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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.

a. What is the height of the triangle when theta = 30°?

b. What is the height of the triangle when theta = 40°?

c. Vance 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.

User Sheepshead
by
7.8k points

2 Answers

1 vote
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
User Zorgan
by
9.0k points
3 votes

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.


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 <strong>height</strong> is:</p><p>[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=(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=(1)/(2)(30)(8.66)\\A=129.9 square feet


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


A=(1)/(2)(30)(12.59)\\A=188.85 square feet


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

User Splact
by
7.3k points
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