Question

PLEASE HELP ITS DUE TODAY AND THIS IS MY LAST QUESTION

A function can be written in different, yet equivalent, forms to model the amount of a radioactive substance, in grams, left after t years.


What aspect of the decay of the substance does each form of the function highlight?


Drag the correct description to the space below each form of the function.

Answer 1

Left: The substance is decreasing by 1/2 every 12 years

Right: The substance is decreasing by 5.61% each year

Explanation:

exponential decay

A = P(1-r)ᵇⁿ, where A is the final amount, P is the initial amount, r is the rate decreased each time period, b is the number of years, and n is the number of times compounded each year

let's write each formula in terms of this

left:

f(t) = 600(1/2)^(t/12)

matching values up...

A = P(1-r)ᵇⁿ

A = f(t)

P = 600

1 - r = 1/2 -> r = 1/2

t/12 = bn -> b = number of years = t, so bn = b/12 -> n = 1/12. Thus, it is compounded 1/12 times each year, so it is compounded every t*12 = 12 years. If it was compounded each month, it would be compounded 12 times a year

Thus, this is decreasing by a rate of 1/2 each 12 years

right:

f(t) = 600(1-0.0561)^(t)

matching values up...

A = P(1-r)ᵇⁿ

A = f(t)

P = 600

1 - r = 1 - 0.0561 -> r = 0.0561 = 5.61%

t = bn -> b = number of years = t, so bn = b -> n = 1. Thus, it is compounded annually (1 time each year)

Thus, this is decreasing by a rate of 5.61% each year