Question

the value of a collector’s item is expected to increase exponentially each year. the item is purchased for $500. after 2 years, the item is worth $551.25. which equation represents y, the value of the item after x years? y = 500(0.05)x y = 500(1.05)x y = 500(0.1025)x y = 500(1.1025)x

Answer 1

Answer: Second option is correct.

Explanation:

Since we have given that

Original price of the item = $500

The value of a collector's item is expected to increase exponentially each year.

According to question, it becomes,

`y=500(1+r)^x\\\\\text{ r denotes rate of growth and x denotes number of years }`

Since we have given that , After 2 years, item is worth of $551.25.

So, our above equation becomes,

`551.25=500(1+r)^2\\\\(551.25)/(500)=(1+r)^2\\\\1.1025=(1+r)^2\\\\√(1.1025)=(1+r)\\\\1.05=1+r\\\\r=1.05-1=0.05`

So, the equation represents y, the value of the item after x years is given by

`y=500(1+0.05)^x\\\\y=500(1.05)^x`

Hence, Second option is correct.

Answer 2

Exponential model implies: y = 500 (a^x)

The data after 2 years drives to 500 (a^2) = 551.25

Then,

a^2 = 551.25/500 = 1.1025

a = √(1.1025) = 1.05

The the equation is y = 500 (1.05)^x [the second option]