Radical Equation Word Problems - Examples & Practice

The key to solving any word problem (whether it contains a radical or not) is to translate the problem from words into math.

That's the biggest step in word problems. Once you've translated the information into numbers, you solve the equation the same way as always.

Let's look at an example to see how this approach works when radicals are involved.

Example:

Hiroki is calculating the speed of a tsunami. The formula is $$\text{speed} = \sqrt{9.8d}$$ where $d$ is ocean depth in meters. In this case, the ocean was $1500$ meters deep. How fast was the wave (in $\text{m}/\text{s}^2$)?

So, in this problem (as with many problems involving radicals), the formula to use has already been given to us. We are interested in the speed of the wave, and we've been given the depth ($d$). So, let's start by plugging that value in: $$\text{speed} = \sqrt{9.8(1500)}$$ Let's solve this problem.

Solution: $$\begin{align} \text{speed} &= \sqrt{9.8(1500)} \\ &= \sqrt{14700} \\ &= \boxed{121\,\text{m}/\text{s}^2}\end{align}$$

Remember to always double check your work! Plug your solved value back into the original equation.

Example walk through: word problem involving radicals

The key to word problems is in the translation. Let's walk through the problem below and solve it.

Lori recently purchased a square painting with an area of $780\text{ cm}^2$. How much wood does Lori need to make the frame? Don't worry about the width and height of the wood, just find the length.

First, find out the essential information in the problem.

A square painting has area $780\,\text{cm}^2$, what is its side length?

What is the relationship between side length and area of a square? $$\text{Area} = \text{side length} \times \text{side length}$$

Denote side length as $x$ for simplicity, plug in the numbers we have: $$780\,\text{cm}^2 = x \times x$$ Solve for $x$: $$\begin{align} 780 &= x^2\\ x&=\sqrt{78}\\ x&\approx27.93 \end{align}$$

Lastly, note that the question is asking how much wood Lori needs to build the frame. A square has $4$ sides so we have to multiply $27.93 \, \text{cm}$ by $4$.

In total, Lori needs at least $27.93 \, \text{cm} \times 4=111.72 \, \text{cm}$ of wood.

Let's look at another problem!

(Video) Problem Solving with Radical Expressions

by Deanna Green

This video by Deanna Green works through word problems with radicals.

$\large{\text{Summary}}$

These problems usually focus around the Pythagorean Theorem. A helpful trick always is to draw a simple picture of what's going on. Then fill in lengths or angles as the problem states.

Let's work through one of the examples she looks at.

A kite is secured to a rope that is tied to the ground. A breeze blows the kite so that the rope is taught while the kite is directly above a flagpole that is $30\text{ ft}$ from where the rope is staked down. Find the altitude of the kite if the rope is $110\text{ ft}$ long.

The first step is to draw a diagram.

The Pythagorean Theorem states:

The length of the hypotenuse (diagonal line) is equal to the square root of the square of the sides added.

As an equation:

$$c=\sqrt{a^2+b^2}$$

where $c$ is the hypotenuse and $a$ and $b$ are the other sides.

Start by plugging in all the information we know.

$$110=\sqrt{30^2+x^2}$$

You cannot split up radicals with a sum underneath into two separate radicals. So, to solve this, we want to take the square of both sides.

$$\begin{align} \left(110\right)^2&=\left(\sqrt{30^2+x^2}\right)^2\\ 12100&=30^2+x^2\\ 12100-30^2&=x^2\\ 12100-900&=x^2\\ 11200&=x^2\\ \sqrt{11200}&=\sqrt{x^2}\\ 105.83&\approx x \end{align}$$

You may need to review the lesson on getting rid of the $x^2$ in with this lesson.