Sure, let's work through the problem step-by-step:
We are given a right triangle where the angles are [tex]\(60^{\circ}\)[/tex], [tex]\(30^{\circ}\)[/tex], and [tex]\(90^{\circ}\)[/tex]. We also know that the side extending across the floor (the base) measures 5 feet.
In a 30-60-90 triangle, the sides have a specific ratio:
- The side opposite the [tex]\(30^{\circ}\)[/tex] angle is [tex]\(x\)[/tex].
- The side opposite the [tex]\(60^{\circ}\)[/tex] angle is [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2x\)[/tex].
In this problem, the given base of 5 feet is the side opposite the [tex]\(60^{\circ}\)[/tex] angle, which corresponds to [tex]\(x\sqrt{3}\)[/tex].
First, we need to determine the value of [tex]\(x\)[/tex]:
[tex]\[ x\sqrt{3} = 5 \implies x = \frac{5}{\sqrt{3}} \][/tex]
Next, we rationalize the denominator of [tex]\(x\)[/tex]:
[tex]\[ x = \frac{5\sqrt{3}}{3} \][/tex]
The height of the mat off the ground is the side opposite the [tex]\(30^{\circ}\)[/tex] angle, which is [tex]\(x\)[/tex]. So, the height can be given as:
[tex]\[ \text{Height} = \frac{5\sqrt{3}}{3} \][/tex]
Now, comparing the height with the given choices:
- [tex]\(\frac{5}{2} \text{ ft}\)[/tex]
- [tex]\(\frac{5\sqrt{3}}{3} \text{ ft}\)[/tex]
- [tex]\(5\sqrt{3} \text{ ft}\)[/tex]
- 10
The corresponding height option is [tex]\(\frac{5\sqrt{3}}{3} \text{ (ft)}\)[/tex].
So the mat is [tex]\(\frac{5\sqrt{3}}{3}\)[/tex] feet off the ground.
