haleywilliams13
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1. Joshua has a ladder that is 12 ft long. He wants to lean the ladder against a vertical wall so that the top of the ladder is 11.8 ft above the ground. For safety reasons, he wants the angle the ladder makes with the ground to be no greater than 75°. Will the ladder be safe at this height?

Respuesta :

budwilkins
So we'll use trig to solve this because it's a right triangle. The hypotenuse is the ladder (h) and the two smaller sides are the ground and the vertical wall (w).
That angle ladder makes with the ground = €
SOH: sin € = opposite/hypotenuse
[tex]sin \: \beta = \frac{opposite}{hypotenuse} = \frac{w}{h} \\ sin \: \beta =\frac{11.8}{12} = 0.983 [/tex]
[tex] \beta = { \sin }^{ - 1} (0.983) = 79.51 \: degrees[/tex]
No the ladder won't be safe!!!

Now let's make it safe:
The ladder's length (w) is constant, so stays 12
So now let's ask in an inequality what height will be safe (75° or less)
[tex] \beta = { \sin }^{ - 1}(\frac{w}{12}) \leqslant 75 \\ \sin({ \sin }^{ - 1}(\frac{w}{12})) \leqslant \sin(75) \\ \frac{w}{12} \leqslant \: 0.9659 [/tex]
[tex](12)\frac{w}{12} \leqslant 0.966(12) \\ w \leqslant 11.59[/tex]
what does that mean?? well as long as you put the ladder against the wall so that the height from ground to top of the ladder is < 11.6 ft!!

Hope that helps! :-D

The ladder is not safe at this height. The height from ground to top of the ladder is < 11.6 ft for safety reasons and this can be determined by using trigonometry functions.

Given :

  • Joshua has a ladder that is 12 ft long.
  • He wants to lean the ladder against a vertical wall so that the top of the ladder is 11.8 ft above the ground.
  • For safety reasons, he wants the angle the ladder makes with the ground to be no greater than 75°.

Check the angle that the ladder makes with the ground:

[tex]\rm sin\theta=\dfrac{Opposite}{Hypotenuse}[/tex]

[tex]\rm sin\theta = \dfrac{11.8}{12}[/tex]

[tex]\rm \theta = sin^{-1}(0.983)[/tex]

[tex]\theta = 79.51^\circ[/tex]

The ladder is safe when the angle the ladder makes with the ground to be no greater than 75° but [tex]\theta[/tex] is [tex]79.51^\circ[/tex] so, the ladder won't be safe.

To make the ladder safe, the height should not be 11.8ft.

[tex]\rm 75^\circ\geq sin^{-1}\dfrac{h }{12}[/tex]

[tex]\rm \dfrac{h}{12}\leq 0.9659[/tex]

[tex]\rm h \leq 0.9659 \times 12[/tex]

[tex]\rm h \leq 11.59\;ft[/tex]

The height from ground to top of the ladder is < 11.6 ft for safety reasons.

For more information, refer to the link given below:

https://brainly.com/question/24853349

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