Problem 119 - College Algebra Essentials

Question

A basic cellphone plan costs \(\$ 20\) per month for 60 calling minutes. Additional time costs \(\$ 0.40\) per minute. The formula $$C=20+0.40(x-60)$$ gives the monthly cost for this plan, \(C\), for \(x\) calling minutes, where \(x>60 .\) How many calling minutes are possible for a monthly cost of at least \(\$ 28\) and at most \(\$ 40 ?\)

Step-by-Step Solution

Answer

So, the number of possible calling minutes for a monthly cost of between $28 and $40 is between 80 and 110 minutes (inclusive).

Step 1: Formulate the inequality for the minimum cost

To solve the problem, it is necessary to set up two inequalities. The first one will express the minimum cost. This inequality will be $20 + 0.4(x - 60) >= 28.

Step 2: Solve the inequality for the minimum cost

Having set up the inequality, solve it for 'x', which represents the number of calling minutes. This can be done by subtracting 20 from both sides to get 0.4(x - 60) >= 8. Then, divide both sides by 0.4 to get x - 60 >= 20. Finally, add 60 to both sides to get x >= 80.

Step 3: Formulate the inequality for the maximum cost

The second inequality expresses the maximum cost: $20 + 0.4(x - 60) <= 40.

Step 4: Solve the inequality for the maximum cost

As in step 2, subtract 20 from both sides to get 0.4(x - 60) <= 20. Divide both sides by 0.4 to get x - 60 <= 50. Finally, add 60 to both sides to yield x <= 110.