Question

problem 4: a shark starts diving the the top of the sea downward at an angle of θ = 33° from the horizontal surface. if the shark travels a diagonal distance of 450 m, find the depth the shark has reached and the horizontal component of the distance.

Explanation:

Step1: Identify the trigonometric relationships

We can model the shark's path as a right triangle, where the diagonal distance is the hypotenuse (\(c = 450\) m), the depth is the opposite side to the angle \(\theta = 33^\circ\), and the horizontal component is the adjacent side to the angle \(\theta\).

For the depth (\(d\), opposite side), we use the sine function: \(\sin\theta=\frac{d}{c}\)

For the horizontal component (\(h\), adjacent side), we use the cosine function: \(\cos\theta=\frac{h}{c}\)

Step2: Calculate the depth

Using \(\sin\theta=\frac{d}{c}\), we can solve for \(d\):

\(d = c\times\sin\theta\)

Substitute \(c = 450\) m and \(\theta = 33^\circ\):

\(d = 450\times\sin(33^\circ)\)

Using a calculator, \(\sin(33^\circ)\approx0.5446\)

\(d\approx450\times0.5446 = 245.07\) m

Step3: Calculate the horizontal component

Using \(\cos\theta=\frac{h}{c}\), we can solve for \(h\):

\(h = c\times\cos\theta\)

Substitute \(c = 450\) m and \(\theta = 33^\circ\):

\(h = 450\times\cos(33^\circ)\)

Using a calculator, \(\cos(33^\circ)\approx0.8387\)

\(h\approx450\times0.8387 = 377.415\) m

Answer:

The depth the shark has reached is approximately \(\boldsymbol{245.07}\) m and the horizontal component of the distance is approximately \(\boldsymbol{377.42}\) m (rounded to two decimal places).