Problem 56 - Algebra 1

Question

A tsunami is a destructive, fast-moving ocean wave that is caused by an undersea earthquake, landslide, or volcano. The Pacific Tsunami Warning Center is responsible for monitoring earthquakes that could potentially cause tsunamis in the Pacific Ocean. Through measuring the water level and calculating the speed of a tsunami, scientists can predict arrival times of tsunamis. The speed \(s\) (in meters per second) at which a tsunami moves is determined by the depth \(d\) (in meters) of the ocean. \(s=\sqrt{g d},\) where \(g\) is 9.8 meters per second per second. Find the speed of a tsunami in a region of the ocean that is 4000 meters deep. Write the result in simplified form.

Step-by-Step Solution

Answer

The speed of a tsunami in a region of the ocean that is 4000 meters deep is approximately 199 meters per second.

Step 1: Identify given variables

First note the variable values we're given: \(d = 4000\) meters (the depth of the ocean) and \(g = 9.8\) meters per second per second (the gravitational acceleration). What we're trying to solve for is \(s\) (the speed of the tsunami).

Step 2: Substitute given variables into formula

Substitute the values of \(g\) and \(d\) into the formula \(s = \sqrt{gd}\): \(s = \sqrt{9.8 \times 4000}\).

Step 3: Calculate and simplify

Doing the calculation and taking the square root gives \(s = 198.997489\) meters per second. Even though we're asked to write the result in simplified form, it's usually best to round to a reasonable number of significant figures based on the precision of the given data. Given that our inputs were to the nearest tenth and whole number, rounding our answer to the nearest whole number gives an adequately precise answer of \(s = 199\) meters per second.

Key Concepts

Square Root Function

The square root function is a key mathematical tool. It helps us determine the original value from a squared one. In other words, if you have a number, finding its square root means you're looking for a value that, when multiplied by itself, gives you the original number.
For instance, using the square root function in the tsunami speed equation, which is written as:

• \( s = \sqrt{gd} \), where \( g \) is gravitational acceleration and \( d \) is ocean depth.

When we say "solve \( \sqrt{9.8 \times 4000} \)", we mean to find a value \( s \) that, squared, equals \( 9.8 \times 4000 \).
The square root function appears as \( \sqrt{ } \), and it is widely used in physics, engineering, and many other fields due to its ability to reverse the process of squaring a number.

Gravitational Acceleration

Gravitational acceleration is a fundamental concept in physics. It describes the rate at which an object accelerates due to the force of gravity. On Earth, this acceleration is approximately \(9.8\) meters per second squared \(m/s^2\).
This means that every second, the speed of a falling object increases by 9.8 meters per second, assuming no other forces act on it.
In the context of the tsunami speed formula, gravitational acceleration \( g \) serves as a constant. It relates to the force driving the tsunami wave as it travels through the ocean.
This constant is crucial in calculating how quickly the tsunami wave moves given any depth of water it travels through.

• Gravitational acceleration is the same worldwide, but minor variations may occur due to altitude and Earth's shape.
• It plays a critical role in various scientific fields, including meteorology, oceanography, and space exploration.

Ocean Depth

Ocean depth is a pivotal factor in calculating tsunami speed. The reason is that the depth directly affects how fast the wave can move across the water. The deeper the ocean, the faster a tsunami wave can travel.
Using the formula \( s = \sqrt{gd} \), the depth \( d \) is one of the variables that, when multiplied by gravitational acceleration \( g \), under the square root, determines the speed \( s \).
A deeper ocean provides more room for the wave's energy to disperse, allowing it to maintain higher speeds over long distances.

• The given depth in the exercise is 4000 meters - a typical depth for deep ocean regions that tsunamis might travel.
• Understanding depth variations can help predict the impact of future tsunamis with more accuracy.

Ocean depth is also important in marine biology and geology, as it affects environmental conditions such as pressure and temperature.