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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 7c
Chapter 4, Problem 7c

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),V(x), where V(x)=2x2−32x+150V(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)V(x) is modeled by V(x)=31x−226V(x)=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 \( x = 8 \).
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.
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.
Calculate \( V(8) \) using the second function: \( V(8) = 31(8) - 226 \). Multiply 31 by 8 and then subtract 226.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Function Composition

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.
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Evaluating Composed Functions

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.
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Equations with Two Variables
Related Practice
Textbook Question

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(x)V(x), where V(x)=2x2−32x+150V(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)V(x) is modeled by V(x)=31x−226V(x)=31x-226. Find the number of volunteers in each of the following months. Sketch a graph of y=V(x)y=V(x) for January through December. In what month are the fewest volunteers available?

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Textbook Question

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(x)V(x), where V(x)=2x2−32x+150V(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)V(x) is modeled by V(x)=31x−226V(x)=31x-226. Find the number of volunteers in each of the following months.

October

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Textbook Question

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(x)V(x), where V(x)=2x2−32x+150V(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)V(x) is modeled by V(x)=31x−226V(x)=31x-226. Find the number of volunteers in each of the following months.

May

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Textbook Question

Solve each problem. If y varies directly as x, and y=20 when x=4, find y when x = -6.

Textbook Question

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(x)V(x), where V(x)=2x2−32x+150V(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)V(x) is modeled by V(x)=31x−226V(x)=31x-226. Find the number of volunteers in each of the following months.

December

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Textbook Question

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(x)V(x), where V(x)=2x2−32x+150V(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 modeled by V(x)=31x−226V(x)=31x-226. Find the number of volunteers in each of the following months.

January

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