Question

lesson 2.8 checkpoint
complete the previous problems, check your solutions, then complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goals.
solve the equation.

1. $k^4 - 100k^2 = 0$
2. $4y^3 - 7y^2 + 28 = 16y$
3. modeling with mathematics

a gift bag shaped like a rectangular prism has a volume of 1152 cubic inches. the dimensions of the gift bag in terms of its width are shown. the height is greater than the width. what are the dimensions of the bag?
(18 − w) in.
(2w + 4) in.
w in.

Explanation:

Problem 24: Step1: Factor out common term

$k^4 - 100k^2 = k^2(k^2 - 100)$

Problem 24: Step2: Factor difference of squares

$k^2(k^2 - 100) = k^2(k - 10)(k + 10)$

Problem 24: Step3: Set factors to 0

$k^2=0, \ k-10=0, \ k+10=0$

Problem 28: Step1: Rearrange all terms to left

$4y^3 - 7y^2 - 16y + 28 = 0$

Problem 28: Step2: Factor by grouping

$y^2(4y - 7) - 4(4y - 7) = 0 \implies (y^2 - 4)(4y - 7) = 0$

Problem 28: Step3: Factor difference of squares

$(y - 2)(y + 2)(4y - 7) = 0$

Problem 28: Step4: Set factors to 0

$y-2=0, \ y+2=0, \ 4y-7=0$

Problem 36: Step1: Use rectangular prism volume formula

Volume $V = l \times w \times h$, so:
$$w(2w+4)(18-w) = 1152$$

Problem 36: Step2: Simplify left-hand side

First expand $(2w+4)(18-w) = -2w^2 + 32w + 72$, then:
$$w(-2w^2 + 32w + 72) = -2w^3 + 32w^2 + 72w = 1152$$

Problem 36: Step3: Rearrange and simplify equation

Divide by $-2$: $w^3 - 16w^2 - 36w + 576 = 0$
Factor by grouping: $w^2(w-16) - 36(w-16)=0 \implies (w^2-36)(w-16)=0$
Factor difference of squares: $(w-6)(w+6)(w-16)=0$

Problem 36: Step4: Validate valid width

$w>0$ and height $18-w > w \implies w<9$. So $w=6$.

Problem 36: Step5: Calculate dimensions

Width $w=6$, length $2(6)+4=16$, height $18-6=12$

Answer:

1. $k = 0$, $k = 10$, $k = -10$
2. $y = 2$, $y = -2$, $y = \frac{7}{4}$
3. Width: 6 inches, Length: 16 inches, Height: 12 inches