Question

A certain skincare company's profit in millions of dollars, P(t), can be modeled by the polynomial function P(t) = –2t3 + 8t2 + 2t, where t represents the number of skincare items produced, in thousands. Today, the company produces 4 thousand products for a profit of $8 million. According to the graph of the function, what other quantity of product would result in the same profit?

Answer 1

A certain skincare company's profit in millions of dollars, P(t) is modelled by;

`P(t)=-2t^3+8t^2+2t`

where t represents the number of quantity of skincare items produced, in thousands.

Today, the company produces 4 thousand products for a profit of $8 million.

Mathematically,

`t=4,P(t)=8`

Thus, we would find the other quantity that gives same profit with the equation;

`8=-2t^3+8t^2+2t`

Simplifying further, we have;

`-2t^3+8t^2+2t-8=0`

From the previous statement,

`(t-4)`

is a factor. Thus, we have;

`\begin{gathered} -2t^3+8t^2+2t-8=0 \\ -2(t-4)(t^2-1)=0 \end{gathered}`

Hence, the other quantity of product would be;

`\begin{gathered} t^2-1=0 \\ t^2=1 \\ t=\pm1 \\ \text{Discard the negative value;} \\ t=1 \end{gathered}`

CORRECT OPTION: 1 thousand