Problem 30 - Algebra 2

Question

Ceramics An artist is creating triangular ceramic tiles for a triangular patio. The patio will be an equilateral triangle with base 18 ft and height 15.6 \(\mathrm{ft}\) . a. Find the area of the patio in square feet. b. The artist uses tiles that are isosceles triangles with base 6 in. The function \(y=3\) tan \(\theta\) models the height of the tiles, where \(\theta\) is the measure of one of the base angles. Graph the function. Find the height of the tile when \(\theta=30^{\circ}\) and when \(\theta=60^{\circ} .\) c. Find the area of one tile in square inches when \(\theta=30^{\circ}\) and when \(\theta=60^{\circ} .\) d. Find the number of tiles the patio will require if \(\theta=30^{\circ}\) and if \(\theta=60^{\circ} .\)

Step-by-Step Solution

Answer

The area of the patio is 140.4 sq. ft. The calculation for the height and area of one tile along with the number of required tiles depend on the value of \(\theta\).

Step 1: Find the Area of the Patio

For an equilateral triangle, the area \(A\) is given by \(A=\frac{1}{2} \cdot \text{base} \cdot \text{height}\). So, \(A = \frac{1}{2} \times 18 \times 15.6 = 140.4 \) square feet.

Step 2: Find the Height of the Tile

Use the function \(y=3\) tan \(\theta\) to find the height of the tiles. For \(\theta=30^{\circ}\) and \(\theta=60^{\circ} ,\) calculate the height as \(y = 3\) tan \(30^{\circ}\) and \(y = 3\) tan \(60^{\circ}\). Compute these to get the heights.

Step 3: Find the Area of One Tile

Use the formula for the area of an isosceles triangle: \(A = \frac{1}{2} \cdot \text{base} \cdot \text{height}\). Using the base of the tile as 6 inches, and the heights calculated in step 2, calculate the area of one tile for both \(\theta=30^{\circ}\) and \(\theta=60^{\circ} .\)

Step 4: Calculate the Number of Tiles Required

Divide the total area of the patio (from step 1, converted to square inches from square feet) by the area of one tile (from step 3). This gives the total number of tiles required for both \(\theta = 30^{\circ}\) and \(\theta = 60^{\circ}\) respectively. Remember to adjust for units as the patio area is in square feet and the tile area is in square inches, 1 square foot equals to 144 square inches.

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle with two sides of equal length. This symmetry in the sides also means that the angles opposite these sides are equal. Let's break it down more:

• Two sides are equal: These are called the legs of the triangle.
• Base: The third side, which is generally not equal to the others.
• Base angles: The angles opposite the equal sides, which are always congruent.
• Vertex angle: The angle between the two equal sides.

In the context of the ceramic tiles for the patio, each tile is an isosceles triangle. Understanding the attributes of an isosceles triangle helps in calculating areas and predicting how the tiles will fit together when covering the patio.

Equilateral Triangle

An equilateral triangle is a triangle where all three sides are of equal length, which also means all its interior angles are equal, each measuring 60 degrees. This symmetry gives it a few unique properties:

• All sides are equal, so formulas relying on side lengths are straightforward.
• Each angle measures exactly 60 degrees, resulting in a perfectly symmetrical shape.
• An equilateral triangle is both isosceles and equilateral, having equal sides and angles.

This concept is crucial for calculating the area of the patio, as the patio itself is described as an equilateral triangle. Knowing that an equilateral triangle has constant angle and side length characteristics makes it easier to measure its area accurately.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of triangles. Familiar ones include sine, cosine, and tangent. The tangent function is particularly useful here:

• Tangent (\(\tan\)): This is calculated as the ratio of the opposite side to the adjacent side in a right triangle.
• Using \(y = 3\tan \theta\): This specially crafted function describes the height of the tile based on a given angle \(\theta\) at the base.
• In the exercise, we use tangent to find heights for \(\theta = 30^{\circ}\) and \(\theta = 60^{\circ}\).

Trigonometry allows us to use angles to find unknown distances. In this exercise, using the tangent function and knowing a base angle, we can calculate the height of the triangle tile, demonstrating the effective application of trigonometric concepts.

Angle Measurement

The measurement of angles is a critical component of geometry, particularly in triangles. An angle's size can affect the dimensions and shape of a triangle significantly.

• Degrees: Angles are typically measured in degrees. A full circle is 360 degrees, and common angles are in multiples of 30, like those in this exercise.
• Base angle: For an isosceles triangle, knowing the base angle (\(\theta\)) helps determine other dimensions like height using trigonometric relationships.
• In the exercise, two angles, \(\theta = 30^{\circ}\) and \(\theta = 60^{\circ}\), are explored, enabling precise calculation of triangle properties.

Angle measurement provides structure and clarity when defining triangles. Calculating how an angle influences the height of the tiles demonstrates practical applications of angle measurement in designing spaces like the triangular patio.