Question

You are considering installing a security system in your new house. You get the following information from two local home security dealers for similar security systems. Dealer \(1: \$ 3560\) to install and \(\$ 15\) per month monitoring fee. Dealer \(2: \$ 2850\) to install and \(\$ 28\) per month for monitoring. Note that the initial cost of the security system from Dealer 1 is much higher than from Dealer 2 , but the monitoring fee is lower. Let \(x\) represent the number of months that you have the security system. a. Write an equation that represents the total cost of the system with Dealer 1 b. Write an equation that represents the total cost of the system with Dealer 2 . c. Write a single equation to determine when the total cost of the systems will be equal. d. Solve the equation in part \(c\) and interpret the result. e. If you plan to live in the house and use the system for 10 years, which system would be less expensive?

Short answer

Answer: The total costs of both security systems will be equal after approximately 54-55 months. After 10 years (120 months), Dealer 1's security system will be less expensive with a total cost of $5360, compared to Dealer 2's total cost of $6210.

Step-by-step solution

Write the total cost equation for Dealer 1

Calculate the total cost of the system for Dealer 1 as the sum of the initial cost and the monthly fee multiplied by the number of months you have the security system, which is represented by variable \(x\). The equation for Dealer 1: $$ C_1(x) = 3560 + 15x $$ #b#.

Write the total cost equation for Dealer 2

In similar way as for Dealer 1, write the equation for Dealer 2: $$ C_2(x) = 2850 + 28x $$ #c#.

Write the equation to determine when the costs are equal

To find when the total costs will be equal, set \( C_1(x) \) equal to \( C_2(x) \): $$ 3560 + 15x = 2850 + 28x $$ #d#.

Solve the equation for when the costs are equal

To solve the equation in part \(c\), first subtract \(15x\) from both sides: $$ 3560 - 2850 = 28x - 15x $$ Keep solving the equation: $$ 710 = 13x $$ Now, divide by 13: $$ x \approx 54.62 $$ This result means that after around 54-55 months, the total cost of both security systems will be equal. #e#.

Determine the less expensive system after 10 years (120 months)

Calculate the total cost for Dealer 1 and Dealer 2 if you plan to have the security system for 120 months: $$ C_1(120) = 3560 + 15 \times 120 = 5360 $$ $$ C_2(120) = 2850 + 28 \times 120 = 6210 $$ Dealer 1's security system is less expensive after 10 years with a total cost of \(5360\) compared to Dealer 2's total cost of \(6210\).