Question

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

Answer 1

Answer:

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

Step-by-step 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:

$V=\dfrac{1}{3}\pi r^2h$

r - radius

h - height

We have:

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

Substitute:

$V=\dfrac{1}{3}\pi(n-2)^2(12n)$

Use (a + b)² = a² + 2ab + b²

$V=\dfrac{1}{3}\pi(n^2-4n+4)(12n)=(4\pi n)(n^2-4n+4)$

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:

$V=\dfrac{1}{3}abh$

a, b - edge of a base

h - height

We have

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

Substitute:

$V=\dfrac{1}{3}(k-2)(2k+3)(3k)$

Use the distributive property and the FOIL.

$V=\dfrac{1}{3}(k-2)(6k^2+9k)=\dfrac{1}{3}(6k^3+9k^2-12k^2-18k)\\\\=\dfrac{1}{3}(6k^3-3k^2-18k)=2k^3-k^2-6k$