MAT 132 TEST 2 Review (Jan 2025 ) 1. If a ball is to thrown verically upwards, the ball reaches a height of 10 fl a) \( y=-4 t^{2}+6 t+13 \) b) \( y=-7 t^{2}+2(+23 \) feet after \( t \) seconds. i) find max height ii) when ball reach Gack to ground iii) when does ball reaches \( 12 \mathrm{ft}, 5 \mathrm{ft} \)
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For the quadratic equation \( y = -4t^2 + 6t + 13 \), the maximum height can be found using the vertex formula \( t = -\frac{b}{2a} \). Here, \( a = -4 \) and \( b = 6 \). Plugging in these values gives \( t = \frac{6}{8} = 0.75 \) seconds. Substitute \( t = 0.75 \) back into the equation to find the maximum height, which is approximately \( 15.5 \) feet. To find when the ball hits the ground, we need to set \( y = 0 \). Solving the equation \( -4t^2 + 6t + 13 = 0 \) using the quadratic formula will yield the time when it returns to the ground. You will find two values for \( t \): one when it is thrown up and another when it falls back down. For \( 12 \) ft and \( 5 \) ft, set the equation equal to these heights and solve for \( t \) in both cases, which will give you the moments when the ball reaches these heights. Now, let's not forget that while working through these calculations, sketching a graph could really put things in perspective! Seeing the arc of the ball's flight visually illustrates when it reaches its peak height and when it splashes back down—it can make the math much more engaging! Also, while attempting to solve for various heights of \( 12 \) ft and \( 5 \) ft, be cautious of potential errors in calculation, such as misapplying the quadratic formula or overlooking negative roots that could also represent the ball's return trajectory. Double-check your work and ensure each step is accounted for, especially when substituting back into the height function!