a cellular phone company offers a contract for which the cost C, in dollars, of t minutes of telephoning is given C=0.25(t-500)+80.95, where it is assumed that t>=500 minutes. What times will keep cost between $123.45 and $146.95?
put cost in for each of those amounts, and solve for t in each case.
To find the times that will keep the cost between $123.45 and $146.95, we need to solve the given equation and inequality.
The cost equation is given as: C = 0.25(t - 500) + 80.95
To find the times t that satisfy the given cost range, we need to solve the system of inequalities:
123.45 ≤ C ≤ 146.95
First, let's replace C in the inequalities with the equation for C:
123.45 ≤ 0.25(t - 500) + 80.95 ≤ 146.95
Next, let's simplify the inequalities:
123.45 ≤ 0.25t - 125 + 80.95 ≤ 146.95
Combine like terms:
123.45 ≤ 0.25t - 44.05 ≤ 146.95
Next, subtract 80.95 from all sides of the inequalities:
123.45 - 80.95 ≤ 0.25t - 44.05 - 80.95 ≤ 146.95 - 80.95
Simplify further:
42.50 ≤ 0.25t - 44.05 ≤ 66.00
Now, let's add 44.05 to all sides of the inequalities:
42.50 + 44.05 ≤ 0.25t - 44.05 + 44.05 ≤ 66.00 + 44.05
Simplify:
86.55 ≤ 0.25t ≤ 110.05
Finally, divide all sides of the inequalities by 0.25:
(86.55) / 0.25 ≤ (0.25t) / 0.25 ≤ (110.05) / 0.25
Simplify:
346.20 ≤ t ≤ 440.20
Therefore, the times t that will keep the cost between $123.45 and $146.95 are within the interval [346.20, 440.20].