Questions: Yaster Inc. is trying to enter the thingamabob market. The research department established the following price-demand, cost, and revenue functions:
- Price-demand function: (p(x)=57-1.14 x)
- Cost function: (C(x)=215+11 x)
- Revenue function: (R(x)=x p(x)=x(57-1.14 x))
where (x) is in thousands of thingamabobs and (C(x)) and (R(x)) are in thousands of dollars. The price (p(x)) is the price in dollars of one thingamabob when the demand is (x) thousand thingamabobs. All three functions have domain (1 leq x leq 50)
Transcript text: Yaster Inc. is trying to enter the thingamabob market. The research department established the following price-demand, cost, and revenue functions:
\begin{tabular}{|c|c|}
\hline$p(x)=57-1.14 x$ & \begin{tabular}{l}
Price- \\
demand \\
function
\end{tabular} \\
\hline$C(x)=215+11 x$ & \begin{tabular}{l}
Cost \\
function
\end{tabular} \\
\hline$R(x)=x p(x)=x(57-1.14 x)$ & \begin{tabular}{l}
Revenue \\
function
\end{tabular} \\
\hline
\end{tabular}
where $x$ is in thousands of thingamabobs and $C(x)$ and $R(x)$ are in thousands of dollars. The price $p(x)$ is the price in dollars of one thingamabob when the demand is $x$ thousand thingamabobs. All three functions have domain $1 \leq x \leq 50$
Solution
Solution Steps
To maximize revenue, we need to find the value of \( x \) that maximizes the revenue function \( R(x) = x(57 - 1.14x) \). This involves finding the vertex of the quadratic function, which can be done using the formula \( x = -\frac{b}{2a} \) for a quadratic equation \( ax^2 + bx + c \).
Step 1: Identify the Revenue Function
The revenue function is given by:
\[ R(x) = x(57 - 1.14x) = -1.14x^2 + 57x \]
Step 2: Find the Vertex of the Quadratic Function
To maximize the revenue, we need to find the vertex of the quadratic function. The formula for the vertex \( x \) of a quadratic equation \( ax^2 + bx + c \) is:
\[ x = -\frac{b}{2a} \]
Step 3: Calculate the Value of \( x \)
Substitute \( a = -1.14 \) and \( b = 57 \) into the vertex formula:
\[ x = -\frac{57}{2 \times -1.14} = 25 \]
Final Answer
The number of thingamabobs Yaster Inc. should produce and sell to maximize revenue is:
\[ \boxed{25} \]