Applying Radical Equations

Learning Objective

• Use radical equations to solve real world problems.

Mathematical problems that include radical equationsAn equation that contains a variable within a radical term. arise in a number of professions, from engineering to nursing. Let’s take a look at some of the contexts within which radical equations arise and consider how to solve them.

Distance and size calculations using the Pythagorean TheoremThe formula used to relate the lengths of the sides in any right triangle, `a^2+b^2=c^2`. often produce radicalsThe math symbol `sqrt`, used to denote the process of taking a root of a quantity. because the formula contains three squared terms: `a^2`, `b^2`, and `c^2`. Consider the following problem:

A rectangular microchip has a length of `5` mm and a diagonal of `5sqrt2` mm. What is the area of the microchip?

Before creating an equation for this problem, it makes sense to sketch a quick diagram. We are told that the microchip is rectangular, and we are also given the values for one length and the diagonal. We can assign any variable to represent the length of the side that we do not know—we’ll choose `b`.

The dotted lines in the sketch above denote the edges of the microchip—it makes it easier to see the triangle that is created by the diagonal, length, and width.

Note that the problem asks for the area of the microchip. In order to find this, we will have to square the length (which we know to be `5` mm) and square the width (currently represented by the variable `b`).

Let’s use the Pythagorean Theorem to find the value for `b`.

We find that `b` has a value of `5`. That means that the area of the microchip is `5text( mm)*5text( mm)`, or `25" mm"^2`. 

Have you ever wondered how far you can see on a clear day? When we stand on the ground, environmental and man-made objects often block the view all the way to the horizon. But the higher we get—for instance, looking out the upper window of a tall building, or sitting at the top of a Ferris wheel—the further away we can see.

One estimate for how far we can see on a clear day is given by the formula `v=1.225sqrta`, where `v =` visibility (in miles) and `a = ` altitude (in feet).

A woman on a hang glider can see `49` miles to the horizon. Using the visibility formula, how far above the ground is she?

To solve this problem, let’s substitute our known values into the visibility formula. We will be left with a one-variable radical equation.

The answer is `1600` feet! She is pretty high up in the air, but this makes sense, because she can see very far. To check our answer, we should substitute `1600` back into our original equation and see if we get a true statement.

`v=1.225sqrta`

`49=1.225sqrt1600`

`49=1.225(40)`

`49=49`

A true statement results when the values `v = 49` and `a = 1600` are inserted into the equation, so we can be confident that we have solved this problem correctly.

One final real world application of radical equations comes from the world of medicine. Before determining the dosage of a drug to give to a patient, doctors will sometimes calculate the patient’s Body Surface Area (or BSA). One way to determine a person’s BSA is to use the following formula: `BSA=sqrt((wh)/3600)`, where `w = ` weight (in pounds), `h = ` height (in centimeters), and BSA is measured in square meters.

Gustav weighs `160` pounds and has a BSA of about `2sqrt2`  `m^2`. How tall (in centimeters) is he?

We find that Gustav is `180` cm tall (this is about `5` feet `11` inches). Notice how this problem had radicals on both sides of the equation. Squaring both sides eliminated both radicals and made this problem much easier to solve.

A number of real world problems can be modeled with radical equations. When faced with a problem where a variable appears underneath a radical, raise both sides to the same power to remove the radical, and then solve for the variable.