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Two objects are moving along separate linear paths where each path is described by position, [tex]d[/tex], and time, [tex]t[/tex]. The variable [tex]d[/tex] is measured in meters, and the variable [tex]t[/tex] is measured in seconds. The equation describing the graph of the position of the first object with respect to time is [tex]d = 2.5t + 22[/tex]. The graph of the position of the second object is a parallel line passing through [tex](t=0, d=1)[/tex]. What is the equation of the second graph?

A. [tex]d = 2.5t + 1[/tex]

B. [tex]d = t + 2.5[/tex]

C. [tex]d = -0.4t + 1[/tex]

D. [tex]d = 2.5t + 3.2[/tex]



Answer :

GinnyAnswer
To determine the equation of the second object's path, we need to find a linear equation that is parallel to the first object's path and passes through the point \((t=0, d=1)\).

1. Identify the slope of the first object's path:

The equation of the first object is given by:
[tex]\[ d = 2.5t + 22 \][/tex]
The slope (m) of this line is 2.5.

2. Determine the slope of the second object's path:

Since the second object's path is parallel to the first object's path, it must have the same slope. Therefore, the slope of the second object's path is also 2.5.

3. Use the point-slope form to find the y-intercept (b) of the second object's path:

The point-slope form of a linear equation is:
[tex]\[ d = mt + b \][/tex]
Since the second object's path passes through the point \((t=0, d=1)\), we can substitute these values into the equation to find \(b\).

Substituting \(t = 0\) and \(d = 1\) into the equation:
[tex]\[ 1 = 2.5 \cdot 0 + b \][/tex]
Simplifying, we get:
[tex]\[ b = 1 \][/tex]

4. Write the final equation:

Combining the slope \(m = 2.5\) and the y-intercept \(b = 1\), we get the equation of the second object's path:
[tex]\[ d = 2.5t + 1 \][/tex]

Therefore, the correct equation of the second object's path is:

A. [tex]\( d = 2.5t + 1 \)[/tex]

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