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.
