Question

25 points! Write and simplify a polynomial expression for the volume of each figure.

DISCLAIMER: IF THE ANSWER IS WRONG IT WILL BE REPORTED

Answer 1

`\large\boxed{1.\ V=3b^4-15b^3+3b^2-15b}\\\boxed{2.\ V=4\pi n^3-16\pi n^2+16\pi n}\\\boxed{3.\ V=2k^3-k^2-6k}`

Explanation:

The picture #1:

It's a rectangular prism. The formula of a volume of a rectangular prism:

`V=lwh`

l - length

w - width

h - height

We have

`l=b+1,\ w=3b,\ h=b-5`

Substitute:

`V=(b+1)(3b)(b-5)`

Use the distributive property a(b + c) = ab + ac

and the FOIL: (a + b)(c + d) = ac + ad + bc + bd

`V=(3b^3+3b)(b-5)=3b^4-15b^3+3b^2-15b`

The picture #2:

It's a cone. The fomula of a volume of a cone:

`V=(1)/(3)\pi r^2h`

r - radius

h - height

We have:

`r=n-2,\ h=12n`

Substitute:

`V=(1)/(3)\pi(n-2)^2(12n)`

Use (a + b)² = a² + 2ab + b²

`V=(1)/(3)\pi(n^2-4n+4)(12n)=(4\pi n)(n^2-4n+4)`

Use the distributive property:

`V=4\pi n^3-16\pi n^2+16\pi n`

The picture #3:

It's a pyramid with a rectangle in the base. The formula of a volume of a rectangular pyramid:

`V=(1)/(3)abh`

a, b - edge of a base

h - height

We have

`a=k-2, b=2k+3, h=3k`

Substitute:

`V=(1)/(3)(k-2)(2k+3)(3k)`

Use the distributive property and the FOIL.

`V=(1)/(3)(k-2)(6k^2+9k)=(1)/(3)(6k^3+9k^2-12k^2-18k)\\\\=(1)/(3)(6k^3-3k^2-18k)=2k^3-k^2-6k`