To find out when Nate and Lucas will be at the same elevation, we need to set up equations for both Nate's and Lucas's elevations and then solve for when these two equations are equal.
Let's denote:
Nate's elevation as N(t)
Lucas's elevation as L(t)
Given:
Nate's elevation starts at -42 feet and he descends at a rate of 10 feet per minute. So, N(t) = -42 - 10t.
Lucas's elevation starts at -12 feet and he descends at a rate of 14 feet per minute. So, L(t) = -12 - 14t.
To find when they are at the same elevation, we set N(t) = L(t) and solve for t:
-42 - 10t = -12 - 14t
Now, let's solve for t:
-42 + 12 = -14t + 10t
-30 = -4t
Divide both sides by -4:
t = 30 / 4
t = 7.5 minutes
So, after 7.5 minutes, Nate and Lucas will be at the same elevation.
To find out what elevation they will be at that time, plug this value of t into either N(t) or L(t). Let's use N(t) for this calculation:
N(7.5) = -42 - 10(7.5)
N(7.5) = -42 - 75
N(7.5) = -117 feet
Therefore, after 7.5 minutes, Nate and Lucas will be at an elevation of -117 feet.
Now, to write an expression to represent Nate's elevation after t minutes, we already have it: N(t) = -42 - 10t. So, Nate's elevation after t minutes can be expressed as -42 - 10t.
