Equation relating x and z is z² = x² + 121.
The horizontal speed of the fish when it is 16 ft from the angler is 4.14 in/s.
How the horizontal speed of the fish is calculated.
Let x be the horizontal distance from the angler to the fish and z be the length of the fishing line, where both x and z are measured in inches.
This arrangement form right triangle
using Pythagorean theorem
z² = x² + 11²
z² = x² + 121 -----------1
Let differentiate the equation with respect to time t.
d/dt(z²) = d(x² + 121)/dt
2zdz/dt = 2xdx/dt + 0
Make dx/dt subject of formula
dx/dt = (z/x)*dz/dt
Given that
The angler hooks a trout and reels in his line at 3 in /s.
The distance between the fish and angler is 16 ft
Substitute into z² = x² + 121
16² = x² + 121
256 = x² + 121
x² = 256 -121
= 135
x = √135 = 11.6 ft
Let us convert to same unit
1ft = 12 inches
z = 16 ft = 192 inches
x = 11.6 ft = 139.2 inches
dz/dt = 3 in/s (given)
Substitute into dx/dt = (z/x)*dz/dt
dx/dt = (192/139.2)*3
= 4.14 in/s
Therefore, the horizontal speed of the fish when it is 16 ft from the angler is 4.14 in/s.