A boy 1.2m tall, stands 6m away from the foot of a vertical lamp pole 4.2m long. If the lamp is at the tip of the pole, calculate (i) the length of the shadow of the body cast by the lamp
(ii) calculate the angle of elevation of the lamp from the boy, correct to the nearest degree
(i) x/1.2 = 6/(4.2-1.2)
(ii) tanθ = (4.2-1.2)/6
I want to know it
To solve this problem, we can divide it into two parts: calculating the length of the shadow of the body cast by the lamp, and calculating the angle of elevation of the lamp from the boy.
(i) Calculating the length of the shadow:
Let's consider the similar triangles formed by the boy, the lamp pole, and the shadow:
1. The height of the boy is 1.2m, and the distance between the boy and the lamp pole is 6m. Therefore, the ratio of the height of the boy to the distance from the boy to the lamp pole is 1.2/6.
2. The length of the lamp pole is 4.2m, and the length of the shadow is unknown. Therefore, the ratio of the length of the shadow to the length of the lamp pole is x/4.2.
Using the concept of similar triangles, we can set up an equation:
1.2/6 = x/4.2
Simplifying the equation:
1.2 * 4.2 = 6x
5.04 = 6x
x = 5.04/6
x ≈ 0.84m
Therefore, the length of the shadow of the body cast by the lamp is approximately 0.84m.
(ii) Calculating the angle of elevation of the lamp from the boy:
The angle of elevation can be determined by calculating the opposite side length and the adjacent side length in a right-angled triangle using trigonometry.
In this case:
Opposite side length = height of the boy = 1.2m
Adjacent side length = distance between the boy and the lamp pole = 6m
Using the tangent function: tan(angle) = opposite/adjacent
tan(angle) = 1.2/6
To find the angle, we can use the inverse tangent or arctan function:
angle = arctan(1.2/6)
Using a calculator or trigonometric table, we find that the angle of elevation of the lamp from the boy is approximately 11.31 degrees (rounded to the nearest degree).
To calculate the length of the shadow of the body cast by the lamp, we can use similar triangles.
Let's assume the length of the shadow is "x" meters.
Using the similar triangles formed by the boy, the lamp pole, and the lamp, we can set up the following proportion:
(Length of Lamp Pole) / (Length of Shadow) = (Distance of Lamp from the Boy) / (Distance of Boy from the Foot of the Pole)
Using the given values:
4.2m / x = 6m / (6m + 1.2m)
Simplifying:
4.2m / x = 6m / 7.2m
Cross-multiplying:
4.2m * 7.2m = 6m * x
x = (4.2m * 7.2m) / 6m
x = 5.04m
Therefore, the length of the shadow of the body cast by the lamp is 5.04 meters.
To calculate the angle of elevation of the lamp from the boy, we can use trigonometry.
Let's assume the angle of elevation is "θ" degrees.
Using the opposite side (height of the lamp pole) and adjacent side (distance of the boy from the foot of the pole), we can set up the tangent function:
tan(θ) = (Height of Lamp Pole) / (Distance of Boy from the Foot of the Pole)
tan(θ) = 4.2m / 6m
Simplifying:
tan(θ) = 0.7
To find the angle, we can take the inverse tangent (arctan) of both sides:
θ = arctan(0.7)
Using a calculator, we find that θ is approximately 35.5 degrees.
Therefore, the angle of elevation of the lamp from the boy is approximately 35.5 degrees.