A ladder 20 m long rest against a vertical wall so that the foot of the ladder is 9 m from the wall
a. Find, correct to the nearest degree, the angle that the ladder makes with the wall
b. Find, correct to 1 decimal place, the height above the ground at which the upper end of the ladder touches the wall
Pls answer with workings I need it very urgent
Solve the following equation
2. sin y = cos ( y +20°)
1.
a.
cos θ = 9 / 20 = 0.18
θ = cos⁻¹ ( 0.18 ) = 79.630240195°
θ = 80° θ to the nearest degree
b.
Pythagorean theorem
h = √ ( 20² - 9² ) = √ ( 400 - 81 ) = √ 319 = 17.8605711
h = 17.9 m correct to 1 decimal place
2.
sin y = cos ( y + 20° )
Use identity:
cos ( y ) = sin ( 90° - y )
sin y = sin [ 90° - ( y + 20° ) ]
sin y = sin ( 90° - y - 20° )
sin y = sin ( 70° - y )
sine is equal so:
y = 70° - y
Add y to both sides
y + y = 70° - y + y
2 y = 70°
y = 70° / 2
y = 35°
sin ( 180° - θ ) = sin θ
so
sin ( 180° - 35° ) = sin 35°
sin 145° = sin 35°
Period of sine = 360°
So, the solutions are:
y = 35° ± n ∙ 360°
and
y = 145° ± n ∙ 360°
where
n = some integer
a. sinB = 9/20.
B = 27o.
b. CosB = h/20.
Cos27 = h/20.
h = ?
a. To find the angle that the ladder makes with the wall, we can use the trigonometric function "arctan". The formula is:
angle = arctan(opposite/adjacent)
In this case, the opposite side is the height of the ladder and the adjacent side is the distance of the foot of the ladder from the wall. So, we have:
angle = arctan(height/distance)
Given that the height of the ladder is unknown, let's denote it as h.
From the given information, we have the following triangle:
|
|
| h
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| /
| /
| /
9m | /
| /
| /
|
|
|
|
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20m
We can solve for h using the Pythagorean theorem:
h^2 + 9^2 = 20^2
h^2 + 81 = 400
h^2 = 319
Taking the square root of both sides, we get:
h ≈ √319
h ≈ 17.9 (rounded to 1 decimal place)
Now we can calculate the angle:
angle = arctan(17.9/9)
angle ≈ arctan(1.9888)
angle ≈ 63.4° (rounded to the nearest degree)
Therefore, the angle that the ladder makes with the wall is approximately 63 degrees.
b. To find the height above the ground at which the upper end of the ladder touches the wall, we can use the trigonometric function "sin". The formula is:
height = opposite side = ladder length * sin(angle)
Using the ladder length of 20 m and the angle we found in part a (63.4°), we can calculate the height:
height = 20 * sin(63.4°)
height ≈ 20 * 0.8958
height ≈ 17.92 (rounded to 1 decimal place)
Therefore, the height above the ground at which the upper end of the ladder touches the wall is approximately 17.9 meters.
a. To find the angle that the ladder makes with the wall, we can use the trigonometric function tangent. Tangent is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, the side opposite the angle is the height of the wall, and the side adjacent to the angle is the distance from the foot of the ladder to the wall.
Let's denote the angle as θ. We have the following equation:
tan(θ) = opposite/adjacent
tan(θ) = height/9
We know that the length of the ladder is 20 m, so the hypotenuse of the right triangle formed by the ladder, the wall, and the ground is 20 m. Using Pythagoras' theorem, we can find the height of the wall.
height^2 + 9^2 = 20^2
height^2 = 20^2 - 9^2
height^2 = 400 - 81
height^2 = 319
height ≈ √319 ≈ 17.87 m
Now, we can substitute the height value into the equation for tangent:
tan(θ) = 17.87/9
To find the angle θ, we can take the inverse tangent (arctan) of both sides:
θ = arctan(17.87/9)
Using a scientific calculator, we can find that θ ≈ 62.8 degrees (rounded to the nearest degree).
b. To find the height above the ground at which the upper end of the ladder touches the wall, we can use the sine function. Sine is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
Let's denote the height above the ground as h. We have the following equation:
sin(θ) = opposite/hypotenuse
sin(θ) = h/20
Using the value of θ we found previously (approximately 62.8 degrees), we can solve for h:
sin(62.8 degrees) = h/20
h = 20 * sin(62.8 degrees)
Using a scientific calculator, we can find that h ≈ 17.35 m (rounded to 1 decimal place).