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You are choosing between two different cell phone plans. The first plan charges a rate of 2222 cents per minute. The second plan charges a monthly fee of $49.95\$49.95 plus 1010 cents per minute.\newlineLet tt be the number of minutes you talk and C1C_{1} and C2C_{2} be the costs (in dollars) of the first and second plans. Give an equation for each in terms of tt, and then find the number of talk minutes that would produce the same cost for both plans (Round your answer to one decimal place).\newline\begin{align*} C_{1}&=\ C_{2}&=\ \end{align*}\newlineIf you talk for minutes the two plans will have the same cost.

Full solution

Q. You are choosing between two different cell phone plans. The first plan charges a rate of 2222 cents per minute. The second plan charges a monthly fee of $49.95\$49.95 plus 1010 cents per minute.\newlineLet tt be the number of minutes you talk and C1C_{1} and C2C_{2} be the costs (in dollars) of the first and second plans. Give an equation for each in terms of tt, and then find the number of talk minutes that would produce the same cost for both plans (Round your answer to one decimal place).\newline\begin{align*} C_{1}&=\ C_{2}&=\ \end{align*}\newlineIf you talk for minutes the two plans will have the same cost.
  1. Write Equations: Let's first write down the equations for the cost of each plan in terms of the number of minutes talked, tt. For the first plan, which charges 2222 cents per minute, the cost C1C_1 in dollars is given by: C1=0.22tC_1 = 0.22t
  2. Cost Calculation: For the second plan, which charges a monthly fee of $49.95\$49.95 plus 1010 cents per minute, the cost C2C_2 in dollars is given by:\newlineC2=49.95+0.10tC_2 = 49.95 + 0.10t
  3. Set Equations Equal: To find the number of minutes that would produce the same cost for both plans, we need to set the two cost equations equal to each other and solve for tt:0.22t=49.95+0.10t0.22t = 49.95 + 0.10t
  4. Subtract and Combine: Subtract 0.10t0.10t from both sides of the equation to get all the terms involving tt on one side:\newline0.22t0.10t=49.950.22t - 0.10t = 49.95
  5. Divide and Solve: Combine like terms on the left side of the equation: 0.12t=49.950.12t = 49.95
  6. Perform Division: Divide both sides of the equation by 0.120.12 to solve for tt: \newlinet=49.950.12t = \frac{49.95}{0.12}
  7. Round to One Decimal: Perform the division to find the value of tt:t416.25t \approx 416.25
  8. Round to One Decimal: Perform the division to find the value of tt:t416.25t \approx 416.25Since we are asked to round the answer to one decimal place, we round tt to:t416.3t \approx 416.3 minutes