Respuesta :

Answer:

9.72 ft (nearest hundredth)

Step-by-step explanation:

  • The ramp is the hypotenuse of a right triangle = 10 ft
  • The ramp makes an angle of 18° with the horizontal
  • Let b = the horizontal

First, calculate the horizontal distance [tex]b[/tex] by using the cosine trig ratio:

[tex]\cos(\theta)=\mathsf{\dfrac{adjacent\ side}{hypotenuse}}[/tex]

[tex]\implies \cos(18)=\dfrac{b}{10}[/tex]

[tex]\implies b=10 \cos(18)[/tex]

Given:

  • [tex]\theta[/tex]  = 12°
  • [tex]b=10 \cos(18)[/tex]
  • The new ramp is the hypotenuse = h

Use the cosine trig ratio to calculate the new hypotenuse:

[tex]\implies \cos(12)=\dfrac{10 \cos(18)}{h}[/tex]

[tex]\implies h=\dfrac{10 \cos(18)}{\cos(12)}[/tex]

⇒ h = 9.723036846...

⇒ h = 9.72 ft (nearest hundredth)

Answer:

  14.86 feet

Step-by-step explanation:

We assume the new ramp arrives at the same height as the old one. That height is found using the sine function:

  h = 10·sin(18°)

We want to find the ramp length x that will give the same height with an angle of 12°:

  h = x·sin(12°)

Substituting for h, we get ...

  10·sin(18°) = x·sin(12°)

  x = 10·sin(18°)/sin(12°) ≈ 14.863 . . . . feet

The new ramp is about 14.86 feet long.

_____

Additional comment

The sine relation is ...

  Sin = Opposite/Hypotenuse

In this use, we rearrange it to ...

  Opposite = Hypotenuse × Sin

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