Set up a system of equations for each word problem
One scuba diver starts at a depth of 12 meters below the water's surface and ascends at a rate of 1 meter per second. A second diver starts at a depth of 15 feet and ascends at a rate of 2 meters per second.
After how many seconds will they be at the same depth? What is this depth?
diver 1
depth = 12 - 1 t
diver 2
depth = 15 - 2 t
so
12 - t = 15 - 2 t
t = 15-12
t = 3 seconds
12 - t = 12 - 3 = 9 meters
Thank you!
To set up a system of equations for this word problem, we need to define the variables and establish the relationships between them.
Let's assume:
- x represents the number of seconds.
- y represents the depth of the first diver (starting at 12 meters below the water's surface) after x seconds.
- z represents the depth of the second diver (starting at 15 meters below the water's surface) after x seconds.
Based on the given information, we can establish the following equations:
For the first diver:
y = -12 + 1x
For the second diver:
z = -15 + 2x
To determine when they will be at the same depth, we need to find the value of x for which y = z. In other words, we need to solve the system of equations:
-12 + 1x = -15 + 2x
Now, let's solve the equation:
Simplifying the equation:
-x = -3
Multiplying both sides by -1:
x = 3
Therefore, after 3 seconds, both divers will be at the same depth. To find this depth, we substitute x = 3 into either of the original equations.
Using the first equation:
y = -12 + 1(3)
y = -12 + 3
y = -9
Thus, after 3 seconds, both divers will be at a depth of -9 meters below the water's surface.