A right prism has height 8 and triangular bases with sides of length 7, 8, and 9. What is the:
a) Volume of the prism?
b) Total surface area of the prism?
1)
Use formula:
V=BH
First, find the perimeter:
2s=7+8+9
s=12
then find the base
b=(12(12-7)(12-8)(12-9))^1/2
=12sqrt5
-->
V=12sqrt5*8
=96sqrt5
or about 214.663 units^3
2)
Use formula:
total surface area=base perimeter*height + 2*area of base
First, find the base perimeter:
p=7+8+9=24
then find the area of base:
s=(7+8+9)/2=12
a=(12*5*4*3)^1/2=12sqrt5
-->
total surface area=24*8+2*12sqrt5
or about 245.666
a) Well, to find the volume of the prism, we need to multiply the base area by the height. Since the base is a triangle, we can use the formula for the area of a triangle (half base times height) to find the base area.
So, the base area would be (1/2)(7)(8) = 28.
Now, we multiply the base area by the height of the prism.
Volume = Base Area x Height
Volume = 28 x 8 = 224.
So, the volume of the prism is 224 cubic units.
b) To find the total surface area of the prism, we need to sum up the areas of all the faces.
The prism has two triangular bases and three rectangular faces.
Let's start with the triangular bases.
The area of one triangle is (1/2)(7)(8) = 28. Since there are two bases, the total area of both bases is 2 x 28 = 56.
Now, let's move on to the rectangular faces.
Each rectangular face has a width of 7 and a height of 8. Since there are three rectangular faces, the total area of all three faces is 3 x (7 x 8) = 168.
Finally, let's add up the areas of all the faces.
Total Surface Area = (Area of Bases) + (Area of Rectangular Faces)
Total Surface Area = 56 + 168 = 224.
So, the total surface area of the prism is 224 square units.
To find the volume of a right prism, we need to multiply the area of the base by the height.
a) Volume of the prism:
The prism has a triangular base with sides of length 7, 8, and 9. To find the area of the base, we can use the Heron's formula, which states that for a triangle with side lengths a, b, and c, the area can be calculated as:
Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2.
For the given triangular base:
s = (7 + 8 + 9) / 2 = 24 / 2 = 12.
Using the formula, the area of the triangular base is:
Area = √[12(12-7)(12-8)(12-9)]
= √[12(5)(4)(3)]
= √[720]
≈ 26.87 (rounded to two decimal places).
Therefore, the volume of the right prism is:
Volume = Base area x Height
= 26.87 x 8
≈ 214.96 (rounded to two decimal places).
b) Total surface area of the prism:
The total surface area of the prism consists of the sum of the areas of the two triangular bases and the area of the four rectangular sides.
The area of the two triangular bases is twice the area of one triangular base, so the total area of the triangular bases is:
2 x Area of the triangular base
= 2 x 26.87
= 53.74.
The rectangular sides of the prism can be found by multiplying the perimeter of the base by the height. The base perimeter is given by the sum of all three sides of the triangular base: 7 + 8 + 9 = 24.
The total surface area of the rectangular sides is:
Perimeter of the base x Height
= 24 x 8
= 192.
Therefore, the total surface area of the prism is:
Total surface area = 2 x Area of the triangular base + Total surface area of the rectangular sides
= 53.74 + 192
≈ 245.74 (rounded to two decimal places).
use Heron's Formula to find the area of the base
Or, find ∠C using the law of cosines. If the sides are a,b,c opposite ∠s A,B,C, and c=9, then
9^2 = 7^2 + 8^2 - 2*7*8*cosC
Now the area of the base is 1/2 ab sinC
The faces are easy, since they are all rectangles.