A scuba diver located 20 feet below the surface of water spots a shipwreck at a 70 degree angle of depression. After descending to a point 45 feet above the ocean floor, the diver sees the shipwreck at a 57 degree angle. Draw a diagram to represent the situation, and determine the depth of the shipwreck.
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To solve this problem, we will need to use trigonometry and create a diagram to represent the situation described.
First, let's draw a rough sketch of the situation.
Diver
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/ |20
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Shipwreck ----------- Ocean Floor (45 feet)
In the diagram, we have a scuba diver located 20 feet below the surface of the water. The diver spots a shipwreck at a 70 degree angle of depression. The diver then descends to a point 45 feet above the ocean floor, where they see the shipwreck at a 57 degree angle.
Let's label the unknown depth of the shipwreck as "x."
Now, let's break down the problem into two right triangles:
Triangle 1: Ocean floor to shipwreck
Triangle 2: Diver to shipwreck
Using the information given, we can set up the following trigonometric equations:
In Triangle 1:
tan(70) = x / (45 + x)
In Triangle 2:
tan(57) = x / 20
Now, we can solve these equations to find the value of x, which represents the depth of the shipwreck:
Step 1: Solve for x in Triangle 1 equation:
tan(70) = x / (45 + x)
To solve for x, we need to isolate it. Cross-multiply and rearrange the equation:
(x / (45 + x)) = tan(70)
x = (tan(70)) * (45 + x)
x = 1.97 * (45 + x)
x = 1.97 * 45 + 1.97 * x
x - 1.97 * x = 1.97 * 45
(1 - 1.97) * x = 1.97 * 45
-0.97 * x = 1.97 * 45
x = (1.97 * 45) / -0.97
x ≈ -91.45
Since depth cannot be negative, we discard the negative value and consider the magnitude, so x ≈ 91.45 feet.
Therefore, the depth of the shipwreck is approximately 91.45 feet.
assuming a level ocean floor, then
if the wreck is at a distance x from the point directly below the diver,
45/x = tan 57°
If the diver started at a point h above the floor, then
h/x = tan 70°
The depth of the ship is 20 + h