Question

PLSSSSSS HELPPPP

Joshua has a ladder that is 23ft long.He wants to lean the ladder against a vertical wall so that the top of the ladder is 21.5ft above the ground. For safety reasons, he wants the angle the ladder makes with the ground to be no greater than 70 degrees. Will the ladder be safe at this height?

Show your work and draw a diagram to support your answer. Round all answers to the nearest hundredth

Answer 1

Answer :

• θ = 69.19°

Solution :

first find the angle formed between the ground and the ladder

• sin θ = opposite/hypotenuse

here,

opposite = 21.5ft

hypotenuse = 23ft

plug in,

sin θ = 21.5ft/23ft

θ = arcsin(21.5/23)

• θ ≈ 69.19° ( to the nearest hundredth )

since the angle formed between the ground and the ladder is less than 70°, thus we can conclude that it's safe for J to climb the ladder at the height of 21.5 ft.

Answer 2

Answer:

Yes, the ladder will be safe at this height, as it makes an angle of 69.19° with the ground, which is less than 70°.

Step-by-step explanation:

To determine if the ladder is safe, we can model the given scenario as a right triangle, where the ladder (23 ft) represents the hypotenuse, and the vertical distance between the top of the ladder and the ground (21.5 ft) represents the height of the triangle.

To find the measure of the angle the ladder makes with the ground, we can use the sine trigonometric ratio:

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

In this case:

• O = 21.5
• H = 23

Substitute the values into the sine ratio and solve for angle θ:

$\sin \theta=\dfrac{21.5}{23}\\\\\\\theta=\sin^{-1}\left(\dfrac{21.5}{23}\right)\\\\\\\theta = 69.193052223...^{\circ}\\\\\\\theta = 69.19^{\circ}\; \sf (nearest\;hundredth)$

Therefore, as the angle the ladder makes with the ground is 69.19°, the ladder will be safe at this height, since the angle is less than 70°.