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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 theta models the height of the triangular garden.

AWhat is the height of the triangle when theta = 30°?

BWhat is the height of the triangle when theta = 40°?

CVance is considering using either theta = 30° or theta = 40° for his garden. Compare the areas of the two possible gardens. Explain how you found the areas.

Verified Answer :

a.
y = 15 tan(30°)
y = 15/√3 ≈ 8.66 ft

b.
y = 15 tan(40°) ≈ 12.59 ft

c.
A = 1/2 base x height ≈ 129.9 ft² and 188.8 ft² for 30° and 40°
hope it helps

TaeKwonDoIsDead

Answer:

A. 8.66 feet

B. 12.59 feet

C. Area of triangle when \(\theta=30\)  is 129.9 square feet. Area of triangle when \(\theta=40\) is 188.85 square feet. Increasing the angle  \(\theta\)  increases the area.


Step-by-step explanation:

The equation that models the height of the triangle is:

\(y=15 Tan \theta\)

Where,

  • \(y\) is the height, and
  • \(\theta\)  is the angle

A.

When \(\theta=30\) , the height is:

\(y=15Tan30\\y=8.66\)


B. When \(\theta=40[/tex\ , the height is:

[tex]y=15Tan40\\y=12.59\)


C. To find the area of the isosceles triangular shaped garden, we use the formula for the area of the triangle:

\(A=\frac{1}{2}bh\)

Where,

  • A is the area
  • b is the base, which is given as 30 feet, and
  • h is the height [8.66 feet when the angle is 30 & 12.59 when angle is 40]

When Vance uses \(\theta=30\) , the area is:

\(A=\frac{1}{2}(30)(8.66)\\A=129.9\)  square feet


When Vance uses \(\theta=40\) , the area is:

\(A=\frac{1}{2}(30)(12.59)\\A=188.85\)  square feet


So we see that when the angle is more, the area is also more.

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