Answered

  1. A container of fuel dispenses fuel at the rate of 5 gallons per second. If y represents the amount of fuel remaining in the container, and x represents the number of seconds that have passed since the fuel started dispensing, then x and y satisfy a linear relationship.



In the coordinate plane, will the slope of the line representing that relationship have a positive, negative, or zero slope? Explain how you know.


  1. A sandwich store charges a delivery fee to bring lunch to an office building. One office pays $33 for 4 turkey sandwiches. Another office pays $61 for 8 turkey sandwiches. How much does each turkey sandwich add to the cost of the delivery? Explain how you know.

Verified Answer :

  • Problem 4: The fuel amount decreases as time passes, so the rate of change is \(-5\) gallons per second, resulting in a negative slope of \(-5\).
  • Problem 5: Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) with points \((4, 33)\) and \((8, 61)\), the cost per sandwich is \(\frac{28}{4} = \boxed{\$7}\).

Explanation

  1. Problem 4 Analysis

For problem 4, we have a container losing fuel over time. Let \(y\) represent the amount of fuel remaining and \(x\) represent the time in seconds. Because the fuel is being dispensed at a constant rate of \(5\) gallons per second, the amount of fuel remaining decreases as time increases. In a linear model \(y = mx + b\), the slope \(m\) represents the rate of change of \(y\) with respect to \(x\). Since the fuel is leaving the container, the rate of change is \(-5\) gallons per second. Consequently, the slope of the line representing this relationship is \(-5\), which is a negative value.

  1. Problem 5 Analysis

For problem 5, we are given two data points representing the cost of delivery for different quantities of turkey sandwiches: \((4, 33)\) and \((8, 61)\), where \(x\) is the number of sandwiches and \(y\) is the total cost in dollars. We assume a linear relationship where the total cost is \(y = mx + b\), with \(m\) being the cost per sandwich. We calculate the slope using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the given values:
\[m = \frac{61 - 33}{8 - 4} = \frac{28}{4} = 7\]
Thus, each turkey sandwich adds \(\$7\) to the total cost.

Examples

Understanding rates of change and linear relationships is essential in everyday life, from calculating fuel consumption for travel to budgeting for business expenses. For instance, if you are planning a road trip, the amount of gas remaining in your tank as you drive is a linear function with a negative slope, determined by your car's fuel efficiency. Similarly, when ordering catering for an event, knowing the per-item cost helps you manage your budget effectively, just like calculating the cost per sandwich in the office delivery example. Mastering these concepts allows you to predict outcomes and make informed financial decisions.

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