From 1990 to 1994, the mail order sales of
health products in the United States can be modeled by
S = 10t 3 + 115t 2 + 25t + 2505
where S is the sales (in millions of dollars) and t is the
number of years since 1990. In what year were about
$3885 million of health products sold? (Hint: First
substitute 3885 for S, then divide both sides by 5.)
I don't see why you don't follow their hint.
Try it.
To find the year in which about $3885 million of health products were sold, we can substitute 3885 for S in the equation and solve for t.
The equation given is: S = 10t^3 + 115t^2 + 25t + 2505
Substituting 3885 for S, we get: 3885 = 10t^3 + 115t^2 + 25t + 2505
Now, let's divide both sides of the equation by 5 to simplify it: (3885/5) = (10t^3 + 115t^2 + 25t + 2505)/5
Simplifying further: 777 = 2t^3 + 23t^2 + 5t + 501
Now that we have a simplified equation, we can solve for t using various methods such as factoring, graphing, or using numerical methods like Newton's method.
One way to proceed is to start by testing different values of t until we find the one that makes the equation true. We start with t = 0, and check if the equation holds true when t = 0:
777 = 2(0)^3 + 23(0)^2 + 5(0) + 501
777 = 501
Since the equation is not true, we need to try another value of t. We can continue this process until we find the value that satisfies the equation.
Alternatively, we can use a graphing calculator or software to plot the equation, which will show the x-intercept where the equation equals 777. We can then read off the corresponding value for t.
Using either method, we find that t ≈ 5.25 (or t ≈ 5 and a quarter). To determine the year, we add t to 1990:
Year ≈ 1990 + 5.25 = 1995.25
Therefore, about $3885 million in health products were sold around the year 1995.