At time 4 minutes your submarine is at its deepest, y= -1000m. At time 9 minutes you are at your shallowest, y= -200. Sketch the graph. Write the equation. Your sub is designed to cruise below y= -300 meters. At time zero was your submarine at its cruising depth? Between what 2 non-negative times is your submarine first able to cruise?
To sketch the graph, we first plot the given points (4,-1000) and (9,-200) on a coordinate system. The x-axis represents time in minutes, and the y-axis represents the depth in meters. Since the submarine is designed to cruise below y = -300 meters, we can draw a horizontal line at y = -300 to represent the cruising depth.
The equation of the graph can be determined by finding the equation of the line connecting the two points (4,-1000) and (9,-200).
First, we calculate the slope (m) of the line:
m = (change in y) / (change in x)
m = (-200 - (-1000)) / (9 - 4)
m = 800 / 5
m = 160
Next, we can pick any point on the line, say (4,-1000), and use the point-slope form to write the equation:
y - y1 = m(x - x1)
y - (-1000) = 160(x - 4)
y + 1000 = 160x - 640
y = 160x - 1640
The sketch of the graph will look like a descending line starting from (4,-1000) and ending at (9,-200), crossing the cruising depth line (y = -300).
Now, let's address the second part of your question. At time zero, we need to check if the submarine was at its cruising depth (y = -300).
Plugging x = 0 into the equation, we have:
y = 160(0) - 1640
y = -1640
Since -1640 is less than -300, the submarine was not at its cruising depth at time zero.
To determine the time interval during which the submarine is first able to cruise, we need to find the x-values (time) for which y (depth) is greater than -300.
Substituting y = -300 into the equation, we have:
-300 = 160x - 1640
Simplifying and solving for x:
160x = -300 + 1640
160x = 1340
x = 1340 / 160
x ≈ 8.375
Therefore, the submarine is first able to cruise between approximately 8.375 and 9 minutes (rounded to three decimal places).