25 dollars to the cost. Hence, this statement is true.

Summary of evaluation:
- Option I is true because a 4-minute increase yields a $1 rise in cost.
- Option II is false because the model is additive per minute, not a multiplicative factor of 1/4 per minute.
- Option III is true because the per-minute increase is exactly 1/4 dollar.

MATH 1151 AU2025 (18443) Calculus Concept Readiness Assessment (CCR)

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