From the side view, a gymnastics mat forms a right triangle with other angles measuring [tex]$60^{\circ}$[/tex] and [tex]$30^{\circ}$[/tex]. The gymnastics mat extends 5 feet across the floor. How high is the mat off the ground?
A. [tex]$\frac{5}{2} ft$[/tex]
B. [tex]$\frac{5 \sqrt{3}}{3} ft$[/tex]
C. [tex]$5 \sqrt{3}$[/tex]
D. 10
Answer (2)
Identify the right triangle formed by the gymnastics mat with angles 3 0 ∘ , 6 0 ∘ , and 9 0 ∘ .
Use the tangent function to relate the height h to the base (5 feet): tan ( 6 0 ∘ ) = 5 h .
Substitute tan ( 6 0 ∘ ) = 3 into the equation: 3 = 5 h .
Solve for h to find the height of the mat: 5 3 .
Explanation
Analyze the problem We are given a right triangle formed by a gymnastics mat. The angles are 9 0 ∘ , 6 0 ∘ , and 3 0 ∘ . The base of the triangle, which is the side extending across the floor, is 5 feet. We need to find the height of the mat, which is the side opposite the 6 0 ∘ angle.
Apply the tangent function Let h be the height of the mat. We can use the tangent function to relate the angle, the adjacent side, and the opposite side. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. tan ( 6 0 ∘ ) = 5 h We know that tan ( 6 0 ∘ ) = 3 . Therefore, 3 = 5 h
Solve for the height To solve for h , we multiply both sides of the equation by 5: h = 5 3 So, the height of the mat off the ground is 5 3 feet.
Examples
Understanding trigonometric relationships, like the one used here, is essential in various real-world applications. For instance, architects use these principles to calculate roof slopes and building heights, ensuring structural integrity and aesthetic appeal. Similarly, surveyors rely on trigonometry to measure distances and elevations, creating accurate maps and land layouts. Even in sports, athletes and coaches use trigonometry to optimize performance, such as determining the optimal angle for throwing a ball or launching a projectile.
The height of the gymnastics mat is calculated using the tangent function in trigonometry. By relating the height to the base using the angle of 60^{oldsymbol{igcirc}} , we find that the height is 5 3 feet. The correct answer is C. 5 3 feet.
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