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A student programs a robot to travel at a constant speed across the classroom floor. The table below represents the relationship between the distance, in feet, the robot travels over a period of time, in seconds.

DISTANCE ROBOT TRAVELED

[tex]\[
\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Time, $t$ \\
(seconds)
\end{tabular} & \begin{tabular}{c}
Distance, $d$ \\
(feet)
\end{tabular} \\
\hline 2 & 1 \\
\hline 4 & 2 \\
\hline 10 & 5 \\
\hline 16 & 8 \\
\hline
\end{tabular}
\][/tex]

Write an equation to represent the distance, [tex]$d$[/tex], in feet, the robot travels in [tex]$t$[/tex] seconds. Using the equation, how many seconds will it take for the robot to travel 11 feet?

Show your work.

Answer:



Answer :

GinnyAnswer
To solve this problem step-by-step, we need to determine the equation that represents the relationship between the distance [tex]\(d\)[/tex] in feet that the robot travels, and the time [tex]\(t\)[/tex] in seconds.

1. Identify the constant speed of the robot:
Since the robot travels at a constant speed, the relationship between distance ([tex]\(d\)[/tex]) and time ([tex]\(t\)[/tex]) can be expressed with a linear equation of the form:
[tex]\[ d = speed \cdot t \][/tex]

2. Calculate the constant speed:
We will use the data points provided to determine this constant speed. We can find the speed by using any two data points. Let's use the first two data points:

When [tex]\( t = 2 \)[/tex] seconds, [tex]\( d = 1 \)[/tex] foot.

When [tex]\( t = 4 \)[/tex] seconds, [tex]\( d = 2 \)[/tex] feet.

The speed (rate of travel) can be calculated by finding the slope between these two points:
[tex]\[ speed = \frac{\Delta d}{\Delta t} = \frac{d_{2} - d_{1}}{t_{2} - t_{1}} \][/tex]
Plug in the values:
[tex]\[ speed = \frac{2 - 1}{4 - 2} = \frac{1}{2} = 0.5 \text{ feet per second} \][/tex]

3. Write the equation:
Knowing the constant speed is 0.5 feet per second, the equation representing the distance [tex]\(d\)[/tex] in feet, the robot travels in [tex]\(t\)[/tex] seconds is:
[tex]\[ d = 0.5 \cdot t \][/tex]

4. Find the time to travel 11 feet:
We need to determine the time [tex]\(t\)[/tex] it takes for the robot to travel 11 feet. Using the equation [tex]\(d = 0.5 \cdot t\)[/tex], we set [tex]\(d = 11\)[/tex] and solve for [tex]\(t\)[/tex]:
[tex]\[ 11 = 0.5 \cdot t \][/tex]
To isolate [tex]\(t\)[/tex], divide both sides by 0.5:
[tex]\[ t = \frac{11}{0.5} = 22 \text{ seconds} \][/tex]

Final Answer:
The equation representing the distance [tex]\(d\)[/tex] in feet that the robot travels in [tex]\(t\)[/tex] seconds is:
[tex]\[ d = 0.5 \cdot t \][/tex]
It will take the robot 22 seconds to travel 11 feet.

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