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The cannon on a battleship can fire a shell a maximum distance of 31.6 km.The ocean is not flat, because the earth is curved. Assume that the radius of the Earth is 6.37 ✕ 103 km. How many meters lower will its surface be 31.6 km from the ship along a horizontal line parallel to the surface at the ship?



Answer :

Answer:

78 m

Explanation:

The cannonball is fired horizontally, at a tangent to the Earth's surface. Therefore, this trajectory forms a right angle (90 degree angle) with the radius line from the Earth's center to the ship. Treating the Earth as a perfect sphere, the radius is the same at all points along the Earth's surface. Therefore, we can use Pythagorean theorem to find the vertical distance between the cannonball and the Earth.

[tex]\Large \text {$ a^2+b^2=c^2 $}\\\\\Large \text {$ (6370-x)^2+(31.6)^2=(6370)^2 $}\\\\\Large \text {$ (6370-x)^2=(6370)^2-(31.6)^2 $}\\\\\Large \text {$ 6370-x=\sqrt{(6370)^2-(31.6)^2} $}\\\\\Large \text {$ 6370-x\approx 6369.922 $}\\\\\Large \text {$ x\approx 0.078 $}[/tex]

The cannonball is about 0.078 km above the Earth's surface, or 78 meters.

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