Decreasing Cube: each dimension decreased by whole number

[SglMomNoBrains]

Decreasing Cube. Each of the three dimensions of a cube with sides of length S centimeters is decreased by a whole number of centimeters. The new volume in cubic centimeter is given my

V(s) = S^3-13s^2+54s-72

a) Find V(10)
b) If the new width is S-6 centimeters, then what are the new length and height
c) Find the volume when S=10 by multiplying the length, width, and height.

s^3-13s^2+54s-72 = 10^3-13(10)^2 + 54(10)-72
= 1000-13(10)^2 + 540-72

This is as far as I can go and I'm not even sure if I'm doing this right

 

[soroban]

Re: Decreasing Cube

Hello, SglMomNoBrains!

A strangely-worded problem . . . They could have made the instructions clearer.

Decreasing Cube.
Each of the three dimensions of a cube with sides of length \(\displaystyle s\) centimeters
is decreased by a different whole number of centimeters.
The new volume in cubic centimeter is given by: .\(\displaystyle V(s) \:= \:s^3-13s^2+54s-72\)

\(\displaystyle \text{a) Find }V(10).\)

Just finish what you started . . .

\(\displaystyle V(10) \:=\:10^3 - 13(10^2) + 54(10) - 72 \:=\:1000 - 1300 + 540 - 72\:=\:168\text{ cm}^3\)

b) If the new width is \(\displaystyle s-6\) centimeters, what are the new length and height?

\(\displaystyle \text{We know that Volume is: }\:V \:=\:L \times W \times H\)


\(\displaystyle \text{If }W \,=\,s-6\text{, we can use long division}\)

. \(\displaystyle \text{and we get: }\:V(s) \:=\s^3 - 13s^2 + 54s - 72 \:=\s-6)(s^2-7s+12)\)

\(\displaystyle \text{This factors further: }\:V(s) \;=\;\underbrace{(s-6)}_{\text{width}}\underbrace{(s-3)}_{\text{length}}\underbrace{(s-4)}_{\text{height}}\)


\(\displaystyle \text{Therefore, the new length and height are: }\s-3)\text{ and }(s-4)\)

c) Find the volume when \(\displaystyle s=10\) by multiplying the length, width, and height.

We already know this volume, but they want us to recalculate it with our new knowledge.


\(\displaystyle \text{If }s = 10\text{, we have: }\;\begin{array}{cccccccc}L &=& s - 3 &=& 10-3 &=& 7 \\ W &=& s-6 &=& 10-6 &=& 4 \\ H &=& s - 4 &=& 10-4 &=& 6\end{array}\)


\(\displaystyle \text{Therefore: }\;V \;=\;7 \times 4 \times 6 \;=\;168\text{ cm}^3\)

 

[SglMomNoBrains]

Thank you soooooooooooo much, I'm just barely passing this class and don't know what else to do. I have to try and find examples for each question just so I can take notes and then work out the question but for this problem I could not find any notes anywhere on how to work this problem. Math just isn't the same since I'd been in high school 20+ years ago.