Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by V(x), where V(x)=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(x) is mod-eled by V(x)=31x-226. Find the number of volunteers in each of the following months. December

Verified step by step guidance

1. Identify the function that models the number of volunteers from August to December: \( V(x) = 31x - 226 \).
2. Determine the value of \( x \) for December. Since \( x = 1 \) represents January, \( x = 12 \) represents December.
3. Substitute \( x = 12 \) into the function \( V(x) = 31x - 226 \).
4. Calculate \( 31 \times 12 \) to find the first part of the expression.
5. Subtract 226 from the result of \( 31 \times 12 \) to find the number of volunteers in December.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form V(x) = ax^2 + bx + c. In this case, V(x) = 2x^2 - 32x + 150 models the number of volunteers from January to August. Understanding how to evaluate quadratic functions is essential for determining the number of volunteers during these months.

Linear Functions

A linear function is a polynomial function of degree one, represented as V(x) = mx + b, where m is the slope and b is the y-intercept. For the months from August to December, the number of volunteers is modeled by V(x) = 31x - 226. Recognizing how to work with linear functions is crucial for calculating the number of volunteers in the later months.

Function Evaluation

Function evaluation involves substituting a specific value of x into a function to find the corresponding output. To find the number of volunteers in December, one must evaluate the appropriate function for that month. Since December corresponds to x = 12, understanding how to correctly substitute and compute the value is key to solving the problem.