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Answer :
To find out how long after the basketball is thrown it goes into the hoop at a height of 10 feet, we need to set up and solve the given equation for when the height [tex]\( h \)[/tex] is 10 feet. The equation that models the height of the basketball is:
[tex]\[ h = -16t^2 + 23t + 7 \][/tex]
We want to find the time [tex]\( t \)[/tex] when the height [tex]\( h \)[/tex] is 10 feet, so let's set the equation equal to 10:
[tex]\[ -16t^2 + 23t + 7 = 10 \][/tex]
To make it easier to solve, we rearrange the equation:
[tex]\[ -16t^2 + 23t + 7 - 10 = 0 \][/tex]
This simplifies to:
[tex]\[ -16t^2 + 23t - 3 = 0 \][/tex]
Now, to solve this quadratic equation for [tex]\( t \)[/tex], we can use the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = -16 \)[/tex], [tex]\( b = 23 \)[/tex], and [tex]\( c = -3 \)[/tex].
Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = 23^2 - 4(-16)(-3) \][/tex]
The discriminant is positive, indicating there are two real solutions. However, since we're looking for the time it takes for the basketball to reach the hoop after being thrown, we only consider the positive solution for [tex]\( t \)[/tex].
After evaluating the quadratic formula, we find that the time [tex]\( t \)[/tex] when the basketball goes into the hoop is approximately 1.29 seconds.
Therefore, the basketball goes into the hoop about 1.29 seconds after it is thrown. This corresponds to the option "1.29 seconds" in the choices provided.
[tex]\[ h = -16t^2 + 23t + 7 \][/tex]
We want to find the time [tex]\( t \)[/tex] when the height [tex]\( h \)[/tex] is 10 feet, so let's set the equation equal to 10:
[tex]\[ -16t^2 + 23t + 7 = 10 \][/tex]
To make it easier to solve, we rearrange the equation:
[tex]\[ -16t^2 + 23t + 7 - 10 = 0 \][/tex]
This simplifies to:
[tex]\[ -16t^2 + 23t - 3 = 0 \][/tex]
Now, to solve this quadratic equation for [tex]\( t \)[/tex], we can use the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = -16 \)[/tex], [tex]\( b = 23 \)[/tex], and [tex]\( c = -3 \)[/tex].
Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = 23^2 - 4(-16)(-3) \][/tex]
The discriminant is positive, indicating there are two real solutions. However, since we're looking for the time it takes for the basketball to reach the hoop after being thrown, we only consider the positive solution for [tex]\( t \)[/tex].
After evaluating the quadratic formula, we find that the time [tex]\( t \)[/tex] when the basketball goes into the hoop is approximately 1.29 seconds.
Therefore, the basketball goes into the hoop about 1.29 seconds after it is thrown. This corresponds to the option "1.29 seconds" in the choices provided.
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