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5. The distance between two buildings is 67 m. From the top of the shorter building, the angle of elevation to the top of the taller building is 27°. From the same position, the angle of depression to the base of the taller building is 39º. Calculate the height of the two buildings.

Sagot :

First, let's draw a scheme representing the measures in the text. Let's call the shorter building as A and the taller building as B.

Using this scheme, we can see that we have 2 right triangles. The height of the taller building(let's call it hB), is given by the following relation

[tex]\frac{h_B_{}}{67}=\tan 39^o[/tex]

Then, calculating the height we have

[tex]\begin{gathered} h_B=67\tan 39^o=67\times0.80978403319\ldots=54.2555302241 \\ h_B\approx54.3 \end{gathered}[/tex]

The difference between the height of the smaller building(let's call it hA) and the taller building, is given by

[tex]\frac{(h_B-h_A)}{67}=\tan 27^o[/tex]

Solving for h_A, we have

[tex]\begin{gathered} \frac{(h_B-h_A)}{67}=\tan 27^o \\ (h_B-h_A)=67\tan 27^o \\ h_A=h_B-67\tan 27^o \\ h_A\approx20.1 \end{gathered}[/tex]

The height of the smaller building is 20.1m and the height of the taller building is 54.3m.

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