Joseph is building a triangular shaped planter in his front lawn. The sides of the planter will be 6 feet, 7 feet and 11 feet. The planter will be filled with soil 2 inches deep. How many cubic feet of soil will he need? Round up to the nearest hundredth.
you can find the area of the triangle using
(a) Heron's formula
or
(b) A = 1/2 ab sinC (and c^2 = a^2+b^2-2ab cosC to find C)
And the volume is just A * 2/12 ft^3
Well, Joseph must really be digging this triangular planter! Let's calculate the volume of soil he needs.
To find the volume of a triangular planter, we can use the formula:
Volume = (Base * Height) / 2
First, let's determine the base and height of the triangular planter. The base would be the longest side, which is 11 feet, and the height can be found by drawing a perpendicular line from the top vertex down to the base, which creates a right angle. The length of this line will be the height.
Now, we can use some geometry magic! Let's use the Pythagorean theorem to find the height:
Height^2 = 11^2 - (6^2 + 7^2)
Height^2 = 121 - (36 + 49)
Height^2 = 121 - 85
Height^2 = 36
Height = √36
Height = 6 feet
Now that we have the base (11 feet) and the height (6 feet), we can calculate the volume:
Volume = (11 * 6) / 2
Volume = 66 / 2
Volume = 33 cubic feet
But wait, there's more! Joseph wants the planter to be filled with soil 2 inches deep. Since there are 12 inches in a foot, 2 inches is equal to 2/12 = 1/6 of a foot deep.
So, we can multiply the volume by the depth in feet:
Volume with soil = 33 * (1/6)
Volume with soil = 33/6
Volume with soil = 5.5 cubic feet
So Joseph will need approximately 5.5 cubic feet of soil to fill his triangular planter.
To calculate the volume of the planter, we need to find the area of the base and multiply it by the height.
First, let's find the area of the base, which is a triangle.
We can use Heron's formula to calculate the area of a triangle when we know the lengths of its sides.
Let's denote the sides as follows:
a = 6 feet
b = 7 feet
c = 11 feet
The semi-perimeter (s) is calculated by adding the lengths of all three sides and dividing by 2:
s = (a + b + c) / 2 = (6 + 7 + 11) / 2 = 12
Now, we can calculate the area (A) using Heron's formula:
A = sqrt(s * (s - a) * (s - b) * (s - c))
A = sqrt(12 * (12 - 6) * (12 - 7) * (12 - 11))
A = sqrt(12 * 6 * 5 * 1)
A = sqrt(360)
A ≈ 18.9737 (rounded to the nearest thousandth)
The height of the planter is given as the depth of the soil, which is 2 inches. To convert this to feet, we divide by 12:
height = 2 / 12 = 1/6 feet
Finally, we can calculate the volume (V) of the planter:
V = A * height
V = 18.9737 * (1/6)
V ≈ 3.1623 cubic feet (rounded to the nearest hundredth)
Therefore, Joseph will need approximately 3.16 cubic feet of soil to fill the planter to a 2-inch depth.
To calculate the volume of soil needed for the triangular planter, we need to find the area of the base of the planter and multiply it by the depth.
1. First, let's calculate the area of the triangular base using Heron's formula. The formula states that the area can be calculated as:
area = sqrt(s * (s - side1) * (s - side2) * (s - side3))
where s is the semiperimeter (half the perimeter) of the triangle, and side1, side2, and side3 are the lengths of the sides of the triangle.
In this case, the sides of the planter are 6 feet, 7 feet, and 11 feet. So the semiperimeter will be:
s = (6 + 7 + 11) / 2 = 24 / 2 = 12 feet
Plugging the values into the formula, we get:
area = sqrt(12 * (12 - 6) * (12 - 7) * (12 - 11))
= sqrt(12 * 6 * 5 * 1)
= sqrt(360)
= 18.97 square feet (rounded to the nearest hundredth)
2. Next, we need to convert the depth of the soil from inches to feet. Since there are 12 inches in a foot, the depth of 2 inches is equal to 2/12 = 1/6 feet.
3. Finally, we can calculate the volume of soil needed by multiplying the area of the base by the depth:
volume = area * depth
= 18.97 square feet * (1/6) feet
= 3.16 cubic feet (rounded to the nearest hundredth)
Therefore, Joseph will need approximately 3.16 cubic feet of soil to fill the planter.