QuestionWhat is the domain of the profit function $P(R)$ if $R(t)=10t$ and the theater holds a max of 200 people? A) $\{0,1,2,\ldots,200\}$ B) $\{0,10,20,\ldots,2000\}$ C) $\{any$ nonnegative real number $\leq 200\}$ D) $\{any$ nonnegative real number $\leq 2000\}$

Studdy Solution

STEP 1

Assumptions1. The revenue function is given by $R(t)=10t$ , where "t" is the number of tickets sold. . The theater holds a maximum of200 people.
3. The profit function is given by $(R)=0.20R$ , where "R" is the revenue.

STEP 2

First, we need to find the maximum possible revenue. This will be achieved when the theater is at full capacity, i.e., when "t" equals200.
$R_{max} = R(t_{max}) = R(200)$

STEP 3

Now, plug in the value for $t_{max}$ into the revenue function to calculate the maximum revenue.
$R_{max} =10 \times200$

STEP 4

Calculate the maximum revenue.
$R_{max} =10 \times200 =2000$

STEP 5

The domain of the profit function $(R)$ is the set of all possible values of $R$ . Since $R$ represents the revenue, and the maximum possible revenue is $2000, the domain of$ (R) $is all nonnegative real numbers less than or equal to$ 2000.
So, the domain of $(R)$ is $\{any$ nonnegative real number $\leq2000\}$ .
Therefore, the correct answer is D.

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