An outdoor circular fish pond has a diameter of 4.00 m and a uniform full depth of 1.60 m . A fish halfway down in the pond and 0.70 m from the near side can just see the full height of a 1.80 m tall person.
How far away from the edge of the pond is the person?
To find the distance between the person and the edge of the pond, we need to calculate the distance using similar triangles.
Step 1: Calculate the height of the person seen by the fish.
Since the fish can see the full height of the person, the height seen by the fish is 1.80 m.
Step 2: Calculate the height of the fish's eye above the bottom of the pond.
The fish is halfway down in the pond, so its eye is at a depth of 1.60 m / 2 = 0.80 m above the bottom.
Step 3: Calculate the distance of the person from the edge of the pond.
Let "x" be the distance from the edge of the pond to the fish's eye. Since we have similar triangles, we can set up the following proportion:
(x + 0.70 m) / x = 1.80 m / 0.80 m
Cross-multiplying, we get:
(x + 0.70 m) * 0.80 m = x * 1.80 m
Simplifying the equation:
0.80x + 0.56 m = 1.80x
0.80x - 1.80x = -0.56 m
-1x = -0.56 m
x = 0.56 m
Therefore, the person is located 0.56 meters away from the edge of the pond.
To find the distance between the person and the edge of the pond, we can use the concept of similar triangles. Let's break down the problem and solve it step by step:
Step 1: Draw a diagram:
Draw a circle to represent the circular fish pond, and label the diameter as 4.00 m. Mark the center of the circle and draw a vertical line from the center to represent the height of the person. Label it as 1.80 m.
Step 2: Find the radius of the pond:
The radius of the pond is half the diameter, so it will be 4.00 m / 2 = 2.00 m.
Step 3: Find the distance from the center of the pond to the person:
Since the person is halfway down the pond, the distance from the center to the person is equal to the radius of the pond plus the depth halfway down. The depth halfway down is 1.60 m / 2 = 0.80 m. Therefore, the distance from the center to the person is 2.00 m + 0.80 m = 2.80 m.
Step 4: Set up the similarity of triangles:
We can form two similar triangles in the diagram. One triangle consists of the radius of the pond, the distance from the center to the person, and the line of sight to the person. The other triangle consists of the height of the person, the distance from the center to the person, and the line of sight to the person.
Step 5: Calculate the ratio of corresponding sides:
In similar triangles, the ratios of the corresponding sides are equal.
Let x be the distance from the edge of the pond to the person.
The ratio of the radius of the pond to the line of sight is 2.00 m / x,
and the ratio of the height of the person to the line of sight is 1.80 m / 0.70 m.
So we have: 2.00 m / x = 1.80 m / 0.70 m.
Step 6: Solve for x:
To solve for x, we can cross multiply: 2.00 m * 0.70 m = 1.80 m * x.
This gives us: 1.40 m = 1.80 m * x.
Step 7: Divide both sides by 1.80 m:
Dividing both sides of the equation by 1.80 m, we get: 1.40 m / 1.80 m = x.
This simplifies to: 0.7778 = x.
So, the person is approximately 0.7778 m or 0.78 m away from the edge of the pond.