Question

A new popular midsize car has a retail price of $25,000. The midsize car's value M(t) is given by the exponential model M of t is equal to 25,000 times the quantity four fifths to the power of t where t represents the time in years. Identify the domain, in yearly intervals, that contains all the years the car's value is less than $1,000.

[13, ∞)

[14, ∞)

[15, ∞) correct

[16, ∞)

Answer 1

Explanation:

M(t) = 25000×(4/5)^t

e.g. the brand new price is for t=0

25000×(4/5)⁰ = 25000×1 = 25000 correct

the car value after 1 year is then

25000×(4/5)¹ = 25000×4/5 = 5000×4 = $20000

now, we have to find the value for t so that the functional value is less than $1000.

1000 > 25000×(4/5)^t

1 > 25×(4/5)^t

1/25 > (4/5)^t

0.04 > (0.8)^t

the simplest way from here on is trying t=13, t=14, t=15 and t=16.

and we find that for t=15

0.04 > (0.8)^t becomes true the first time, as

0.8¹⁵ = 0.0352, which is smaller than 0.04.

so,

[15, infinity) is the correct answer.