Computer Sales in Canada: 1985-1995, 1.5M Reached

[Veronica_Oles]

Homework Statement

Between 1985 and 1995 the number of computers in thousands sold in Canada is estimated by c(t) = 0.92(t^3 + 8t^2 + 40t +400)
In what year did home computers sale reach 1.5 million?

Homework Equations

The Attempt at a Solution

I know I have to isolate the t to obtain time. So far this is what I have gotten up to

1230.43 = t^3 + 8^2 + 40t
Here is where I am stuck. Would I have to factor out a t in order to move on to th next step?

 

[member 587159]

Homework Statement

Between 1985 and 1995 the number of computers in thousands sold in Canada is estimated by c(t) = 0.92(t^3 + 8t^2 + 40t +400)
In what year did home computers sale reach 1.5 million?

Homework Equations

The Attempt at a Solution

I know I have to isolate the t to obtain time. So far this is what I have gotten up to

1230.43 = t^3 + 8^2 + 40t
Here is where I am stuck. Would I have to factor out a t in order to move on to th next step?

There are formulas to find the roots of any third degree polynomial. However, I assume that you did not cover those formulas in class. In exercises where you are not allowed to use a calculator, you can 'guess' one of the roots and then use something like synthetic division to find the remaining quadratic polynomial. In this case however, this seems quite impossible so I would use a calculator. Note too that you wrote 8^2 instead of 8t^2.

 

[Mark44]

Homework Statement

Between 1985 and 1995 the number of computers in thousands sold in Canada is estimated by c(t) = 0.92(t^3 + 8t^2 + 40t +400)
In what year did home computers sale reach 1.5 million?

Homework Equations

The Attempt at a Solution

I know I have to isolate the t to obtain time. So far this is what I have gotten up to

1230.43 = t^3 + 8^2 + 40t

The second term on the right should be 8t². I'm assuming, but didn't verify, that the 1230.43 number has the 400 term already folded in.

Although there is a technique for solving third-degree polynomials, it's not something I have committed to memory, and it's very complicated. One approach would be to graph the equation C = t³ + 8t² + 40t, and find the point at which the C value is at or close to 1230.43, then read off the t value at that point.

Another approach is to start with an education guess, say t = 10, and see what C value you get, adjusting t up or down in successive calculations.

Here is where I am stuck. Would I have to factor out a t in order to move on to th next step?

 

[SammyS]

Homework Statement

Between 1985 and 1995 the number of computers in thousands sold in Canada is estimated by c(t) = 0.92(t^3 + 8t^2 + 40t +400)
In what year did home computers sale reach 1.5 million?

Homework Equations

The Attempt at a Solution

I know I have to isolate the t to obtain time. So far this is what I have gotten up to

1230.43 = t^3 + 8^2 + 40t
Here is where I am stuck. Would I have to factor out a t in order to move on to th next step?

Factoring is only useful for the case where the othe side of the equation is zero.

By the way, what is the definition of the variable, t ?

 

[Veronica_Oles]

There are formulas to find the roots of any third degree polynomial. However, I assume that you did not cover those formulas in class. In exercises where you are not allowed to use a calculator, you can 'guess' one of the roots and then use something like synthetic division to find the remaining quadratic polynomial. In this case however, this seems quite impossible so I would use a calculator. Note too that you wrote 8^2 instead of 8t^2.

Sorry about that I meant 8t^2. The book uses a graphing calculator however we were meant to solve it algebraically:/

 

[Veronica_Oles]

Factoring is only useful for the case where the othe side of the equation is zero.

By the way, what is the definition of the variable, t ?

T is for time.

 

[Ray Vickson]

Sorry about that I meant 8t^2. The book uses a graphing calculator however we were meant to solve it algebraically:/

The exact solution is very complicated to get. Typically in such a problem, we unashamedly employ numerical methods, such as graphical analysis, resorting to a spreadsheet or computer algebra system, or using a decent scientific calculator.

For an exact method, see, eg.,
http://www.math.vanderbilt.edu/~schectex/courses/cubic/

 

[SammyS]

T is for time.

Yes, time. But how is that time measured?
For the year 1985 is t = 1985 ?