Question

suppose that a wave forms in shallow water. then the depth ( d ) of the water (in meters) and the velocity ( v ) of the wave (in meters per second) are related by the equation ( v = sqrt{9.8d} ). if a wave formed in shallow water has a velocity of 5.3 meters per second, what is the water’s depth? carry your intermediate computations to at least four decimal places, and round your answer to the nearest tenth.

Explanation:

Step1: Start with the given equation

We have the equation \( v = \sqrt{9.8d} \), and we know that \( v = 5.3 \) m/s. We need to solve for \( d \). First, square both sides of the equation to eliminate the square root.

$$ v^2 = (\sqrt{9.8d})^2 $$

$$ v^2 = 9.8d $$

Step2: Solve for \( d \)

Now, we can solve for \( d \) by dividing both sides of the equation by \( 9.8 \).

$$ d=\frac{v^2}{9.8} $$

Substitute \( v = 5.3 \) into the equation:

$$ d=\frac{(5.3)^2}{9.8} $$

First, calculate \( (5.3)^2 \): \( 5.3\times5.3 = 28.09 \)
Then, divide by \( 9.8 \): \( d=\frac{28.09}{9.8}\approx2.8663 \)

Step3: Round to the nearest tenth

The value of \( d\approx2.8663 \). Rounding to the nearest tenth means looking at the hundredth place. The hundredth digit is 6, which is greater than or equal to 5, so we round up the tenth place.
\( 2.8663\approx2.9 \)

Answer:

\( 2.9 \)