A reconnaissance airplane P, flying at 11,000 feet above a point R on the surface of the water, spots a submarine S at an angle of depression of β = 21° and a tanker T at an angle of depression of α = 39°, as shown in the figure. In addition, ∠SPT is found to be γ = 99°. Approximate the distance between the submarine and the tanker. (Round your answer to the nearest whole number.)
I don't see how ∠SPT can be 99°.
∠SPT = ∠SPR + ∠RPT = 51°+69° = 120°
Did I miss something in the explanation?
39 + 21 + 39= 99
Hmm. Not having the diagram is a bit of a problem.
In my diagram, extend the sea-level line TR to intersect PS at Q. Then, I have the angle of depression α = ∠PTR and β = ∠PQR.
Apparently that is wrong, so please describe the diagram using only labeled points. Where does the 2nd 39° angle come from? Apparently α and β are not really the angles of depression from P to S and T.
To find the distance between the submarine and the tanker, we can use trigonometry and the given angles of depression.
Let's name the distance between the reconnaissance airplane P and the submarine S as x and the distance between the reconnaissance airplane P and the tanker T as y.
Now, we can use the tangent function to relate the angle of depression β and the distance x. The tangent of an angle is equal to the opposite side divided by the adjacent side in a right triangle. In this case, the opposite side is the height of the airplane (11,000 feet) and the adjacent side is the distance x. So, we have:
tan(β) = opposite/adjacent
tan(21°) = 11,000 / x
To isolate x, we can rearrange the equation:
x = 11,000 / tan(21°)
Similarly, we can use the tangent function to relate the angle of depression α and the distance y:
tan(α) = 11,000 / y
Again, isolating y:
y = 11,000 / tan(39°)
Now, we want to find the distance between the submarine and the tanker, which is the length of the line segment ST. To find ST, we can use the Law of Cosines.
The Law of Cosines states that in a triangle with sides a, b, and c and angle C opposite to side c, we have:
c^2 = a^2 + b^2 - 2ab * cos(C)
In triangle PST, we can use this formula to find the side ST:
ST^2 = x^2 + y^2 - 2xy * cos(γ)
Substituting the values we found for x and y:
ST^2 = (11,000 / tan(21°))^2 + (11,000 / tan(39°))^2 - 2 * (11,000 / tan(21°)) * (11,000 / tan(39°)) * cos(99°)
Now, we can calculate the value of ST using these trigonometric calculations.