Use cylindrical coordinates. Evaluate 9x(x2 + y2) dV, where E is the solid in the first octant that lies beneath the paraboloid
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The image provided is the shape of a triangular prism.
The base of the prism (triangles) are both equilateral. This means that all sides measure 3 centimeters.
Find the area of one triangle based on the information given:
A = 1/2bh
**Notice that we cannot put in any number for the height of the triangular base because we are only given three side. As a result, we have to take a different approach to finding the area of the triangles. We can use another method known as Heron's Formula.**
You can use this formula to find the area of a triangle using the 3 side lengths.
Heron's Formula :
1) First, find the semi-perimeter of the triangle. We can find this by adding all the sides and dividing the sum by 2.
= 3 + 3 + 3 / 2
= 4.5
Use this formula to find the area:
A = √S(S - A)(S - B)(S - C)
**S, in this case, represents the semi-perimeter.**
Substitute the side lengths of the triangle and semi-perimeter.
A = √4.5(4.5 - 3)(4.5 - 3)(4.5 - 3)
A = √4.5(1.5)(1.5)(1.5)
A = √15.1875
Then, we simplify the number by turning the decimal into a fraction and further reducing it.
We would get:
A = √243 / √16
Simplify √16 into 4 and simplify the radical √243 into 9√3 / 4.
Now, multiply this number by two because we have another triangular base.
A = 9√3 / 4 × 2
A = 9√3 / 2 or 7.79422863406
2) Now that we have the surface areas of both triangles, we can find the surface areas of the rectangles.
This is easier to find because the triangles are equilateral and the height of the prism is simply 15.
A = lw
A = (15)(3)
A = 45
45 is the area of one rectangle. Since there are 3, multiply 45 by 3.
A = 45 × 3
A = 135
3) Add up all your values to find the surface area.
S.A. = 135 + 7.79422863406
S.A. = 142.794228634 cm²
Solution : 142.794228634 cm²
The base of the prism (triangles) are both equilateral. This means that all sides measure 3 centimeters.
Find the area of one triangle based on the information given:
A = 1/2bh
**Notice that we cannot put in any number for the height of the triangular base because we are only given three side. As a result, we have to take a different approach to finding the area of the triangles. We can use another method known as Heron's Formula.**
You can use this formula to find the area of a triangle using the 3 side lengths.
Heron's Formula :
1) First, find the semi-perimeter of the triangle. We can find this by adding all the sides and dividing the sum by 2.
= 3 + 3 + 3 / 2
= 4.5
Use this formula to find the area:
A = √S(S - A)(S - B)(S - C)
**S, in this case, represents the semi-perimeter.**
Substitute the side lengths of the triangle and semi-perimeter.
A = √4.5(4.5 - 3)(4.5 - 3)(4.5 - 3)
A = √4.5(1.5)(1.5)(1.5)
A = √15.1875
Then, we simplify the number by turning the decimal into a fraction and further reducing it.
We would get:
A = √243 / √16
Simplify √16 into 4 and simplify the radical √243 into 9√3 / 4.
Now, multiply this number by two because we have another triangular base.
A = 9√3 / 4 × 2
A = 9√3 / 2 or 7.79422863406
2) Now that we have the surface areas of both triangles, we can find the surface areas of the rectangles.
This is easier to find because the triangles are equilateral and the height of the prism is simply 15.
A = lw
A = (15)(3)
A = 45
45 is the area of one rectangle. Since there are 3, multiply 45 by 3.
A = 45 × 3
A = 135
3) Add up all your values to find the surface area.
S.A. = 135 + 7.79422863406
S.A. = 142.794228634 cm²
Solution : 142.794228634 cm²
