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To determine how long it takes for the football to hit the ground, we need to solve the equation [tex]\( h = -16t^2 + 36t + 4 \)[/tex]. This equation models the height [tex]\( h \)[/tex] of the football as a function of time [tex]\( t \)[/tex]. When the football hits the ground, the height [tex]\( h \)[/tex] is 0, so we need to solve:
[tex]\[ -16t^2 + 36t + 4 = 0 \][/tex]
This is a quadratic equation of the form [tex]\( at^2 + bt + c = 0 \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 36 \)[/tex], and [tex]\( c = 4 \)[/tex]. To solve for [tex]\( t \)[/tex], we use the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. Calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 36^2 - 4(-16)(4) \][/tex]
[tex]\[ \Delta = 1296 + 256 \][/tex]
[tex]\[ \Delta = 1552 \][/tex]
2. Compute the square root of the discriminant:
[tex]\[ \sqrt{1552} = \sqrt{4 \times 388} = 2\sqrt{388} = 2 \times 19.697715603592208 \approx 39.395431207184416 \][/tex]
3. Apply the quadratic formula:
[tex]\[ t = \frac{-36 \pm 39.395431207184416}{-32} \][/tex]
We now have two solutions:
[tex]\[ t_1 = \frac{-36 + 39.395431207184416}{-32} \approx \frac{3.395431207184416}{-32} \approx -0.10610722522451299 \][/tex]
[tex]\[ t_2 = \frac{-36 - 39.395431207184416}{-32} \approx \frac{-75.395431207184416}{-32} \approx 2.356107225224513 \][/tex]
Since time cannot be negative, we discard the first solution [tex]\( t_1 \)[/tex]:
[tex]\[ t_2 \approx 2.356107225224513 \][/tex]
Therefore, it will take approximately [tex]\( 2.356 \)[/tex] seconds for the football to hit the ground.
Given the options, none of them precisely match our numerical solution derived from solving the quadratic equation. We choose:
[tex]\[ \frac{9 \pm \sqrt{65}}{8} \][/tex] is closest in structure, but the discriminant and coefficients don't match our original quadratic formula perfectly, indicating a misalignment between our exact numerical solution and the choices provided.
The direct match of the specific number solutions is not found in the multiple-choice options, but this is how the correct answer is reached using the quadratic formula step-by-step.
[tex]\[ -16t^2 + 36t + 4 = 0 \][/tex]
This is a quadratic equation of the form [tex]\( at^2 + bt + c = 0 \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 36 \)[/tex], and [tex]\( c = 4 \)[/tex]. To solve for [tex]\( t \)[/tex], we use the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. Calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 36^2 - 4(-16)(4) \][/tex]
[tex]\[ \Delta = 1296 + 256 \][/tex]
[tex]\[ \Delta = 1552 \][/tex]
2. Compute the square root of the discriminant:
[tex]\[ \sqrt{1552} = \sqrt{4 \times 388} = 2\sqrt{388} = 2 \times 19.697715603592208 \approx 39.395431207184416 \][/tex]
3. Apply the quadratic formula:
[tex]\[ t = \frac{-36 \pm 39.395431207184416}{-32} \][/tex]
We now have two solutions:
[tex]\[ t_1 = \frac{-36 + 39.395431207184416}{-32} \approx \frac{3.395431207184416}{-32} \approx -0.10610722522451299 \][/tex]
[tex]\[ t_2 = \frac{-36 - 39.395431207184416}{-32} \approx \frac{-75.395431207184416}{-32} \approx 2.356107225224513 \][/tex]
Since time cannot be negative, we discard the first solution [tex]\( t_1 \)[/tex]:
[tex]\[ t_2 \approx 2.356107225224513 \][/tex]
Therefore, it will take approximately [tex]\( 2.356 \)[/tex] seconds for the football to hit the ground.
Given the options, none of them precisely match our numerical solution derived from solving the quadratic equation. We choose:
[tex]\[ \frac{9 \pm \sqrt{65}}{8} \][/tex] is closest in structure, but the discriminant and coefficients don't match our original quadratic formula perfectly, indicating a misalignment between our exact numerical solution and the choices provided.
The direct match of the specific number solutions is not found in the multiple-choice options, but this is how the correct answer is reached using the quadratic formula step-by-step.
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