Take the earth as a sphere with radius 6371 km. Jack is standing on the Cliffs of Moher at the point J which is 214 metres above sea level. He is looking out to sea at a point H on the horizon. Taking A as the centre of the earth, find |JH|, the distance from Jack to the horizon. Give your answer correct to the nearest km. The Cliffs of Moher, at point C, are at latitude 53° north of the equator. On the diagram, s1 represents the circle that is at latitude 53°. s2 represents the equator (which is at latitude 0°). A is the centre of the earth. s1 and s2 are on parallel planes. Find the length of the circle s1. Give your answer correct to the nearest km.

Photo AI

Take the earth as a sphere with radius 6371 km - Leaving Cert Mathematics - Question 6 - 2017

Question 6

Take the earth as a sphere with radius 6371 km. Jack is standing on the Cliffs of Moher at the point J which is 214 metres above sea level. He is looking out to sea ... show full transcript

Worked Solution & Example Answer:Take the earth as a sphere with radius 6371 km - Leaving Cert Mathematics - Question 6 - 2017

Step 1

|JH| = 6371 + 0.214

96%

114 rated

Answer

To find the distance from Jack to the horizon, we start with the formula for the distance to the horizon from a height: $|JH| = \sqrt{|A|^2 - |AH|^2}$ Where:

|A| is the radius of the Earth plus the height of Jack: $|A| = 6371 + 0.214$
|AH| is the radius of the Earth: $|AH| = 6371$

Step 2

|JH|^2 = |A|^2 - |AH|^2

99%

104 rated

Answer

Substituting in the values: $|JH|^2 = (6371 + 0.214)^2 - 6371^2$ Calculating this gives: $|JH|^2 = (6371.214)^2 - 6371^2$ $= 52.21$

Step 3

|JH| = \sqrt{(6371 + 0.214)^2 - 6371^2}

96%

101 rated

Answer

Taking the square root: $|JH| = \sqrt{52.21}$ Calculating this will yield: $|JH| \approx 7.22$ km, rounded to the nearest km gives: $|JH| \approx 7 km$ .

Step 4

Find the length of the circle s1

98%

120 rated

Answer

To find the length of the circle at latitude 53°, we use the radius: $r_{s1} = 6371 \times \cos(53°)$ Calculate the cosine: $r_{s1} = 6371 \times 0.6018 \approx 3834.1635$ km. Now, find the length: $L_{s1} = 2\pi r_{s1} = 2\pi (3834.1635) \approx 24091.226$ km, which rounds to: $L_{s1} \approx 24100$ km.

Join the Leaving Cert students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;