A right prism has volume 3645 and bases that are regular hexagons with sides of length 9. Find the height of the prism.
[Formula for finding the volume of a prism]
Volume = Area of base * Height
Using algebra...
Height = Volume / Area of base
This is what we will use to find the height of the prism.
Given that the volume of the prism is 3645, all we need to find is the area of the base. The hexagonal base can be split into six congruent equilateral triangles. The side length of the base is 9; so, all we need to do is to this:
Area of base = 6 * ((9^2)*sqrt(3)) / 4 = 243 * sqrt(3) / 2
Here, I am using the formula of an equilateral triangle’s area: A = (s^2)*sqrt(3) / 4. In this problem, the side length is 9.
Now we have everything we need to find the height of the prism.
Height = Volume / Area of base = 3645 / (243 * sqrt(3) / 2) = 10 * sqrt(3)
Answer: The height of the prism is 10*sqrt(3) units.
Why did the clown go to school? To brush up on his triangle skills! Now, let's solve this problem together.
The volume of a prism is given by the formula V = Bh, where B is the area of the base and h is the height.
In this case, we know that the base is a regular hexagon with sides of length 9. To find the area of a regular hexagon, you can use the formula A = (3√3/2) * s^2, where s is the length of one side.
Plugging in the values, we have B = (3√3/2) * 9^2.
Now, let's plug in the given volume, V = 3645, and the base area, B, into the formula V = Bh. We get:
3645 = (3√3/2) * 9^2 * h.
To find the height, let's isolate h:
h = 3645 / [(3√3/2) * 9^2].
Calculating this out, the height of the prism is approximately h = 15.691.
So, the height of the prism is approximately 15.691 units.
To find the height of the prism, we need to use the formula for the volume of a right prism, which is given by:
Volume = Base Area * Height
We are given that the volume is 3645. Let's first find the base area of the prism.
The base of the prism is a regular hexagon with sides of length 9. To find the area of a regular hexagon, we can use the formula:
Area = (3 * √3 * s²) / 2
where 's' is the length of each side of the hexagon.
In this case, the length of each side of the hexagon is 9, so we can substitute this value into the formula to find the base area:
Area = (3 * √3 * 9²) / 2
= (3 * √3 * 81) / 2
= (243 * √3) / 2
= (121.5 * √3)
Now, let's substitute the values into the volume formula:
3645 = (121.5 * √3) * Height
To find the height, divide both sides of the equation by (121.5 * √3):
Height = 3645 / (121.5 * √3)
Now, let's simplify the expression:
Height = 3645 / (121.5 * √3)
= 30 / √3
To rationalize the denominator, we multiply both the numerator and denominator by √3:
Height = (30 / √3) * (√3 / √3)
= 30√3 / 3
= 10√3
Therefore, the height of the prism is 10√3.
area of base = (3 sqrt3) /2 * 9^2 = 210.4
so 210.4 h = 3645
h = 3645/210.4
The tricky part is to find the area of the hexagon
It is made of 6 equilateral triangles with sides 9 and angles of 60°
The area of one of them is (1/2)(9)(9)sin60
= 81/2(√3/2) = (81/4)√3
but we have 6 of them, so the area of the base = 486√3/4
then
486√3/4 h = 3645
243√3/2 h = 3645
h = 2(3645)/(243√3) = ....