Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.