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.
October

Verified step by step guidance

1. Identify which piece of the piecewise function applies to October. Since October is the 10th month, and the function V(x) = 31x - 226 applies from August (x=8) to December (x=12), use this formula for x = 10.
2. Substitute x = 10 into the function V(x) = 31x - 226 to set up the expression for the number of volunteers in October.
3. Write the expression as V(10) = 31 \times 10 - 226.
4. Simplify the multiplication part first: calculate 31 \times 10.
5. Subtract 226 from the result of the multiplication to find the number of volunteers in October.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions depending on the input value's 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 is essential.

Substitution in Functions

Substitution involves replacing the variable in a function with a specific value to find the output. Here, to find the number of volunteers in October, substitute x = 10 into the appropriate piece of the function.

Linear and Quadratic Functions

The problem involves both quadratic (2x² - 32x + 150) and linear (31x - 226) functions. Recognizing the type of function helps in understanding the shape and behavior of the volunteer count over time and correctly evaluating the expressions.