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In the diagram, S, T and K lie in the same horizontal plane - NSC Mathematics - Question 7 - 2023 - Paper 2

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In the diagram, S, T and K lie in the same horizontal plane. RS is a vertical tower. The angle of depression from R to K is \( \beta \). TSK = \( \alpha \), TS = \( ... show full transcript

Worked Solution & Example Answer:In the diagram, S, T and K lie in the same horizontal plane - NSC Mathematics - Question 7 - 2023 - Paper 2

Step 1

Determine the length of SK in terms of p, q and a.

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Answer

To find the length of ( SK ), we can use the formula for the area of a triangle:

q=12p(SK)sin⁡(α)q = \frac{1}{2} p (SK) \sin(\alpha)

Rearranging this gives: SK=2qpsin⁡(α)SK = \frac{2q}{p \sin(\alpha)}

Step 2

Show that RS = \( \frac{2q \tan \beta}{p \sin \alpha} \).

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Answer

From triangle ( RKS ), we know:

RSSK=sin⁡(β)\frac{RS}{SK} = \sin(\beta)

Substituting for ( SK ):

RS=SKsin⁡(β)=(2qpsin⁡(α))sin⁡(β)RS = SK \sin(\beta) = \left(\frac{2q}{p \sin(\alpha)}\right) \sin(\beta)

This simplifies to: RS=2qtan⁡βpsin⁡(α)RS = \frac{2q \tan \beta}{p \sin(\alpha)}

Step 3

Calculate the size of a if \( \alpha < 90° \) and \( RS = 70 \) m, \( p = 80 \) m, \( q = 2500 \) m² and \( \beta = 42° \).

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Answer

Substituting the values into the equation from part 7.2:

70=2⋅2500⋅tan⁡(42°)80⋅sin⁡(α)70 = \frac{2 \cdot 2500 \cdot \tan(42°)}{80 \cdot \sin(\alpha)}

Solving for ( \sin(\alpha) ):

( \sin(\alpha) = \frac{2 \cdot 2500 \cdot \tan(42°)}{70 \cdot 80} )

Calculating this gives ( \sin(\alpha) \approx 0.80 ), and therefore:

( a \approx 53.51° ).

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