Answered

In the diagram, AB and ED represent two
vertical buildings, 27 m apart, on level MO
ground. The heights of AB and ED are 15 m and 24 m respectively. A car C on BD is at equal distance from A and E.
Find
(a) the distances BC and AC
(b) the angle of elevation of E from C.
​

In the diagram AB and ED represent twovertical buildings 27 m apart on level MOground The heights of AB and ED are 15 m and 24 m respectively A car C on BD is a class=


Answer :

zarahaider4211

Answer:

(a) BC = 20 AC = 25

(b) <C = 74 degrees

Step-by-step explanation:

Triangles

Pythagorean Theorem

According to the Pythagorean theorem, one side of a right triangle can be calculated using the other two.

                                          [tex]a^2+b^2=c^2[/tex],

where a and b are the legs and c is the hypotenuse. This can be rearranged and simplified further to find certain lengths of a right triangle.

[tex]\dotfill[/tex]

Inverse Trigonometric Functions

The basic trig functions are

  • [tex]\sin(x)=\dfrac{\text{opp}}{\text{hyp}}[/tex]
  • [tex]\cos(x)=\dfrac{\text{adj}}{\text{hyp}}[/tex]
  • [tex]\tan(x)=\dfrac{\text{opp}}{\text{adj}}[/tex]

where "opp", "adj", and "hyp" are determined by the location of the reference angle x.

Their inverses use the ratio of side lengths to find the reference angle.

  • [tex]\sin^{-1}\left(\dfrac{\text{opp}}{\text{hyp}}\right)=x[/tex]
  • [tex]\cos^{-1}\left(\dfrac{\text{adj}}{\text{hyp}}\right)=x[/tex]
  • [tex]\tan^{-1}\left(\dfrac{\text{opp}}{\text{adj}}\right)=x[/tex]

[tex]\hrulefill[/tex]

Solving the Problem

We're told

  • AB = 15
  • ED = 24
  • AC = EC
  • BC + CD = 27 (BD = 27)

and we need to find the length of AC and the elevation of E from C or angle DCE angle (bottom left angle).

We can start by letting the length of CD by y and BC (27-y).

Since the two triangles' hypotenuse are the same, their Pythagorean equations must also be.

So,

                                [tex]AB^2+BC^2=ED^2+CD^2[/tex]

                                 [tex]15^2+(27-y)^2=24^2+y^2[/tex]

if we solve for y we can find the length of CD and BC.

                                [tex]15^2+(27-y)^2=24^2+y^2[/tex]

                            [tex]225+729-54y+y^2=576+y^2[/tex]

                                         [tex]954-54y=576[/tex]

                                              [tex]378=54y[/tex]

                                                   [tex]7=y[/tex]

                                [tex]BC=27-y=27-7=20[/tex]

Knowing the lengths of each triangle's side legs, we can find their hypotenuse. We can use either pair of legs since the hypotenuse is the same.

Recalling Pythagorean triples, the hypotenuse length of a triangle with side lengths 24 and 7 is 25, so AC = 25. (This can also be solved by just doing the calculation by hand/calculator)

Lastly, we need to find the angle of DCE. We know its legs and hypotenuse length, so we can use any of the inverse trig functions to determine the angle measure.

Using the inverse sine function, the angle measure is

                                     [tex]\sin^{-1}\left(\dfrac{24}{25}\right) \approx 74^\circ[/tex].

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