Solve each problem. During the course of a year, the number of volunteers available to run a food bank each month is modeled by $V\left(x\right)$, where $V\left(x\right)=2x^2-32x+150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, $V\left(x\right)$ is modeled by $V\left(x\right)=31x-226$. Find the number of volunteers in each of the following months.
December

Verified step by step guidance

1. Identify the correct function to use for December. Since December corresponds to month 12, and the problem states that for months from August (x=8) to December (x=12), the function is \(V(x) = 31x - 226\), we will use this linear function.
2. Substitute \(x = 12\) into the function \(V(x) = 31x - 226\) to find the number of volunteers in December.
3. Write the substitution explicitly: \(V(12) = 31 \times 12 - 226\).
4. Perform the multiplication part first: calculate \$31 \times 12$ (do not compute the final value here, just set up the expression).
5. Then subtract 226 from the product to get the number of volunteers in December.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. In this problem, V(x) uses one formula from January to August and another from August to December, so understanding how to apply the correct formula based on the month (x-value) is essential.

Function Evaluation

Function evaluation involves substituting a specific input value into the function's formula to find the output. Here, to find the number of volunteers in December (x=12), you substitute 12 into the appropriate piece of V(x) and simplify.

Linear and Quadratic Functions

The problem involves both quadratic (2x² - 32x + 150) and linear (31x - 226) functions. Recognizing the difference helps in correctly evaluating and interpreting the number of volunteers, as quadratic functions have curved graphs and linear functions have straight-line graphs.