Mastery Check Solutions

Show What You Know

A baker is making a diamond-shaped cake for an engagement party. The baker will use a template to cut larger cakes into an equilateral triangle and an equilateral hexagon.

The formula for the area of an equilateral triangle is $A = \frac{\sqrt{3}}{4} s^{2}$ where s represents the side length.

The formula for the area of a hexagon is $A = \frac{1}{2} a n s$ where a is the apothem (which is the perpendicular distance from the side of the figure to its center), and n is the number of sides of the figure.

Find the total area of the cake.

$A = \frac{\sqrt{3}}{4} s^{2}$

$\begin{array}{rcl} A & = & \frac{\sqrt{3}}{4} {(4 \sqrt{2})}^{2} \\ = & \frac{\sqrt{3}}{4} (16 \sqrt{4}) \\ = & \frac{\sqrt{3}}{4} (16 (2)) \\ = & \frac{\sqrt{3}}{4} (32) \\ = & 8 \sqrt{3} \end{array}$

$A = \frac{1}{2} a n s$

$\begin{array}{rcl} A & = & \frac{1}{2} (2 \sqrt{6}) (6) (4 \sqrt{2}) \\ = & \frac{1}{2} (48 \sqrt{12}) \\ = & 24 \sqrt{12} \\ = & 24 \sqrt{2^{2} \cdot 3} \\ = & 24 (2) \sqrt{3} \\ = & 48 \sqrt{3} \end{array}$

$\begin{array}{rcl} T A & = & 48 \sqrt{3} + 8 \sqrt{3} \\ = & 56 \sqrt{3} {units}^{2} \end{array}$

Note

You need to find the area of each figure and then add the two areas together.

The area of the rectangular box for the cake is $200 + 12 \sqrt{6}$ square units.

If the length of the cake box is $6 \sqrt{6} + 2$ units, what is the width? Write your answer in simplified radical form.

$A = l w w = \frac{A}{l}$

$A = 200 + 12 \sqrt{6} l = 6 \sqrt{6} + 2$

$\begin{array}{rcl} w & = & \frac{200 + 12 \sqrt{6}}{6 \sqrt{6} + 2} \\ = & \frac{200 + 12 \sqrt{6}}{6 \sqrt{6} + 2} (\frac{6 \sqrt{6} - 2}{6 \sqrt{6} - 2}) \\ = & \frac{1200 \sqrt{6} - 400 + 72 \sqrt{36} - 24 \sqrt{6}}{36 (6) - 4} \\ = & \frac{1176 \sqrt{6} - 400 + 72 (6)}{216 - 4} \\ = & \frac{1176 \sqrt{6} + 32}{212} \\ = & \frac{4 (294 \sqrt{6} + 8)}{4 (53)} \\ = & \frac{4 (294 \sqrt{6} + 8)}{4 (53)} \end{array}$

$w = \frac{294 \sqrt{6} + 8}{53} units$

Note

Q: What must be simplified out of the denominator in order for it to be simplified?

A: The radicals

Q: When the denominator contains a binomial expression with at least one radical, how do you simplify the radical out?

A: Use the conjugate.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Rationalize the denominator of a radical expression.
Determine the conjugate of a given expression.
Use conjugates to simplify radicals with a binomial in the denominator.
Simplify radical expressions using more than one operation.