The sales of new products over time are found to follow a quadratic model before dropping off to zero at the end of the product’s lifetime. A new product is released with sales modeled by the equation S=−(t−3)^2+81, where S represents the sales in millions of dollars and t represents the number of years the product will be sold. According to the equation, what will be the lifetime of this product?
To find the lifetime of the product, we need to determine the value of t when the sales, S, drop to zero.
Set S=0 in the equation:
0 = -(t-3)^2 + 81
-(t-3)^2 = -81
(t-3)^2 = 81
Taking the square root of both sides:
t-3 = ±√81
t-3 = ±9
t = 3 + 9 or t = 3 - 9
t = 12 or t = -6
Since the product cannot be sold for a negative number of years, the lifetime of the product is 12 years.