Question

The value of the TV is $1500. Its value decreases by 14% each year. (A) write a function that represents the value y (in dollars) of the TV after t years. (B) find the approximate monthly percent decrease in value. Round to the nearest tenth. (C) Graph of the function from part (a). Use the graph to estimate the value of TV after 3 years.

Answer 1

Answer:

Part A) $y=1,500(0.86)^t$

Part B) The monthly percent decrease is about 1.3%

Part C) The value of TV after 3 years is about $950

Step-by-step explanation:

Part A) write a function that represents the value y (in dollars) of the TV after t years.

we know that

The exponential decay function is given by

$y=a(1-r)^t$

where

y is the value in dollars of the TV

t is the number of years

a is the initial value

r is the rate of decay

In this problem we have

$a=\ 1,500\\r=14\%=14/100=0.14$

substitute

$y=1,500(1-0.14)^t$

$y=1,500(0.86)^t$

Part B) Find the approximate monthly percent decrease in value

we know that

$1\ year=12\ months$

I can rewrite the variable t as

$\frac{1}{12}(12t)$

Rewrite the function as

$y=1,500(0.86)^{\frac{1}{12}(12t)}$

using Power of a Power Property

$y=1,500(0.86^{\frac{1}{12}})^{(12t)}$

$y=1,500(0.9875)^{(12t)}$

Find the decay rate

$1-r=0.9875\\r=1-0.9875\\r=0.0125=1.25\%$

therefore

The monthly percent decrease is about 1.3%

Part C) Graph of the function from part (a). Use the graph to estimate the value of TV after 3 years

we have

$y=1,500(0.86)^t$

using a graphing tool

see the attached figure

From the graph

For t=3 years

The value of y is approximate 950

therefore

The value of TV after 3 years is about $950