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 5.2 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. meters

Explanation:

Step1: Start with the given formula

We have the formula \( v = \sqrt{9.8d} \), and we know that \( v = 5.2 \) 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: Substitute the value of \( v \)

Substitute \( v = 5.2 \) into the equation:
\( (5.2)^2 = 9.8d \)
Calculate \( (5.2)^2 \): \( 5.2\times5.2 = 27.04 \)
So, \( 27.04 = 9.8d \)

Step3: Solve for \( d \)

Divide both sides of the equation by 9.8 to isolate \( d \):
\( d=\frac{27.04}{9.8} \)
Calculate the division: \( \frac{27.04}{9.8}\approx2.759184 \)

Step4: Round to the nearest tenth

The tenths place is the first decimal place. The number \( 2.759184 \) has a 7 in the tenths place and a 5 in the hundredths place. Since 5 is greater than or equal to 5, we round up the tenths place:
\( 2.759184\approx2.8 \)

Answer:

\( 2.8 \)