A barge is pulled by two tugboats. The first tugboat is traveling at a speed of 16 knots with heading 135°, and the second tugboat is traveling at a speed of 15 knots with heading 195°. Find the resulting speed and direction of the barge. (Round your answers to the nearest whole number.)

V = 16Knots @ 135o+15Knots @ 195o.

X = 16*cos135 + 15*cos195 = -25.80 Knots
Y = 16*sin135 + 15*sin195 = 7.43 Knots.

tanA = Y/X = 7.43/-25.80 = -0.28798
Ar = -16.07o = Reference angle.
A = -16.07 + 180 = 164o = Direction.

V = X/cosA = -25.80/cos164 = 27 Knots.

To find the resulting speed and direction of the barge, we need to use vector addition.

Step 1: Convert the speeds from knots to miles per hour (mph), as this is a more commonly used unit for speed calculations.

1 knot = 1.151 mph

Speed of the first tugboat = 16 knots × 1.151 mph/knot ≈ 18.416 mph
Speed of the second tugboat = 15 knots × 1.151 mph/knot ≈ 17.265 mph

Step 2: Convert the headings to Cartesian coordinates, where north is the positive y-axis and east is the positive x-axis.

Heading 135°:
x-component = cos(135°) ≈ -0.707
y-component = sin(135°) ≈ 0.707

Heading 195°:
x-component = cos(195°) ≈ -0.984
y-component = sin(195°) ≈ -0.174

Step 3: Calculate the Cartesian components for each tugboat's motion by multiplying each component by the corresponding speed.

First tugboat's x-component = -0.707 × 18.416 mph ≈ -13.018 mph
First tugboat's y-component = 0.707 × 18.416 mph ≈ 13.018 mph

Second tugboat's x-component = -0.984 × 17.265 mph ≈ -16.998 mph
Second tugboat's y-component = -0.174 × 17.265 mph ≈ -3.003 mph

Step 4: Add the components of both tugboats to get the resulting x-component and y-component of the barge's motion.

Resulting x-component = (-13.018 mph) + (-16.998 mph) ≈ -30.016 mph
Resulting y-component = (13.018 mph) + (-3.003 mph) ≈ 10.015 mph

Step 5: Use the Pythagorean theorem to find the magnitude of the resulting motion.

Magnitude = √(Resulting x-component² + Resulting y-component²)
Magnitude = √((-30.016 mph)² + (10.015 mph)²) ≈ 31.623 mph

Step 6: Use the inverse tangent function to find the angle (direction) of the resulting motion.

Direction = arctan(Resulting y-component / Resulting x-component)
Direction = arctan(10.015 mph / -30.016 mph) ≈ -18.065°

Step 7: Convert the magnitude back to knots.

Magnitude ≈ 31.623 mph ÷ 1.151 mph/knot ≈ 27.453 knots

Therefore, the resulting speed of the barge is approximately 27 knots, and the direction is approximately 18° counter-clockwise from west.