Question

algebra 2 2b2403 - 6 fy - 1
benchmark test 2
due feb 5 - 1:45 pm late
lucy cuts 4 squares with side length x in. from the corners of a 12 in. by 18 in. cardboard rectangle. she folds the remaining cardboard to make a tray that is x in. high. write and simplify a polynomial function for the volume v of the tray in terms of x.
a. $4x^3 - 30x^2 + 54x$
b. $4x^3 - 60x^2 + 216x$
c. $4x^2 - 30x + 54$
d. $4x^3 - 30x^2 + 15x + 54$
question 14 of 30

Explanation:

Step1: Define tray dimensions

When squares of side $x$ are cut from each corner, the length of the tray becomes $18-2x$, the width becomes $12-2x$, and the height is $x$.

Step2: Apply volume formula

Volume $V = \text{length} \times \text{width} \times \text{height}$, so:
$$V = x(18-2x)(12-2x)$$

Step3: Multiply binomials first

Multiply $(18-2x)(12-2x)$:
$$(18)(12) + 18(-2x) -2x(12) + (-2x)(-2x) = 216 - 36x -24x +4x^2 = 4x^2 -60x +216$$

Step4: Multiply by $x$

$$V = x(4x^2 -60x +216) = 4x^3 -60x^2 +216x$$

Answer:

B. $4x^3 - 60x^2 + 216x$