the angle of depression of a ship from a top of a tower is 46deree. a. If a ship is 126m from the foot of the tower, calculate the height of the tower. b. If the height of the tower is 72m tall, calculate the distance between the (i) ship and the foot of the tower (ii) ship and to top of the tower
(a) h/126 = tan46°
(b)
(i) 72/x = tan46°
(ii) 72/z = sin46°
To solve these problems, we can use trigonometric ratios, specifically the tangent function.
a. To find the height of the tower, we know the angle of depression and the distance from the foot of the tower to the ship. The tangent of an angle of depression is given by the opposite side divided by the adjacent side.
Let's assign variables:
Let h be the height of the tower.
Let d be the distance from the foot of the tower to the ship.
Using the tangent function: tan(46 degrees) = h / d.
Rearranging the equation, we have h = tan(46 degrees) * d.
Given that d = 126m, we can substitute it into the equation:
h = tan(46 degrees) * 126 m.
Use a calculator to find the tangent of 46 degrees, which is approximately 1.0355 (rounded to four decimal places):
h = 1.0355 * 126 m.
h ≈ 130.46 m (rounded to two decimal places).
Therefore, the height of the tower is approximately 130.46 meters.
b. i. To find the distance between the ship and the foot of the tower, we use the same trigonometric ratio, but rearrange the formula to solve for d.
Using the tangent function again: tan(46 degrees) = h / d.
Rearranging the equation, we get d = h / tan(46 degrees).
Given that h = 72m, we can substitute it into the equation:
d = 72m / tan(46 degrees).
Approximating the tangent of 46 degrees to 1.0355 (as done before), we have:
d ≈ 72m / 1.0355.
d ≈ 69.54 m (rounded to two decimal places).
Therefore, the distance between the ship and the foot of the tower is approximately 69.54 meters.
b. ii. To find the distance between the ship and the top of the tower, we can use the Pythagorean theorem. The distance between the ship and the top of the tower is the hypotenuse of a right triangle with the height of the tower and the distance between the ship and the foot of the tower as its sides.
Using the Pythagorean theorem:
h² = d² + x²,
where h is the height of the tower, d is the distance between the ship and the foot of the tower, and x is the distance between the ship and the top of the tower.
Given h = 72m and d = 126m, we can substitute the values into the equation:
72² = 126² + x².
5184 = 15876 + x²,
Subtracting 15876 from both sides:
x² = 5184 - 15876,
x² = -10692.
We encounter a problem here. The result is negative, which means there is no real solution. This implies that the distance between the ship and the top of the tower is not possible based on the given information.