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Math  /  Geometry

Question5) Albert and Belle are part of a scientific team studying thunderclouds. The team is about to launch a weather balloon into an active part of a cloud. Albert's rope is 7.8 m long and makes an angle of 36 degrees with the ground. Belle's rope is 5.9 m long. Draw a sketch and determine the angle that Belle makes with the ground. (6 marks)

Studdy Solution

STEP 1

1. The problem involves a right triangle formed by the ropes and the ground.
2. Albert's and Belle's ropes are hypotenuses of their respective right triangles.
3. The angle given for Albert's rope is with respect to the ground.
4. We need to find the angle Belle's rope makes with the ground.

STEP 2

1. Draw a diagram.
2. Use trigonometric relationships to find the height of the balloon.
3. Use the height to find the angle Belle's rope makes with the ground.

STEP 3

Draw a right triangle for Albert with the ground as the base, the rope as the hypotenuse, and the height of the balloon as the opposite side. Label the angle between the rope and the ground as 36∘ 36^\circ .
Draw another right triangle for Belle with the same height of the balloon, Belle's rope as the hypotenuse, and the angle with the ground as the unknown angle we need to find.

STEP 4

Use trigonometry to find the height of the balloon. For Albert's triangle, use the sine function:
sin⁡(36∘)=oppositehypotenuse=h7.8 \sin(36^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{7.8}
Solve for h h :
h=7.8⋅sin⁡(36∘) h = 7.8 \cdot \sin(36^\circ)

STEP 5

Calculate the height h h :
h=7.8⋅sin⁡(36∘)≈7.8⋅0.5878≈4.586 m h = 7.8 \cdot \sin(36^\circ) \approx 7.8 \cdot 0.5878 \approx 4.586 \, \text{m}

STEP 6

Use the height to find the angle Belle's rope makes with the ground. For Belle's triangle, use the sine function:
sin⁡(θ)=oppositehypotenuse=4.5865.9 \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4.586}{5.9}
Solve for θ \theta :
θ=sin⁡−1(4.5865.9) \theta = \sin^{-1}\left(\frac{4.586}{5.9}\right)

STEP 7

Calculate the angle θ \theta :
θ=sin⁡−1(4.5865.9)≈sin⁡−1(0.7773)≈51.1∘ \theta = \sin^{-1}\left(\frac{4.586}{5.9}\right) \approx \sin^{-1}(0.7773) \approx 51.1^\circ
The angle that Belle's rope makes with the ground is approximately:
51.1∘ \boxed{51.1^\circ}

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