How long is the shadow cast by the 540-foot tower when the sun's rays follow the vector [6,10,-27]? Also at what angle do the rays of the sun strike the ground?
To determine the length of the shadow cast by the tower, we need to find the length of the shadow vector. The tower's height is given as 540 feet, which corresponds to the vertical component of the shadow vector.
The vertical component can be found by multiplying the height (540 ft) by the vertical component of the sun's rays vector (10):
Vertical Component = 540 ft × 10 = 5400 ft
Next, we need to find the horizontal distance covered by the shadow. The horizontal component can be found by multiplying the vertical component (5400 ft) by the horizontal component of the sun's rays vector (6), then dividing it by the additional vertical component (27):
Horizontal Component = (5400 ft × 6) / 27 = 1200 ft
Now, we can calculate the length of the shadow vector using the Pythagorean theorem:
Shadow Vector = √(Vertical Component^2 + Horizontal Component^2)
Shadow Vector = √(5400^2 + 1200^2) ft
Shadow Vector ≈ √(29,160,000 + 1,440,000) ft
Shadow Vector ≈ √30,600,000 ft
Shadow Vector ≈ 5,524.74 ft
Therefore, the length of the shadow cast by the 540-foot tower is approximately 5,524.74 feet.
To determine the angle at which the rays of the sun strike the ground, we can use the dot product formula:
Dot Product = |a| |b| cosθ
Where |a| and |b| are the magnitudes of the two vectors and θ is the angle between them.
In this case, |a| is the magnitude of the sun's rays vector, given by:
|a| = √(6^2 + 10^2 + (-27)^2) = √(36 + 100 + 729) = √865 = 29.41
We know that |b| is the magnitude of the shadow vector, calculated to be approximately 5,524.74 feet.
So, the dot product equation becomes:
29.41 × 5,524.74 × cosθ = 6 × 10 × -27
By solving this equation for cosθ, we can find the angle:
cosθ = (6 × 10 × -27) / (29.41 × 5,524.74)
cosθ ≈ -0.00073082
To find the angle θ, we can take the inverse cosine (arccos) of -0.00073082:
θ ≈ arccos(-0.00073082) ≈ 90.08°
Therefore, the rays of the sun strike the ground at an angle of approximately 90.08 degrees.
To find the length of the shadow cast by the tower, we can use similar triangles. Let's assume the length of the shadow is 'x.'
Given:
Height of the tower = 540 feet
Direction vector of the sun's rays = [6, 10, -27]
We can calculate the angle between the sun's rays and the ground using the dot product formula:
cosθ = (a · b) / (|a| * |b|),
where a = [0, 0, 1] (vector perpendicular to the ground)
b = [6, 10, -27]
|a| = sqrt(0^2 + 0^2 + 1^2) = 1 (since |a| = 1)
|b| = sqrt(6^2 + 10^2 + (-27)^2) = sqrt(736) ≈ 27.13
a · b = (0 * 6) + (0 * 10) + (1 * -27) = -27
cosθ = (-27) / (1 * 27.13) = -0.996
θ ≈ acos(-0.996) ≈ 175.22 degrees (approximately)
Therefore, the sun's rays strike the ground at an angle of approximately 175.22 degrees.
To find the length of the shadow, we can use the property of similar triangles. The height of the tower (540 feet) is to the length of the shadow (x) as -27 (from the direction vector) is to 1.
540 / x = -27 / 1
Cross-multiplying, we get:
x = (540 * 1) / -27 = -20 feet
Since the length cannot be negative, the magnitude of the shadow is 20 feet.