The top of a particular lighthouse is in the shape of a hemisphere on top of a cylinder - Leaving Cert Mathematics - Question 8 - 2022

The volume of the cylinder is given by:

$V = \pi r^2 h$

Substituting the known values,( r = 3 m ):

$36\pi = \pi (3)^2 h$

This simplifies to:

$36 = 9h$

Solving for (h):

$h = \frac{36}{9} = 4 m$

Thus, the height of the cylinder (h) is (4 m).

To find the angle (A), we use the tangent function:

$\tan A = \frac{47}{7.5}$

Thus:

$A = \tan^{-1}\left(\frac{47}{7.5}\right) \approx 81°$

So, the angle at the base of the cone is approximately (81°).

Using the right triangle formed by the cone:

Let the height of the cone after the top part is removed be (k). We know:

$k = 47 - \left(\frac{7.5 - 3}{7.5} \cdot 47\right)$

Calculating the value:

$k = 47 - \left(\frac{4.5}{7.5} \cdot 47\right) \approx 47 - 28.2 = 18.8 m$

So, the height after the top part is removed is approximately (18.8 m).

The area (A) of a circle is calculated using:

$A = \pi r^2$

Using a radius of 50 km:

$A = \pi (50)^2 = 7853.98... \approx 7854 km²$

Thus, the area is approximately (7854 km²).

To convert nautical miles to kilometers, we have:

$50 km = 27 nautical miles$

Thus, for 1 nautical mile:

$1 nautical mile = \frac{50}{27} \approx 1.8519 km \approx 1.852 km$

So, there are approximately (1.852 km) in 1 nautical mile.