Question

a third option
the college student who is looking for a data plan for her cell phone found an option which offers a 2 gigabyte monthly allowance for data.
the monthly cost of this option can be represented by function c, defined by: c(x) = 30x + 5 where x is the gigabytes of data used over the monthly allowance.

1. find c(0.7) and explain what it means in this situation.
2. the student’s budget is $50 a month. to find out how many gigabytes of data she could use if she chose this plan, she writes c(x) = 50 and solves for x. what value of makes this equation true?

explain your reasoning.

Explanation:

Step1: Calculate \( C(0.7) \)

Substitute \( x = 0.7 \) into \( C(x)=30x + 5 \)

$$
LATEXBLOCK0
$$

Step2: Interpret \( C(0.7) \)

Since \( x \) is the gigabytes of data used over the monthly allowance, when \( x = 0.7 \), \( C(0.7)=26 \) means that when the student uses \( 0.7 \) gigabytes of data over the 2 - gigabyte monthly allowance, the monthly cost of the data plan is \( \$26 \).

Step3: Solve \( C(x)=50 \)

Set \( 30x + 5=50 \)
Subtract 5 from both sides: \( 30x=50 - 5=45 \)
Divide both sides by 30: \( x=\frac{45}{30}=1.5 \)

Step4: Interpret the solution for \( x \) in \( C(x) = 50 \)

The equation \( C(x)=50 \) is used to find the amount of data \( x \) (over the 2 - gigabyte allowance) that the student can use within her \( \$50 \) budget. When \( x = 1.5 \), it means the student can use \( 1.5 \) gigabytes of data over the 2 - gigabyte monthly allowance. So in total, she can use \( 2+1.5 = 3.5 \) gigabytes of data.

Answer:

1. \( C(0.7)=26 \). It means when the student uses \( 0.7 \) gigabytes of data over the monthly allowance, the cost is \( \$26 \).
2. \( x = 1.5 \). When solving \( 30x+5 = 50 \), we first isolate the term with \( x \) (\( 30x=45 \)) and then find \( x=\frac{45}{30}=1.5 \). This \( x \) value represents the gigabytes of data over the allowance that the student can use with a \( \$50 \) budget.