Question

question 2 (multiple choice, worth 1 points)\\(\\(08.04\\) mc)\the length of a rectangular frame is represented by the expression \\(2x + 4\\), and the width of the rectangular frame is represented by the expression \\(2x + 10\\). write an equation to solve for the width of a rectangular frame that has a total area of 120 square inches\\(\boldsymbol{2x^2 + 20x - 80 = 0}\\)\\(4x^2 + 28x + 40 = 0\\)\\(4x^2 + 28x - 80 = 0\\)\\(x^2 + 6x + 20 = 0\\)\question 3 (multiple choice, worth 1 points)\\(\\(08.04\\) lc)\sicario kicked a soccer ball off the ground at a speed of 48 feet per second. the height of the ball can be represented by the function \\(h(t) = -16t^2 + 48t\\), where \\(t\\) is the time in seconds. how many seconds did the ball travel before returning to the ground?\\(t = 0.25\\) seconds\\(t = 3\\) seconds\\(t = 16\\) seconds\\(t = 48\\) seconds

Explanation:

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Question 2

Step1: Recall area formula for rectangle

Area = Length × Width

Step2: Substitute given expressions

$\text{Area} = (2x+4)(2x+10) = 120$

Step3: Expand the left-hand side

$(2x)(2x) + (2x)(10) + 4(2x) + 4(10) = 4x^2 + 20x + 8x + 40 = 4x^2 + 28x + 40$

Step4: Set equal to 120 and simplify

$4x^2 + 28x + 40 - 120 = 0$
$4x^2 + 28x - 80 = 0$

Step1: Set height to 0 (ground level)

$H(t) = -16t^2 + 48t = 0$

Step2: Factor out common term

$-8t(2t - 6) = 0$

Step3: Solve for t

Solutions: $-8t=0 \implies t=0$ (launch time), and $2t-6=0 \implies t=3$

Step4: Select valid time (post-launch)

Choose $t=3$ seconds (time to return to ground)

Answer:

$4x^2 + 28x - 80 = 0$

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Question 3