Question

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.

Answer 1

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

Answer 2

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.