Question

Because unintended lane changes by distracted drivers are responsible for 43% of all highway fatalities, Ford Motor Co. and Volvo launched a program to jointly develop a technology to prevent accidents by sleepy or distracted drivers. A device costing $260 tracks lane markings and sounds an alert during lane change. If these devices are included in 100,000 new cars per year beginning 3 years from now, determine the present worth of the cost over a 10-year period at an interest rate of 10% per year.

Answer 1

present cost of the device: $ 114,634,777.81

Step-by-step explanation:

we are going to add a device worth 260 dollars on 100,000 cars three years from now, every year.

The cash outflow will be for: 260 x 100,000 = 26,000,000 every year

Timeline:

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th)

<----/----/----/(/----/----/----/----/----/----/----/)---->

The first step is to calcualte the value of the annuity that begings in 3 years:

Notice during a seven years period it has 8 payment so it will be an annuity due:

`C * (1-(1+r)^(-time) )/(rate)(1+r) = PV\\`

C 26,000,000

time 8 years

rate 0.1

`26000000 * (1-(1+0.1)^(-8) )/(0.1) = PV\\`

PV $152,578,889.2600

Now, we calcualte the present value of this annuity which begins 3 years from now:

1st 2nd Annuity-due

<----(/----/----/)--------->

We discount as a lump sum:

`(Nominal)/((1 + rate)^(time) ) = PV`

Nominal: 152,578,889.2600

time 3 years

rate 0.1

`(152,578,889.2600)/((1 + 0.1)^(3) ) = PV`

PV 114,634,777.81