Vance is designing a garden in the shape of an isosceles triangle. The base of the garden is 30 feet long. The function
y
=
15
tan
θ
y
=
15
tan
θ
models the height of the triangular garden.
What is the height of the triangle when
θ
=
30
°
θ
=
30
°
?
What is the height of the triangle when
θ
=
40
°
θ
=
40
°
?
Vance is considering using either
θ
=
30
°
θ
=
30
°
or
θ
=
40
°
θ
=
40
°
for his garden. Compare the areas of the two possible gardens. Explain how you found the areas.
To find the height of the triangle when θ = 30°, we use the formula y = 15tanθ and substitute 30° for θ:
y = 15tan30°
y = 15(0.577) (using the value of tan30° from a trigonometric table or calculator)
y ≈ 8.66
So the height of the triangle when θ = 30° is approximately 8.66 feet.
To find the height of the triangle when θ = 40°, we again use the formula y = 15tanθ and substitute 40° for θ:
y = 15tan40°
y = 15(0.839) (using the value of tan40° from a trigonometric table or calculator)
y ≈ 12.59
So the height of the triangle when θ = 40° is approximately 12.59 feet.
To compare the areas of the two possible gardens, we use the formula for the area of a triangle: A = (1/2)bh, where b is the base length (30 feet) and h is the height of the triangle.
For θ = 30°:
A = (1/2)(30)(8.66)
A ≈ 130.98
For θ = 40°:
A = (1/2)(30)(12.59)
A ≈ 188.85
So the garden with θ = 40° has a larger area than the garden with θ = 30°. This is because a larger angle leads to a taller triangle, which increases the area.