Problem 19 - Algebra Form and Function

Question

The number of gallons of gas in a car's tank, \(d\) miles after stopping for gas, is given by \(15-d / 20\). (a) Write an equation whose solution is the number of miles it takes for the amount of gas in the tank to reach 10 gallons. (b) Make a plot of the balance for \(d=40,60,80,100,\) \(120,140,\) and indicate the solution \(m=100\) to the equation in part (a).

Step-by-Step Solution

Answer

a) Using the function \(15-\frac{d}{20}\), we find that it takes 100 miles for the amount of gas in the tank to reach 10 gallons. b) The graph should show a curve connecting the points (40, 13), (60, 12), (80, 11), (100, 10), (120, 9), and (140, 8). A vertical line should be drawn at \(d=100\) miles, indicating the solution to part (a).

Step 1: (a) Find the equation for 10 gallons of gas in the tank

To find the number of miles it takes for the gas in the tank to reach 10 gallons, we need to set the given function equal to 10 and solve for \(d\). The function is given as \(15-d / 20\). Therefore, we must solve the equation \(15-d / 20 = 10\) for \(d\).

Step 2: (a) Solve the equation for d

Start by subtracting 15 from both sides of the equation. This gives: \[-\frac{d}{20}=-5\] Now, we can multiply both sides of the equation by -20 to isolate \(d\): \[d = 100\] So, it takes 100 miles for the amount of gas in the tank to reach 10 gallons.

Step 3: (b) Select the points to plot

Now, we need to plot the function for the given values of \(d=40,60,80,100,120,140\) and indicate the solution to part (a) on the plot. To do this, we first need to calculate the amount of gas in the tank for each value of \(d\). Using the function \(15-d / 20\), we get the following gas amounts: - \(d= 40\): \(15-\frac{40}{20}=15-2=13\) gallons - \(d= 60\): \(15-\frac{60}{20}=15-3=12\) gallons - \(d= 80\): \(15-\frac{80}{20}=15-4=11\) gallons - \(d=100\): \(15-\frac{100}{20}=15-5=10\) gallons - \(d=120\): \(15-\frac{120}{20}=15-6=9\) gallons - \(d=140\): \(15-\frac{140}{20}=15-7=8\) gallons

Step 4: (b) Plot the points on a graph and indicate the solution for part (a)

On a graph with the x-axis representing \(d\) (miles) and the y-axis representing the amount of gas in the tank (gallons), plot the points: - \((40, 13)\) - \((60, 12)\) - \((80, 11)\) - \((100, 10)\) - \((120, 9)\) - \((140, 8)\) Connect the points with a smooth curve and draw a vertical line at \(d=100\) miles, indicating the solution to part (a).

Key Concepts

Linear Equations

Linear equations are fundamental in algebra. They are equations of the form \(ax + b = c\), where \(x\) represents the variable. Here, they define a straight line when graphed. In our problem, the equation is \(15 - \frac{d}{20} = 10\). This can be simplified to find \(d\), the number of miles needed for the fuel to drop to 10 gallons.

To solve this, first rearrange the given equation. Start by subtracting 15 from both sides, obtaining \(-\frac{d}{20} = -5\). Then, multiply both sides by -20 to isolate \(d\), yielding \(d = 100\).

Linear equations often require simplifying through basic algebraic operations. They provide a straightforward way to find solutions for unknown variables by balancing operations on both sides of the equation. Understanding how to manipulate and solve these equations is key to mastering algebra.

Graphing Functions

Graphing functions is a powerful way to visualize relationships between variables. It helps in understanding how one variable affects another. In this exercise, we use graphing to check how many gallons of gas remain after driving certain miles.

First, calculate the gas left for each given \(d\) value (namely 40, 60, 80, 100, 120, and 140). By plugging these into the equation, you find the remaining gallons. For instance:

• At \(d = 40\), the gas left is 13 gallons.
• At \(d = 60\), it is 12 gallons.

Once calculated, plot these points on a graph. The x-axis represents the miles \(d\), while the y-axis shows the gas in the tank. Connect the dots to see a declining trend, illustrating the gas usage over distance.

Graphing helps in visual verification of calculated values, and here it clearly shows the fuel depletion pattern across miles. When you locate \(d = 100\), you can visually confirm that 10 gallons remain, affirming the solution from the linear equation.

Problem Solving

Problem-solving involves breaking down complex problems into manageable steps, making it easier to understand and solve. In this scenario, the task was to determine when the gas tank reaches 10 gallons and visualize it on a graph.

Here’s a structured approach:

• Identify the problem: We need to find out when the gas levels reach 10 gallons.
• Set up the equation: Use the given formula \(15 - \frac{d}{20} = 10\) and solve for \(d\).
• Solve the equation: Rearrange and calculate to find \(d = 100\).
• Visualize the data: Plot the corresponding points on a graph for different values of \(d\).

Breaking down problems into smaller steps makes complex tasks approachable and ensures that each component is thoroughly understood. Problem-solving is an essential skill, not just in math, but in everyday decision-making.