Solve each problem. During the course of ayear, 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.
August

Verified step by step guidance

1. Identify the month for which you need to find the number of volunteers. Since August corresponds to the 8th month, we have \( x = 8 \).
2. Determine which piece of the piecewise function applies for \( x = 8 \). The problem states that for months January to August (\( 1 \leq x \leq 8 \)), the function is \( V(x) = 2x^2 - 32x + 150 \). For months August to December (\( 8 \leq x \leq 12 \)), the function is \( V(x) = 31x - 226 \). Since August is the boundary month, check both expressions to confirm consistency.
3. Calculate \( V(8) \) using the first function: \( V(8) = 2(8)^2 - 32(8) + 150 \). This involves squaring 8, multiplying by 2, then subtracting 32 times 8, and finally adding 150.
4. Calculate \( V(8) \) using the second function: \( V(8) = 31(8) - 226 \). Multiply 31 by 8 and then subtract 226.
5. Compare the two results to ensure the model is consistent at \( x = 8 \). Either value can be used as the number of volunteers in August, depending on the problem's context.

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. Understanding how to apply the correct formula based on the input value is essential for accurate evaluation.

Evaluating Quadratic and Linear Functions

Evaluating a function means substituting a given input value into the function's formula and simplifying to find the output. Here, V(x) is quadratic for the first interval and linear for the second. Knowing how to correctly substitute and simplify expressions like 2x² - 32x + 150 or 31x - 226 is crucial.

Interpreting the Variable and Domain

The variable x represents months, with x=1 for January, so x=8 corresponds to August. Recognizing the domain and correctly identifying which formula applies at x=8 ensures the right function is used to find the number of volunteers for August.