Question

Online jewelry sales have increased steadily over the last 10 years. In 2003, sales were approximately 2 billion dollars, and in 2010 they were approximately 14.8 billion. (i) Find the unit rate at which online jewelry sales have been increasing. (Express your answer rounded correctly to the nearest hundredth of a billion per year.) billion dollars per year (ii) Construct a model to predict online jewelery sales. According to your model, what should the online jewelery sales be in 2019? (Express your answer rounded correctly to the nearest tenth of a billion.) billion dollars

Answer 1

Answer: (i) The unit rate at which online jewelry sales have been increasing is 1.33 billion dollars per year.

(ii) The online jewelry sales in 2019 will be 191.7 billion dollars.

Explanation:

In 2003, sales were approximately 2 billion dollars and in 2010, they were approximately 14.8 billion dollars.

(i) If
`x` is the number of years after 2003 and
`y` is the amount of sales....

then the equation will be:
`y= ab^x` , where
`a` is the initial amount and
`b` is the growth rate.

for 2003,
`x=0` and for 2010,
`x=7`

So, the two points in form of (x, y) will be:
`(0,2)` and
`(7,14.8)`

Now plugging these two points int the above equation....

`2= ab^0\\ \\ a= 2\\ \\ and\\ \\ 14.8=ab^7\\ \\ 14.8=2*b^7\\ \\ b^7=7.4\\ \\b= \sqrt[7]{7.4}=1.3309.... \approx 1.33`

Thus, the online jewelry sales have been increasing at a rate of 1.33 billion dollars per year.

(ii) As we got
`a=2` and
`b=1.33`, so the equation will be now:
`y= 2(1.33)^x`

For the year 2019, the value of
`x` will be: (2019-2003) = 16

So plugging
`x=16` into the above equation, we will get.....

`y=2(1.33)^16\\ \\ y=191.7150... \approx 191.7`

(Rounded to the nearest tenth)

Thus, the online jewelry sales in 2019 will be 191.7 billion dollars.