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Linear Functions in Real-World Contexts (Hard) - SAT Math Practice Problems with Solutions

Linear Functions in Real-World Contexts (Hard) - SAT Math Practice Problems with Solutions

1 / 15

A company’s monthly profit \(P(x)\) from selling x units of a product is modeled by the function \(P(x) = -2x^2 + 80x - 500\). Determine the number of units that must be sold to achieve maximum profit and calculate the maximum profit.

2 / 15

A small business has a revenue model given by \(R(x) = 10x - 0.05x^2\), where \(x\) is the number of units sold. The cost function is \(C(x) = 5x + 100\). Find the profit function \(P(x)\) and determine the number of units that must be sold to maximize profit.

3 / 15

A gym membership has a monthly fee plus an additional fee for personal training sessions. The total cost in dollars for x personal training sessions in a month is given by \(C(x) = 0.50x + 40\). If a member has a budget of $100 for personal training sessions in a month, how many sessions can they afford?

4 / 15

A water tank is being filled at a constant rate and then drained at a different constant rate. The volume of water in the tank after t minutes can be modeled by \(V(t) = \begin{cases} 2t + 50 & \text{if } 0 \leq t \leq 20 \\ -3t + 90 & \text{if } t > 20 \end{cases}\). After how many minutes will the tank be empty?

5 / 15

A car rental company offers a discount on rentals exceeding 100 miles. The cost function for renting a car and driving m miles is given by \(C(m) = \begin{cases} 0.40m + 50 & \text{if } m \leq 100 \\ 0.30m + 60 & \text{if } m > 100 \end{cases}\). What is the total cost for driving 120 miles?

6 / 15

A landscaping service charges a flat fee for equipment and an additional charge per square foot of lawn maintained. The function \(l(s) = 0.02s + 75\) represents the total cost in dollars for maintaining s square feet of lawn. What does the y-intercept signify?

7 / 15

A cleaning service charges a base fee plus an additional fee per hour worked. The function \(c(h) = 15h + 35\) represents the total cost in dollars for h hours of cleaning. What does the y-intercept represent?

8 / 15

A printing company charges a setup fee and an additional cost per page printed. The function \(p(p) = 0.05p + 20\) represents the total cost in dollars of printing p pages. What does the slope of the line represented by \(p(p)\) signify?

9 / 15

A delivery service charges a fixed fee for picking up a package and an additional fee based on the distance traveled. The function \(d(x) = 0.25x + 10\) represents the total cost in dollars of delivering a package over x miles. What does the y-intercept represent?

10 / 15

A mobile phone plan has a monthly subscription fee and an additional charge per minute of usage. The function \(p(m) = 0.10m + 30\) represents the total monthly cost in dollars for m minutes of usage. What does the y-intercept represent?

11 / 15

A taxi company charges a base fare plus an additional charge per mile traveled. The function \(t(m) = 1.5m + 5\) represents the total cost in dollars of traveling m miles. What does the slope of the line represented by \(t(m)\) represent?

12 / 15

A water tank is being drained at a constant rate. The function \(d(t) = -2t + 100\) models the volume of water in liters remaining in the tank after t minutes. What does the y-intercept represent?

13 / 15

A plumber charges a service call fee plus an hourly rate for labor. If the function \(p(h) = 45h + 90\) represents the total cost of h hours of work, what does the y-intercept indicate?

14 / 15

An online store sells books at a fixed price plus shipping costs. The function \(g(b) = 12b + 7\) gives the total cost in dollars of buying b books. What does the y-intercept of the graph of \(y = g(b)\) represent?

15 / 15

A car rental company charges a flat fee plus an additional amount per mile driven. The function \(f(m) = 0.35m + 50\) models the total cost in dollars of renting a car and driving m miles. What does the slope of the line represented by \(f(m)\) signify in this context?

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About This Quiz

Foundational Concept Explanation

The core concept tested in these questions is the interpretation and application of linear functions in real-world scenarios. Linear functions are typically represented in the form [latex]y = mx + b[/latex], where:

  • m is the slope, representing the rate of change or the incremental increase/decrease in the dependent variable for each unit increase in the independent variable.
  • b is the y-intercept, representing the value of the dependent variable when the independent variable is zero.

In practical terms, the slope often corresponds to a rate (e.g., cost per unit, speed, etc.), while the y-intercept often represents an initial value or a fixed cost. Understanding these components helps in analyzing and solving problems related to various real-world situations such as cost calculations, rates of change, and optimization.

Additionally, some questions involve piecewise functions, where the behavior of the function changes based on different intervals of the independent variable. Recognizing and applying the correct segment of the piecewise function is crucial for accurate problem-solving.

Detailed Success Tips

  1. Identify Key Components: Always identify the slope (rate of change) and the y-intercept (initial value) in the given linear function. These are critical for understanding the context and solving the problem.
  2. Understand the Context: Pay close attention to the context provided in the word problem. Translate the real-world scenario into mathematical terms using the given function.
  3. Analyze Piecewise Functions Carefully: For problems involving piecewise functions, carefully determine which part of the function applies to the given situation. Check the conditions for each segment and apply the appropriate formula.
  4. Solve Step-by-Step: Break down complex problems into smaller, manageable steps. Solve for unknowns step-by-step, ensuring you correctly interpret and use the function in each step.
  5. Check Your Work: After solving, verify your answer by substituting back into the original function or context to ensure it makes logical sense.
  6. Practice Regularly: Regular practice with similar problems will enhance your ability to quickly recognize patterns and apply the necessary mathematical concepts effectively.

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