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.

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TomShelby TomShelby · Answer 1 · 2019-10-12T20:06:05+02:00

Answer:

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

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:

[tex]C \times \frac{1-(1+r)^{-time} }{rate}(1+r) = PV\\[/tex]

C 26,000,000

time 8 years

rate 0.1

[tex]26000000 \times \frac{1-(1+0.1)^{-8} }{0.1} = PV\\[/tex]

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:

[tex]\frac{Nominal}{(1 + rate)^{time} } = PV[/tex]

Nominal: 152,578,889.2600

time 3 years

rate 0.1

[tex]\frac{152,578,889.2600}{(1 + 0.1)^{3} } = PV[/tex]

PV 114,634,777.81