Let's solve this step-by-step to determine the equation of the line representing the depth of the dolphin, [tex]\(y\)[/tex], after [tex]\(x\)[/tex] seconds.
1. Initial Depth:
The dolphin starts 10 feet below sea level. Below sea level is taken as negative, so the initial depth [tex]\(y_0\)[/tex] is [tex]\(-10\)[/tex] feet.
2. Rate of Descent:
The dolphin dives at a rate of 9 feet per second. Diving deeper means the depth is increasing in the negative direction. Therefore, the slope [tex]\(m\)[/tex] is [tex]\(-9\)[/tex] (negative because the depth is increasing downward).
3. Equation of the Line:
The general form of a linear equation is:
[tex]\[
y = mx + c
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept.
4. Substitute the Values:
Here, [tex]\(m = -9\)[/tex] and [tex]\(c = -10\)[/tex]. Substituting these values into the equation gives:
[tex]\[
y = -9x - 10
\][/tex]
Now we need to match this equation with the provided options:
A. [tex]\(y = 9x - 10\)[/tex]
B. [tex]\(y = -9x - 10\)[/tex]
C. [tex]\(y = 9x + 10\)[/tex]
D. [tex]\(y = -9x + 10\)[/tex]
The correct option aligns with our derived equation [tex]\(y = -9x - 10\)[/tex], which is option B.
Thus, the correct answer is B. [tex]\(y = -9x - 10\)[/tex].
