Question

The volume of a square pyramid with a height equal to four less that the length of the base, is given by `V(x) = 1/3(x^3+8x^2+16x)` where `x` is the height in `cm`. If the length of the side of the base is `9 cm`, what is the volume of the pyramid?

Answer 1

`V = 135 cm^3`

Explanation:

Wow there is a lot of information given in these 2 sentences.!

The first thing is to break down the information which is given into smaller parts.

The phrase:
"the volume is given by " means "there is a formula and here it is"

So, if you want to calculate the volume of the pyramid, the formula you will have to use is:

`Vcolor(red)((x)) = 1/3(color(red)x^3+8color(red)x^2+16color(red)x)" "larr` the variables are all `color(red)x`

`V(color(red)x)` means that the value of the Volume depends on the value of `color(red)x`

You are told `color(red)x` is the height of the pyramid.

So the height is needed for the formula. `color(red)(x = " height ")`

The base of the pyramid is a square

You are also told that the height, `color(red)(x,)` is based on the length of the side of the square. If you know the length of the side of the square, you can work out the height.

The height is "4 less than the length of the side"

`rarr color(red)x = side -4`

(So, If the side is 12, then `color(red)(x=8)`, If the side is 25, then `color(red)x=21` etc)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now we are ready to find the answer!

The side of the square is `9` cm `rarr color(red)(x= 9-4 =5cm)" "larr` (4 less)

If `color(red)(x=5)` you can find the volume by using `color(red)(5) " for "color(red)x " in " V(color(red)x)`

`Vcolor(red)((x)) = 1/3(color(red)x^3+8color(red)x^2+16color(red)x)`

`Vcolor(red)((5)) = 1/3(color(red)5^3+8color(red)((5))^2+16color(red)((5)))" "larr` now calculate!

`V(5) = 1/3(125+200+80) = 1/3(405)`

`V = 135 cm^3`