A pilot of a small plane was flying over an oil spill at an altitude of 10,000 feet. He found that the near edge of the spill had an angle of depression of 58 degrees and the far edge of the spill had an angle of depression of 44 degrees. What is the width of the spill?
review the definition of cotθ. Draw a diagram, and it is clear that the width w is given by
w = 10000(cot44°-cot58°)
To find the width of the spill, we can use trigonometry and the concept of angle of depression.
Let's look at the problem setup: The pilot is flying over the oil spill at an altitude of 10,000 feet. The near edge of the spill has an angle of depression of 58 degrees, while the far edge has an angle of depression of 44 degrees.
First, we need to understand the concept of angle of depression. The angle of depression is the angle between the line of sight from the observer (pilot) to the object (near or far edge of the spill) and the horizontal plane (ground level).
To find the width of the spill, we need to calculate the horizontal distance between the two lines of sight (from the pilot to the near and far edges of the spill).
Let's denote the horizontal distance as "x."
Using Trigonometry:
We can use the tangent function to relate the angle of depression to the horizontal distance.
For the near edge of the spill:
tan(58 degrees) = height of observer / x
Since the height of the observer is 10,000 feet, we can write the equation as:
tan(58 degrees) = 10,000 / x
Similarly, for the far edge of the spill:
tan(44 degrees) = 10,000 / (x + width)
Since we want to find the width of the spill, let's solve these two equations simultaneously to find the value of "x."
Solving the equations:
Step 1: Rearrange the first equation to solve for "x":
x = 10,000 / tan(58 degrees)
Step 2: Substitute the expression for "x" into the second equation:
tan(44 degrees) = 10,000 / (10,000 / tan(58 degrees) + width)
Step 3: Simplify the expression:
tan(44 degrees) = tan(58 degrees) / (1 + width / (10,000 * tan(58 degrees)))
Step 4: Cross multiply and rearrange the equation:
tan(44 degrees) * (1 + width / (10,000 * tan(58 degrees))) = tan(58 degrees)
Step 5: Divide both sides by tan(44 degrees):
1 + width / (10,000 * tan(58 degrees)) = tan(58 degrees) / tan(44 degrees)
Step 6: Subtract 1 from both sides:
width / (10,000 * tan(58 degrees)) = tan(58 degrees) / tan(44 degrees) - 1
Step 7: Simplify the right side:
width / (10,000 * tan(58 degrees)) = (tan(58 degrees) - tan(44 degrees)) / tan(44 degrees)
Step 8: Multiply both sides by (10,000 * tan(58 degrees)):
width = (10,000 * tan(58 degrees)) * (tan(58 degrees) - tan(44 degrees)) / tan(44 degrees)
Now, using a calculator, we can find the value of the width using the given angles (58 degrees and 44 degrees).