## 15.1 Speed of a Tsunami

A tsunami is generally referred to is a series of waves on the ocean caused by earthquakes or other events that cause sudden displacements of large volumes of water. In ideal situation, the velocity $$v$$ of a wave at where the water depth is $$d$$ meters is approximately $v=\sqrt{9.8d}.$ The wave will slow down when closer to the coast but will be higher.

Suppose a tsunami was caused by earthquake somewhere 10000 meters away the coast of California. The depth of the water where the tsunami was generated is 5000 meter.

• What’s the initial speed of the tsunami?
• What’s the speed of the tsunami at where the water depth is 2000.
• Suppose the speed wouldn’t decrease, how long it takes the tsunami reach the coast?

## 15.2 The Domain of a Radical Function

A radical function $$f$$ is defined by an equation $$f(x)=\sqrt[n]{r(x)}$$, where $$r(x)$$ is an algebraic expression. For example $$f(x)=\sqrt{x+1}$$. When $$n$$ is odd number, $$r(x)$$ can be any real number. When $$n$$ is even, $$r(x)$$ has to be nonnegative, that is $$r(x)\geq 0$$ so that $$f(x)$$ is a real number.

Example 15.1 Find the domain of the function $$f(x)=\sqrt{x+1}$$.

Solution.

Since the index is $$2$$ which is even, the function has real outputs only if the radicand $$x+1\geq 0$$. Solve the inequality, we get $$x\geq -1$$. In interval notation, the domain is $[-1,\infty).$

Example 15.2 For a pendulum clock, the period T of the pendulum is approximately modeled by the following function of the length L of the pendulum: $T=2\sqrt{L}$ where $$L$$ and $$T$$ are measured in meters and seconds respectively.

• If the length of the pendulum is 4 meters, what is the period?
• If the period of a pendulum clock is 1 second, how long should be the pendulum?
• Solution.
• Because the length of the pendulum is 4 meters, that is $$L=4$$. Then the period is $$T=2\cdot\sqrt{4}=4$$ seconds.
• Because the period of the pendulum is 1 second, that is $$T=1$$. Then the length $$L$$ is the solution of the equations $$2\sqrt{L}=1$$. \begin{aligned} 2\sqrt{L}&=1\\ \sqrt{L}&=\frac12\\ L&=\left(\frac12\right)^2\\ L&=\frac14. \end{aligned} So the length of the pendulum should be 1/4 in so that the period is 1 second.

## 15.3 Practice

Problem 15.1 Find the domain of each function. Write in interval notation.

1. $$f(x)=1-\frac{2x}{x+3}$$
2. $$f(x)=\frac{x-2}{x^2-4}$$
3. $$f(x)=\sqrt{1-x^2}$$
4. $$f(x)=-\sqrt{\frac{1}{x-5}}$$