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Chapter 3: Problem 2

You can use the equation \(d=24-45 t\) to model the distance from a destination for someone driving down the highway, where distance \(d\) is measured in miles and time \(t\) is measured in hours. Graph the equation and use the trace function to find the approximate time for each distance given in \(2 \mathrm{a}\) and \(\mathrm{b}\). a. \(d=16 \mathrm{mi}\) (a) b. \(d=3 \mathrm{mi}\) c. What is the real-world meaning of 24 ? (a) d. What is the real-world meaning of 45 ? e. Solve the equation \(24-45 t=16\).

Short Answer

Expert verified
a. About 0.178 hours; b. About 0.467 hours; c. Initial distance (24 miles); d. Travel rate (45 mph); e. Time is approximately 0.178 hours for 16 miles.

Step by step solution

01

Solve for Time with Distance = 16 miles

First, substitute \(d = 16\) into the equation \(d = 24 - 45t\). This gives us the equation: \(16 = 24 - 45t\).
02

Rearrange Equation to Solve for t

Rearrange the equation from Step 1 by subtracting 24 from both sides: \(16 - 24 = -45t\). Simplifying this, we get: \(-8 = -45t\).
03

Solve for t

Divide both sides of the equation from Step 2 by \(-45\) to isolate \(t\): \(t = \frac{-8}{-45}\). Simplify the fraction to get \(t \approx 0.178\) hours.
04

Solve for Time with Distance = 3 miles

Substitute \(d = 3\) into the equation \(d = 24 - 45t\). This gives us the equation: \(3 = 24 - 45t\).
05

Rearrange and Solve for t

Rearrange the equation from Step 4 by subtracting 24 from both sides: \(3 - 24 = -45t\). Simplifying this, we get: \(-21 = -45t\). Divide both sides by \(-45\) to isolate \(t\): \(t = \frac{-21}{-45}\). Simplify the fraction to get \(t \approx 0.467\) hours.
06

Interpret the Meaning of 24 in Real World Context

In the equation \(d = 24 - 45t\), the number 24 represents the initial distance from the destination when \(t = 0\) hours.
07

Interpret the Meaning of 45 in Real World Context

The number 45 in the equation represents the rate of travel in miles per hour after moving away from the initial point.
08

Solve the Equation 24-45t=16

To solve the equation \(24 - 45t = 16\), subtract 24 from both sides: \(16 - 24 = -45t\). Simplify to get \(-8 = -45t\) and divide by \(-45\) to find \(t = \frac{-8}{-45} \approx 0.178\) hours.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Equations
When we talk about graphing linear equations, like the one given by \( d = 24 - 45t \), we're using a graph to visually represent the relationship between two variables. In this case, the variables are distance (\(d\)) and time (\(t\)). Here's how you can graph this equation on a standard Cartesian coordinate plane.

  • Identify the intercepts to help plot the line. The intercepts are where the line crosses the axes.
  • The \(y\)-intercept is found when \(t = 0\). Substitute \(t = 0\) into the equation to get \(d = 24\), so the \(y\)-intercept is point (0, 24).
  • Another point can be found by setting \(d\) to 0, giving \(0 = 24 - 45t\), which solves to \(t = \frac{24}{45}\).
  • Connect these points on the graph to draw the line representing the equation.
The slope of the line indicates how the distance changes over time, which can be calculated or observed by the steepness of the line.

To answer questions using the graph, you can trace along the line to find the time values at specific distances such as 16 miles and 3 miles, which gives practical insight into the travel time for those distances.
Rate of Change
The rate of change in this context refers to how quickly distance changes as time progresses. In the equation \(d = 24 - 45t\), the rate of change is represented by \(-45\), which is effectively the slope of the equation.

  • In mathematical terms, the rate of change tells us that for every hour that passes, the distance decreases by 45 miles.
  • A negative rate of change indicates that the distance from the destination is reducing, which means the driver is getting closer to the endpoint.
  • The steeper the slope, the faster the rate of change.
Understanding the rate of change is crucial as it gives insight into the driving dynamics, determining how fast the driver is moving towards the destination.

In practical terms, knowing the rate of change helps in planning trips. For instance, if the rate is consistent, you can estimate arrival times or calculate delays.
Distance-Time Relationship
The distance-time relationship depicted in the equation \(d = 24 - 45t\) is a fundamental aspect of linear equations that describe motion scenarios. Here鈥檚 what you'll need to grasp about this linear relationship.

  • The equation shows how distance \(d\) decreases as time \(t\) increases. This tells us that the vehicle is nearing its destination over time.
  • The initial distance of 24 miles (when \(t = 0\)) represents where the traveler starts concerning the destination.
  • The negative sign in front of the 45 indicates the direction of travel towards the destination as time progresses.
  • By altering \(t\), we can determine different distance values, providing a clear picture of travel progress at any given time.
This linear relationship is pivotal, especially when evaluating scenarios like travel or analyzing speeds, giving a clear and calculated method to predict future distances based on given travel time frames.

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Most popular questions from this chapter

Describe what the rate of change looks like in each graph. a. the graph of a person walking at a steady rate toward a motion sensor (a) b. the graph of a person standing still c. the graph of a person walking at a steady rate away from a motion sensor d. the graph of one person walking at a steady rate faster than another person

5\. Give the additive inverse of each number. a. \(\frac{1}{5}\) a b. 17 c. \(-23\) d. \(-x\)

APPLICATION Amber makes \(S 6\) an hour at a sandwich shop. She wants to know how many hours she needs to work to save \(\$ 500\) in her bank account. On her first paycheck, she notices that her net pay is about \(75 \%\) of her gross pay. a. How many hours must she work to earn \(\$ 500\) in gross pay? b. How many hours must she work to earn \(\$ 500\) in net pay?

For each table, write a formula for list \(\mathrm{L}_{2}\) in terms of list \(\mathrm{Ll}\). $$ \begin{aligned} &\text { a. }\\\ &\begin{array}{|r|r|} \hline \text { L1 } & \text { L2 } \\ \hline 0 & -5.7 \\ \hline 1 & -3.4 \\ \hline 2 & -1.1 \\ \hline 3 & 1.2 \\ \hline 4 & 3.5 \\ \hline 5 & 5.8 \\ \hline \end{array}\\\ &\begin{array}{|r|r|} \hline \text { b. } & \text { L2 } \\ \hline-3 & 19 \\ \hline-1 & 3 \\ \hline 0 & -5 \\ \hline 2 & -21 \\ \hline 5 & -45 \\ \hline 6 & -53 \\ \hline \end{array}\\\ &\begin{array}{|r|c|} \hline \text { c. } & \text { L2 } \\ \hline 3 & 13.5 \\ \hline-2 & 11 \\ \hline-9 & 7.5 \\ \hline 0 & 12 \\ \hline 6 & 15 \\ \hline-5 & 9.5 \\ \hline \end{array} \end{aligned} $$

Draw a scatter plot on your paper picturing (time, distance) at \(1 \mathrm{~s}\) intervals if you start timing Carol's walk as she walks toward her partner starting at a distance of \(5.9 \mathrm{~m}\) and moving at a constant speed of \(0.6 \mathrm{~m} / \mathrm{s}\).
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