12. Vance is designing a garden in the shape of an isosceles triangle. The base of the garden is 36 feet long. The function models the height of the triangular garden.
a. What is the height of the triangle when theta=45 degree?
b. What is the height of the triangle when theta=55 degree?
c. Vance is considering using either theta =45degree or theta=55 degree for his garden. Compare the areas of the two possible gardens. Explain how you found the areas.
To calculate the height of the triangular garden, we need to use the trigonometric function given in the question, which models the height as a function of the angle theta. However, the specific form of the function is not provided, so we cannot calculate the exact heights without it.
a. In order to find the height of the triangle when theta is 45 degrees, we need the specific function given in the question. Once we have that, we can plug in the value of theta (45 degrees) and evaluate the function to find the height.
b. Similarly, to find the height of the triangle when theta is 55 degrees, we need the specific function given in the question. Once we have the function, we can plug in the value of theta (55 degrees) and evaluate the function to find the height.
c. To compare the areas of the two possible gardens, we need to know the length of the base of the triangle for both cases. The question states that the base of the garden is 36 feet long, so the base length is the same for both 45-degree and 55-degree triangles.
To find the area of a triangle, we can use the formula:
Area = 0.5 * base * height
Once we know the height of the triangle for both cases (a and b), we can calculate the areas of the two possible gardens by plugging the values into the above formula. The garden with the larger area would be considered to be bigger.
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.
You don't state where Ø is, I will assume it is one of the two equal angles at the base.
a) tan 45 = height/18
height = 18tan45 = 18
b) repeat with tan 55 = height/18
c) since area = (1/2)(36)(height)
the larger the height, the larger the area.
you do the calculations.