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a rectangular box is 24 in. long, 12 in wide, and 18 in high. If each dimension is increased by x in .. write a polynomial function in standard form modeling the volume of the box

Respuesta :

Answer:

V = x³ + 54x² + 936x + 5,184

Step-by-step explanation:

If we add a value of 'x' to each side of the box, the new dimensions can be represented as

x + 24

x + 12 and

x + 18

To find the volume of the new box, multiply all of the dimensions together

V = (x + 24)(x + 12)(x + 18)    

Foil the first and second binomial....

V = (x² + 36x + 288)(x + 18)

 Now multiply the two polynomials together...

V = x²(x) + 36x(x) + 288x + x²(18) + 36x(18) + 288(18)

V = x³ + 36x² + 288x + 18x² + 648x + 5,184

which simplifies to

V = x³ + 54x² + 936x + 5,184     where x represents the increase in inches

The polynomial function in standard form modelling the volume of the box is x³ + 54x² + 936x + 5184

The previous dimension of the rectangular box is as follows;

length = 24 inches

width = 12 inches

height = 18 inches

Then the dimension were increased by x. Therefore,

length = x + 24 inches

width = x + 12 inches

height = x + 18 inches

Therefore,

Volume of a rectangular box = lwh

where

l = length

w = width

h = height

Therefore,

volume = (x + 24)(x+12)(x+18)

volume = (x² + 12x + 24x + 288)(x + 18)

volume = x³ + 18x² + 12x² + 216x + 24x² + 432x + 288x + 5184

volume = x³ + 54x² + 936x + 5184

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