To find the equation of the second graph given that it is parallel to the first and passes through the point [tex]\((t=0, d=1)\)[/tex], follow these detailed steps:
1. Identify the slope of the first object:
The equation for the first object's position is given by:
[tex]\[
d = 25t + 2.2
\][/tex]
Here, the slope (or gradient) of the line is 25.
2. Slope of the second object:
Since the second object's path is parallel to the first, it will have the same slope. Therefore, the slope of the second object's graph is also 25.
3. Determine the y-intercept of the second object's graph:
The second object's graph passes through the point [tex]\((t=0, d=1)\)[/tex]. We use this information to find the y-intercept, [tex]\(b\)[/tex], in the slope-intercept form of the line equation [tex]\(d = mt + b\)[/tex].
We know:
[tex]\[
d = 25t + b
\][/tex]
Substituting the point [tex]\((0, 1)\)[/tex] into this equation:
[tex]\[
1 = 25 \cdot 0 + b \implies b = 1
\][/tex]
4. Write the equation of the second object's position:
Now that we have both the slope (25) and the y-intercept (1), we can construct the equation of the second object's path:
[tex]\[
d = 25t + 1
\][/tex]
5. Verify the correct answer:
We now compare this equation to the options given:
- Option A: [tex]\(d = 25t + 1\)[/tex]
- Option B: [tex]\(d = t + 25\)[/tex]
- Option C: [tex]\(d = -0.4t + 1\)[/tex]
- Option D: [tex]\(d = 25t + 3.2\)[/tex]
The correct match is Option A: [tex]\(d = 25t + 1\)[/tex].
Therefore, the correct answer is:
- A: [tex]\(d = 25t + 1\)[/tex]
