To find the values of the height of the triangle when theta is given, we need to understand the relationship between the given values and the function.
In this case, we are given that the base of the garden, which is also the length of the two equal sides of the isosceles triangle, is 36 feet.
The function that models the height of the triangular garden is not given, so we cannot directly calculate the heights. However, it is common for isosceles triangles to have a special property, where the height can be calculated using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In an isosceles triangle, the height is one of the legs of the right triangle formed by dropping a perpendicular line from the apex to the base.
To find the height of the triangle when theta is given, we can use the following steps:
Step 1: Calculate the length of the other two sides of the triangle (the equal sides) using the given base length of 36 feet. Since the triangle is isosceles, the other two sides will also have a length of 36 feet.
Step 2: Calculate the length of the height using the Pythagorean theorem. In this case, the height is the unknown side, the length of the other two sides is 36 feet, and the base is the hypotenuse.
Step 3: Substitute the given value of theta into the function to find the height.
a. When theta = 45 degrees:
- Calculate the length of the height using the Pythagorean theorem.
- Substitute 45 degrees into the function to find the height.
b. When theta = 55 degrees:
- Calculate the length of the height using the Pythagorean theorem.
- Substitute 55 degrees into the function to find the height.
To compare the areas of the two possible gardens (with theta = 45 degrees and theta = 55 degrees), use the formula for the area of a triangle: A = 0.5 * base * height. Substitute the respective values of base and height for each case, and calculate their areas. Then compare the two areas to see which one is larger.
This is how you can find the heights of the triangle for each given theta and compare the areas of the two possible gardens.