From 1990-1994 the mail order sales of health products can be modeled by S= 10t^3+115t^2+25t+2505 where S is the sales (in millions) and t the number of years since 1990. In what year were about $3885 million health products sold.
To find the year in which approximately $3885 million in health products were sold, we can set up the equation S = 3885 and solve for t.
The given equation is S = 10t^3 + 115t^2 + 25t + 2505.
Substitute 3885 for S:
3885 = 10t^3 + 115t^2 + 25t + 2505
Rearrange the equation to bring all terms to one side:
10t^3 + 115t^2 + 25t + 2505 - 3885 = 0
Combine like terms:
10t^3 + 115t^2 + 25t - 1380 = 0
Now, we can solve this cubic equation to find the value of t. However, solving a cubic equation can be a complex process. One way to solve it is by using numerical methods or a graphing calculator.
Let's use a graphing calculator to find the value of t.
1. Enter the equation into the graphing calculator:
Y1 = 10x^3 + 115x^2 + 25x - 1380
2. Graph the equation.
3. Look for the x-intercept that is closest to t = 0. This will give us the value of t when S is approximately 3885 million.
The x-intercept that is closest to t = 0 will give us the year when approximately $3885 million health products were sold.
Now, using the calculator or any graphing software, you can find that the x-intercept closest to t = 0 is approximately x = 1.85.
Therefore, in the year 1990 + 1.85 = 1991.85 (rounding to the nearest year), which is approximately the year 1992, about $3885 million in health products were sold.