Answer :

rvkacademic

Answer:

Blank 1: Ground =  4.9

Blank 2: Ladder = 7.7

Step-by-step explanation:

This is a problem involving a right triangle and trig ratios

Couple of ways to approach this

I prefer the law of sines approach

Law of Sines

In any given triangle the ratio of the side of the triangle to the since of the angle opposite is constant

If

What does that mean?
a, b and c are sides of a triangle

and

A, B and C are angles.

a, b and c are sides.

A, B and C are angles.

(Side a faces angle A, side b faces angle B and  side c faces angle C).

then
[tex]\dfrac{a}{sin A} =\dfrac{b}{sin B} =\dfrac{c}{sin C}[/tex]

(Math is Fun: mathsisfun.com/algebra/trig-sine-law.html)

If we consider the figure and utilizing the sides represented,

the missing angle ∠ACB measures 90 - 51 = 39°

and we get


[tex]\dfrac{ladder}{sin \;90^\circ} =\dfrac{6}{sin 51} =\dfrac{ground}{sin 39}[/tex]

Using a calculator
sin 90° = 1
sin 51° = 0.777
sin 39° = 0.629

we get

[tex]\dfrac{ladder}1} =\dfrac{6}{0.777} =\dfrac{ground}{0.629}[/tex]

Using the first two terms we get
ladder = 6/0.777= 7.722

Using the second two terms we get
[tex]\dfrac{6}{0.777} =\dfrac{ground}{0.629}\\\\ground = \dfrac{6}{0.777} * 0.629\\\\ground = 4.8571[/tex]

Rounding nearest tenth

Blank 1: Ground =  4.9

Blank 2: Ladder = 7.7



semsee45

Answer:

Blank 1:  4.9

Blank 2:  7.7

Step-by-step explanation:

Question 1

To find the distance along the ground from the bottom of the ladder to the wall, we need to find the measure of side AB of right triangle ABC.

The measure of angle A as 51° and the length of the side opposite angle A as 6 m. As we need to find the length of the side adjacent to angle A, we can use the tangent trigonometric ratio:

[tex]\boxed{\begin{array}{l}\underline{\textsf{Tangent trigonometric ratio}}\\\\\sf \tan(\theta)=\dfrac{O}{A}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$O$ is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\end{array}}[/tex]

In this case:

  • θ = 51°
  • O = BC = 6 m
  • A = AB

Substitute the values into the tangent ratio and solve for AB:

[tex]\tan 51^{\circ}=\dfrac{6}{AB}\\\\\\AB=\dfrac{6}{\tan 51^{\circ}}\\\\\\AB=4.85870419917...\\\\\\AB=4.9\; \sf (nearest\;tenth)[/tex]

Therefore, the distance along the ground from the bottom of the ladder to the wall rounded to the nearest tenth is:

[tex]\Large\boxed{\boxed{4.9 \; \sf m}}[/tex]

[tex]\dotfill[/tex]

Question 2

The ladder is represented by the hypotenuse AC of right triangle ABC.

The measure of angle A as 51° and the length of the side opposite angle A as 6 m. To find the length of the hypotenuse, we can use the sine trigonometric ratio:

[tex]\boxed{\begin{array}{l}\underline{\textsf{Sine trigonometric ratio}}\\\\\sf \sin(\theta)=\dfrac{O}{H}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{O is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{H is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]

In this case:

  • θ = 51°
  • O = BC = 6 m
  • H = AC

Substitute the values into the sine ratio and solve for AC:

[tex]\sin 51^{\circ}=\dfrac{6}{AC}\\\\\\AC=\dfrac{6}{\sin51^{\circ}}\\\\\\AC=7.720557395359...\\\\\\AC=7.7\; \sf (nearest\;tenth)[/tex]

Therefore, the length of the ladder rounded to the nearest tenth is:

[tex]\Large\boxed{\boxed{7.7 \; \sf m}}[/tex]

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