A triangular prism of cheese is measured and found to be 2.0 inches tall. The edges of its base are 9.0, 9.0, and 4.0 inches long. Several congruent prisms are to be arranged around a common 2.0- inch segment, as shown. How many prisms can be accommodated? What is their total volume? PLEASE INCLUDE WORK please SOMEONE HELPME BRO

14 prisms can be accomodated

Their total volume would be 504 in³

Step-by-step explanation:

First, we have to figure out the angle at the vertex.  For that, we need the Law of cosines. c²=a²+b²-2ab*cos(c)

4²=9²+9²-2(9)(9)cos(c)

16=81+81-162cos(c)

16-81-81=-162cos(c)

-142=-162cos(c)

-142/-162=cos(c)

.9=cos(c)

use the inverse

cos^-1(.9)=c

25.8=c

Now that we know c=25.8, we can figure out how many cheese wedges there would be in the circle.

36÷25.8=14

Now we figure out volume.

1/2(9)(4)(2)=36 in³

The total volume would be

14x36=504 in³

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.