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.
Let 1990 be 1, 1991 be 2, 1992 be 3, and 1994 be 4
3885=10(3^3)+115(3^2)+25(3)+2505
3885= 270 + 1035 + 75 +2505
3885=3885
1992 is the answer
3885=10t^3+115t^2+25t+2505
−10t3−115t2−25t+1380=0
−5(t−3)(2t2+29t+92)=0
t−3=0 2t2+29t+92=0
t=3
1992
To find the year when about $3885 million health products were sold, we need to solve the equation 10t^3 + 115t^2 + 25t + 2505 = 3885, where S represents the sales.
1. Rearrange the equation to isolate t on one side:
10t^3 + 115t^2 + 25t + 2505 - 3885 = 0
2. Combine like terms and simplify:
10t^3 + 115t^2 + 25t - 1380 = 0
3. Since this is a cubic equation, it may not have a simple algebraic solution. We will need to use numerical methods or technology to find the approximate solution.
Using a calculator, numerical methods, or software, we find that t ≈ 2.46.
4. Since t represents the number of years since 1990, we can calculate the year:
1990 + t ≈ 1990 + 2.46 ≈ 1992.46
Therefore, about $3885 million health products were sold in the year 1992.
To find the year when $3885 million health products were sold, you need to solve the given equation:
S = 10t^3 + 115t^2 + 25t + 2505
where S represents the sales in millions and t is the number of years since 1990.
Since we are looking for the year when sales reach $3885 million, we can set the equation equal to 3885 and solve for t:
10t^3 + 115t^2 + 25t + 2505 = 3885
To continue solving for t, we need to rearrange the equation:
10t^3 + 115t^2 + 25t + 2505 - 3885 = 0
Simplifying:
10t^3 + 115t^2 + 25t - 1380 = 0
Now we have a cubic equation in the form ax^3 + bx^2 + cx + d = 0. To solve this, you can use various methods, such as factoring, synthetic division, or the rational root theorem if applicable.
Once you find the values of t that satisfy the equation, you can add the value of t to 1990 to determine the year when $3885 million health products were sold.