Question

4. Assume that the sales of a certain appliance dealer are approximated by a linear function. Suppose that sales were $850,000 in 1992 and $1,262,500 in 1997. X = 0 represents 1992. (a) Find an equation giving the dealer's yearly sales. (b) What were the dealer's approximate sales in 1995? (c) Estimate sales in 1999.

Answer 1

The answer are:

a. S(x) = $82,500(x - 1992) + $850,000

b. Sales in 1995: $1,097,500

c. Sales in 1999: $1,427,500

To solve this, we need to crea a linear function. A linear function is of the form:

`L(x)=ax+b`

Where a is the slope and b the y-intercept.

We want the sales in function of the year. If we call the year x, and teh sale S, we call the line S(x)

The first thing we need to do is to find the slope of the line. We can calculate the slope of the line with two points like:

`\begin{gathered} P=(x_p,y_p)\text{ and Q}=(x_q,y_q) \\ \text{slope}=(y_q-y_p)/(x_q-x_p) \end{gathered}`

For two points P and Q. We know the value of two points:

In 1992, the sales was $850,000

In 1997, the sales was $1,262,500

Then we call P = (1992, $850,000) and Q = (1997, $1,262,500)

Now let's calculate the slope:

`slope=(1,262,500-850,000)/(1997-1992)=(412,500)/(5)=82,500`

Now that we know the slope, we need to find the term b of the form of a line.

b is the value of the function when x = 0. In this case, the problem says that when x = 0, we need to use the value of 1992. The sales in 1992 was $850,000. then b = $850,000

We almost done with a. the only thing remaining is that we want the year 1992 to be the starting point. Then, to our value of x, we need to rest 1992.

let's see:

`\begin{gathered} \text{slope}=82,500 \\ b=850,000 \\ S(x)=82,500\cdot x+850,000 \end{gathered}`

If we just left the equation like that, when we want to know what happens in 1992:

`S(1992)=82,500\cdot1992+850,000=165,190,000`

Which we know is wrong because the sales in 1992 were 850,000. To fix this, we need to rest 1992 from x. This way:

`S(x)=82,500(x-1992)+850,000`

If we try this equation for 1992:

`\begin{gathered} S(1992)=82,500(1992-1992)+850,00 \\ S(1992)=82,500\cdot0+850,000 \\ S(1992)=850,000 \end{gathered}`

And that's the correct value of the sales for 1992.

This was the hard part, all we have to do now is evaluate the function for x = 1995 and x = 1999:

Part b.

`\begin{gathered} S(x)=82,500(x-1992)+850,000 \\ S(1995)=82,500(1995-1992)+850,000 \\ S(1995)=82,500\cdot3+850,000 \\ S(1995)=1,097,500 \end{gathered}`

Part c.

`\begin{gathered} S(1999)=82,500(1999-1992)+850,000 \\ S(1999)=82,500\cdot7+850,000 \\ S(1999)=1,427,500 \end{gathered}`

And the problem is solved