Question

1) The dimensions of an irregular solid are as shown in the figure below.

a) Write a polynomial function, in standard form, to model the volume of this solid.

b) If the volume of the solid is 208 cubic inches, what is the value of x?

Answer 1

Hello,

a) Volume of the figure = Volume of light blue rectangular parallelepiped - volume of dark blue rectangular parallelepiped

Volume of rectangular parallelepiped = lengh × high × height

Volume of light blue rectangular parallelepiped :

V = (x + 2) × (x + 5) × (2x + 1)

V = (x² + 5x + 2x + 10) × (2x + 1)

V = (x² + 7x + 10) × (2x + 1)

V = 2x³ + 15x² + 27x + 10

Volume of dark blue rectangular parallelepiped :

V = x × 3 × (x + 5)

V = 3x(x + 5)

V = 3x² + 15x

Polynomial function, in standard form, to model the volume of this solid :

V = 2x³ + 15x² + 27x + 10 - (3x² + 15x)

V = 2x³ + 15x² - 3x² + 27x - 15x + 10

V = 2x³ + 12x² + 12x + 10

b) We have to solve 2x³ + 12x² + 12x + 10 = 208

⇔ 2x³ + 12x² + 12x - 198 = 0

⇔ 2(x - 3)(x² + 9x + 33) = 0

⇔ x = 3 or x² + 9x + 33 = 0

a = 1 ; b = 9 ; c = 33

Δ = b² - 4ac = 9² - 4 × 1 × 33 = -51 < 0 ⇔ Δ < 0 ⇒ no solution

So if the volume of the solid is 208 cubic inches, the value of x is 3