the sales of new products over time are found to follow a quadratic model before dropping off to zero at the…

the sales of new products over time are found to follow a quadratic model before dropping off to zero at the end of the products 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?\na 6 years\nb 12 years\nc 18 years\nd 24 years

the sales of new products over time are found to follow a quadratic model before dropping off to zero at the end of the products 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?\na 6 years\nb 12 years\nc 18 years\nd 24 years

Answer

Explanation:

Step1: Set sales to zero

We know that at the end of the product's lifetime, $S = 0$. So we set up the equation $0=-(t - 3)^2+81$.

Step2: Rearrange the equation

First, move $(t - 3)^2$ to one - side: $(t - 3)^2=81$.

Step3: Take the square root of both sides

We get $t - 3=\pm\sqrt{81}=\pm9$.

Step4: Solve for t

Case 1: When $t - 3 = 9$, then $t=9 + 3=12$. Case 2: When $t - 3=-9$, then $t=-9 + 3=-6$. Since the number of years $t$ cannot be negative in this context, we discard $t=-6$.

Answer:

B. 12 years