To find the height of the gymnastics mat, we can use the properties of a 30-60-90 triangle. In such a triangle, the sides are in a consistent ratio based on the angles. The ratios are as follows:
- The side opposite the 30° angle (the shorter leg) is [tex]\( x \)[/tex]
- The side opposite the 60° angle (the longer leg) is [tex]\( x \sqrt{3} \)[/tex]
- The hypotenuse (the side opposite the 90° angle) is [tex]\( 2x \)[/tex]
In this problem, the gym mat forms a right triangle with one of the non-right angles being [tex]\(60^{\circ}\)[/tex] and the other being [tex]\(30^{\circ}\)[/tex]. The given side, which is 5 feet long, lies along the floor and is opposite the 30° angle. Therefore, the side that lies on the floor is the shorter leg, [tex]\( x \)[/tex], of the 30-60-90 triangle.
Based on the properties of the 30-60-90 triangle, we know that:
- The length of the side opposite the 60° angle (the height of the mat) can be found using the ratio [tex]\( x \sqrt{3} \)[/tex].
Given that the side opposite the 30° angle (the shorter leg) is 5 feet, we can calculate the height of the mat as follows:
[tex]\[ \text{height} = x \sqrt{3} = 5 \sqrt{3} \][/tex]
Therefore, the height of the mat off the ground is [tex]\( 5 \sqrt{3} \)[/tex] feet.
The correct answer to the problem is [tex]\( 5 \sqrt{3} \)[/tex].
