A fisherman on a wharf is pulling a rope tied to a boat. If his hand is 5 meters above sea level, how long is the rope if it makes an angle of 20°40' with water?
Y = 5 m. = ht.
A = 20.67o
L = Length of rope.
sin A = Y/L
Solve for L.
14.16
To determine the length of the rope, we can use trigonometry. In this case, the angle provided is in degrees and minutes, so we need to convert it to decimal form.
First, let's convert the angle from degrees, minutes to decimal form.
20°40' = 20 + (40/60) = 20.67°
Now we can use the sine function to find the length of the rope.
Sine(angle) = opposite/hypotenuse
In this case, the opposite side is the vertical distance from the fisherman's hand to the water level, which is 5 meters.
Sine(20.67°) = 5/hypotenuse
Next, rearrange the equation to solve for the hypotenuse:
hypotenuse = 5 / Sine(20.67°)
Now, let's calculate the length of the rope.
hypotenuse ≈ 5 / Sine(20.67°)
hypotenuse ≈ 5 / 0.354
hypotenuse ≈ 14.12
Therefore, the length of the rope is approximately 14.12 meters.
To find the length of the rope, we can use trigonometry and the given angle. Let's break down the given angle of 20°40' into decimal degrees.
20°40' = 20 + (40/60) = 20 + 0.6667 = 20.6667 degrees
Now, let's consider the situation. We have a right triangle where the hypotenuse is the length of the rope, the vertical side is 5 meters (the height of the fisherman's hand above sea level), and the angle between the vertical side and the hypotenuse is 20.6667 degrees.
Using the trigonometric function sine (sin), we can write:
sin(angle) = opposite/hypotenuse
In this case, the opposite side is the vertical side with a length of 5 meters, and the angle is 20.6667 degrees. Therefore, the equation becomes:
sin(20.6667°) = 5 meters / hypotenuse
Now, to find the length of the rope (hypotenuse), we rearrange the equation:
hypotenuse = 5 meters / sin(20.6667°)
Using a scientific calculator, we can evaluate sin(20.6667°) to find its value, and then divide 5 meters by that value to get the length of the rope.
Calculating the value of sin(20.6667°), we find:
sin(20.6667°) ≈ 0.359
Dividing 5 meters by 0.359, we can determine the length of the rope:
Length of the rope = 5 meters / 0.359 ≈ 13.928 meters
Therefore, the length of the rope is approximately 13.928 meters.