Let's start by examining the function [tex]\( P(d) \)[/tex] which is given by:
[tex]\[ P(d) = \frac{1}{33}d + 1 \][/tex]
Here, [tex]\( P(d) \)[/tex] represents the pressure in atmospheres (atm) at a depth [tex]\( d \)[/tex] in feet under the sea.
### Finding [tex]\( P(0) \)[/tex]
To determine [tex]\( P(0) \)[/tex], we substitute [tex]\( d = 0 \)[/tex] into the function:
[tex]\[ P(0) = \frac{1}{33}(0) + 1 \][/tex]
Simplifying the expression inside the brackets:
[tex]\[ P(0) = 0 + 1 \][/tex]
[tex]\[ P(0) = 1 \][/tex]
So, the pressure at a depth of 0 feet under the sea is [tex]\( 1 \)[/tex] atm.
### Finding [tex]\( P(33) \)[/tex]
Next, we find [tex]\( P(33) \)[/tex] by substituting [tex]\( d = 33 \)[/tex] into the function:
[tex]\[ P(33) = \frac{1}{33}(33) + 1 \][/tex]
Simplifying the expression inside the brackets:
[tex]\[ P(33) = 1 + 1 \][/tex]
[tex]\[ P(33) = 2 \][/tex]
So, the pressure at a depth of 33 feet under the sea is [tex]\( 2 \)[/tex] atm.
### Summary of the Calculated Values
[tex]\[
\begin{array}{l}
P(0) = 1 \\
P(33) = 2
\end{array}
\][/tex]
### Graphing the Function
To graph the function [tex]\( P(d) = \frac{1}{33}d + 1 \)[/tex]:
1. Identify Key Points:
- We already know two points on the graph: [tex]\( (0, 1) \)[/tex] and [tex]\( (33, 2) \)[/tex].
2. Plot the Points:
- Start with the point [tex]\( (0, 1) \)[/tex].
- Next plot the point [tex]\( (33, 2) \)[/tex].
3. Draw the Line:
- Connect these points with a straight line, since the function is linear.
Your graph should show an increasing linear relationship where the pressure increases as the depth increases. The line passes through the points [tex]\( (0, 1) \)[/tex] and [tex]\( (33, 2) \)[/tex].
The graph helps to visualize how the pressure changes with the depth under the sea. As observed from the values:
- At a depth of 0 feet, the pressure is 1 atm.
- At a depth of 33 feet, the pressure is 2 atm.
