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.

C. Area of triangle when [tex]\theta=30[/tex] is 129.9 square feet. Area of triangle when [tex]\theta=40[/tex] is 188.85 square feet. Increasing the angle [tex]\theta[/tex] increases the area.

Step-by-step explanation:

The equation that models the height of the triangle is:

[tex]y=15 Tan \theta[/tex]

Where,

[tex]y[/tex] is the height, and

[tex]\theta[/tex] is the angle

A.

When [tex]\theta=30[/tex] , the height is:

[tex]y=15Tan30\\y=8.66[/tex]

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

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

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

[tex]A=\frac{1}{2}bh[/tex]

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 [tex]\theta=30[/tex] , the area is:

[tex]A=\frac{1}{2}(30)(8.66)\\A=129.9[/tex] square feet

When Vance uses [tex]\theta=40[/tex] , the area is:

[tex]A=\frac{1}{2}(30)(12.59)\\A=188.85[/tex] square feet

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