Question

Leslie is making a cardboard box that has a volume of 2x3 + 3x2 − 11x − 6 cubic feet. the base has an area of x2 + x − 6 square feet. what will the height of the box be? (hint: volume = base area · height)

Answer 1

Height of the box = 2x+1

Explanation:

Leslie is making a cardboard box that has a

volume of cubic feet.
`2x^3 + 3x^2 − 11x − 6`

the base has an area of
`x^2 + x − 6` square feet.

Volume of a box = Base area * height of the box

we are given with volume and base area

Replace it in the formula. LEts factor the given expression

`2x^3 + 3x^2 - 11x - 6`

`\left(x-2\right)\left(2x+1\right)\left(x+3\right)`

Now we factor
`x^2 +x -6 = (x+3))x-2)`

Volume of a box = Base area * height of the box

`\left(x-2\right)\left(2x+1\right)\left(x+3\right)= (x+3)(x-2) * height`

Divide both sides by (x+3)(x-2)

Height of the box = 2x+1

Answer 2

we know that

The volume of the box is equal to

V=area of the base*height

solve for the height

height=Volume/(area of the base)

`V=2x^(3)+3x^(2)- 11x- 6\ ft^(3) \\ Area\ of\ the\ base=x^(2)+x-6\ ft^(2)`

using a graph tool-----> find the roots

see the attached figures

so

`2x^(3)+3x^(2)- 11x- 6=2*(x+3)*(x+0.5)*(x-2)`

`x^(2)+x-6=(x+3)*(x-2)`

substitute

height=Volume/(area of the base)

`height= (2*(x+3)*(x+0.5)*(x-2))/((x+3)*(x-2)) = 2*(x+0.5)`

`height=2x+1`

therefore

the answer is

`height=2x+1`