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An 18 meters ladder is resting against a wall and makes an angle of 30° with the ground.
If the ladder is pushed vertically along the wall by a distance of x meters, the angle is
changed to 60° with the ground. The value of x is
A)9(√3 −1)
B) 2√3
c) ³(√3-1)
D) 9-√3

An 18 meters ladder is resting against a wall and makes an angle of 30 with the ground If the ladder is pushed vertically along the wall by a distance of x mete class=


Answer :

Answer:

A

Step-by-step explanation:

using the sine ratio and the exact values

sin30° = [tex]\frac{1}{2}[/tex] and sin60° = [tex]\frac{\sqrt{3} }{2}[/tex]

calculate the height of the wall below x , call it h

the hypotenuse is the length of the ladder

using the sine ratio in the lower right triangle.

sin30° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{h}{18}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )

2h = 18 ( divide both sides by 2 )

h = 9

lower part of the wall is 9 meters

using the sine ratio in the right triangle on the right

sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{9+x}{18}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )

2(9 + x) = 18[tex]\sqrt{3}[/tex] ( divide both sides by 2 )

9 + x = 9[tex]\sqrt{3}[/tex] ( subtract 9 from both sides )

x = 9[tex]\sqrt{3}[/tex] - 9 = 9([tex]\sqrt{3}[/tex] - 1)

semsee45

Answer:

[tex]\textsf{A)} \quad x=9(\sqrt{3}-1)[/tex]

Step-by-step explanation:

Interior angles of a triangle sum to 180°.  Therefore, if the ladder makes an angle of 30° with the ground, the angle it makes with the wall is 60°.

Therefore, the two triangles are congruent as their corresponding angles are the same and the length of their longest side (hypotenuse) is the same.

30-60-90 Triangle Theorem

A 30-60-90 triangle is a special right triangle where the measures of its angles are in the ratio 1 : 2 : 3 and the measure of its sides are in the ratio 1 : √3 : 2.

The formula for the sides is  b : b√3 : 2b  where:

  • b = the side opposite the 30° angle (shortest leg).
  • b√3 = the side opposite the 60° angle.
  • 2b = the side opposite the right angle (hypotenuse).

Therefore, as the length of the hypotenuse is 18:

[tex]\implies 2b=18[/tex]

[tex]\implies b=9[/tex]

[tex]\implies b\sqrt{3}=9\sqrt{3}[/tex]

To find x, subtract the length of the shortest leg of the right triangle from the length of the other leg of the right triangle:

[tex]\implies x=9\sqrt{3}-9[/tex]

[tex]\implies x=9(\sqrt{3}-1)[/tex]

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