Vance is designing a garden in the shape of an isosceles triangle. The base of the garden is 36 feet long. The function y=18 tan theta 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 thete=45 degree or theta=55 degree for his garden. Compare the areas of the two possible gardens. Explain how you found the areas.
i need the solution pls how to solve
To find the height of the triangle when theta = 45 degrees, we can substitute this value in the given function and solve for y.
a) Theta = 45 degrees:
y = 18 * tan(45)
Using the trigonometric identity tan(45) = 1, we have:
y = 18 * 1
y = 18 feet
So, the height of the triangle when theta = 45 degrees is 18 feet.
Similarly, to find the height of the triangle when theta = 55 degrees, we can substitute this value in the given function and solve for y.
b) Theta = 55 degrees:
y = 18 * tan(55)
Using a calculator or trigonometric table, we find that tan(55) ≈ 1.4281.
y ≈ 18 * 1.4281
y ≈ 25.7058 feet (approximately)
So, the height of the triangle when theta = 55 degrees is approximately 25.7058 feet.
To compare the areas of the two possible gardens (with theta = 45 degrees and theta = 55 degrees), we need to use the formula for the area of a triangle, which is 0.5 * base * height.
c) Area of garden with theta = 45 degrees:
Area = 0.5 * 36 * 18
Area = 324 square feet
Area of garden with theta = 55 degrees:
Area = 0.5 * 36 * 25.7058
Area ≈ 462.725 square feet (approximately)
Comparing the areas, we find that the garden with theta = 55 degrees has a larger area than the garden with theta = 45 degrees.
you saying that you cannot evaluate 18tanθ when given θ?
the area is 1/2 bh = 18y
since tanθ is increasing, the area will be greater when θ is greater.