An angler has a fish at the end of his line, which is being reeled in at the rate of 2 feet per second from a bridge 30 feet above the water. At what speed is the fish moving through the water towards the bridge when the amount of line out is 50 feet? (Assume the fish is at the surface of the water and that there is no sag in the line.)
the answer is 2.5 ft/s
ask me for the solution!
Well, the fish is probably swimming away as fast as it can, trying to escape being caught. So, let's say it's swimming at a speed of 100 miles per hour. Now, imagine the angler reeling in the line at 2 feet per second. The poor fish doesn't stand a chance! It's like being chased by a snail on a bicycle. So, to answer your question, the speed at which the fish is moving through the water towards the bridge is probably somewhere between "not very fast" and "ridiculously slow". But hey, at least the angler will have a funny story to tell about the time they caught a fish that was slower than a snail on a bicycle!
To find the speed at which the fish is moving through the water towards the bridge, we need to first find the rate at which the line is being reeled in when the amount of line out is 50 feet.
Let's assume:
L = length of line out
H = height of the bridge above the water
R = rate at which the line is being reeled in
S = speed at which the fish is moving through the water towards the bridge
Using similar triangles, we can set up the following proportion:
(L + H) / L = S / R
Substituting the given values:
(50 + 30) / 50 = S / R
80 / 50 = S / R
Simplifying, we get:
8/5 = S / R
Now, we can find the rate at which the line is being reeled in by rearranging the equation:
R = (5/8) * S
Since we know that the line is being reeled in at a rate of 2 feet per second, we can substitute this value into the equation:
2 = (5/8) * S
Multiplying both sides of the equation by 8/5, we get:
(8/5) * 2 = S
Simplifying, we find:
S = 16/5
Therefore, the fish is moving through the water towards the bridge at a speed of 16/5 feet per second when the amount of line out is 50 feet.
To find the speed at which the fish is moving through the water towards the bridge, we need to consider the relationships between the different moving parts: the speed at which the line is being reeled in, the height of the bridge, and the length of the line.
Let's break down the problem into two parts:
1. Determine how fast the line is being reeled in.
2. Calculate the rate at which the fish is moving through the water.
Part 1: Finding the speed at which the line is being reeled in.
Given that the line is being reeled in at a rate of 2 feet per second, we can say that the speed at which the line is being reeled in is 2 ft/s. This value remains constant throughout the problem.
Part 2: Calculating the rate at which the fish is moving through the water.
When the amount of line out is 50 feet, we can create a right-angled triangle with the following sides:
- The height of the bridge: 30 feet
- The length of the line: 50 feet
To find the rate at which the fish is moving through the water, we'll use a formula from Trigonometry known as the Pythagorean Theorem. The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Using the formula, we can calculate the length of the hypotenuse (the speed at which the fish is moving through the water):
hypotenuse^2 = height^2 + line length^2
hypotenuse^2 = 30^2 + 50^2
hypotenuse^2 = 900 + 2500
hypotenuse^2 = 3400
To solve for the hypotenuse, we take the square root of both sides:
hypotenuse = sqrt(3400)
hypotenuse ≈ 58.31
Therefore, the rate at which the fish is moving through the water towards the bridge when the amount of line out is 50 feet is approximately 58.31 feet per second.