QuestionA ball is thrown from 7 feet high, modeled by $f(x)=-0.4 x^{2}+2.7 x+7$ . Find its max height and distance from release.

Studdy Solution

STEP 1

Assumptions1. The height of the ball, $f(x)$ , can be modeled by the equation $f(x)=-0.4 x^{}+.7 x+7$ . $x$ is the ball's horizontal distance, in feet, from where it was thrown3. We need to find the maximum height of the ball and the distance from where it was thrown when this maximum height occurs

STEP 2

The equation given is a quadratic equation in the form of $ax^2 + bx + c$ . The maximum or minimum value of a quadratic function occurs at its vertex. The x-coordinate of the vertex of a quadratic function given in the form $f(x) = ax^2 + bx + c$ is given by $-\frac{b}{2a}$ .

STEP 3

Now, plug in the given values for $a$ and $b$ to calculate the x-coordinate of the vertex.
$x = -\frac{b}{2a} = -\frac{2.7}{2(-0.)}$

STEP 4

Calculate the x-coordinate of the vertex.
$x = -\frac{2.7}{-0.8} =3.375$

STEP 5

Now that we have the x-coordinate of the vertex, we can find the maximum height of the ball by plugging this value into the equation for $f(x)$ .
$f(x) = -0.4(3.375)^2 +2.7(3.375) +7$

STEP 6

Calculate the maximum height of the ball.
$f(3.375) = -0.4(3.375)^2 +2.(3.375) + =11.35625$ The maximum height of the ball is approximately11.4 feet, which occurs3.4 feet from the point of release.

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QuestionA ball is thrown from 7 feet high, modeled by $f(x)=-0.4 x^{2}+2.7 x+7$ . Find its max height and distance from release.

Studdy Solution

STEP 1

Assumptions1. The height of the ball, $f(x)$ , can be modeled by the equation $f(x)=-0.4 x^{}+.7 x+7$ . $x$ is the ball's horizontal distance, in feet, from where it was thrown3. We need to find the maximum height of the ball and the distance from where it was thrown when this maximum height occurs

STEP 2

The equation given is a quadratic equation in the form of $ax^2 + bx + c$ . The maximum or minimum value of a quadratic function occurs at its vertex. The x-coordinate of the vertex of a quadratic function given in the form $f(x) = ax^2 + bx + c$ is given by $-\frac{b}{2a}$ .

STEP 3

Now, plug in the given values for $a$ and $b$ to calculate the x-coordinate of the vertex.
$x = -\frac{b}{2a} = -\frac{2.7}{2(-0.)}$

STEP 4

Calculate the x-coordinate of the vertex.
$x = -\frac{2.7}{-0.8} =3.375$

STEP 5

Now that we have the x-coordinate of the vertex, we can find the maximum height of the ball by plugging this value into the equation for $f(x)$ .
$f(x) = -0.4(3.375)^2 +2.7(3.375) +7$

STEP 6

Calculate the maximum height of the ball.
$f(3.375) = -0.4(3.375)^2 +2.(3.375) + =11.35625$ The maximum height of the ball is approximately11.4 feet, which occurs3.4 feet from the point of release.

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QuestionA ball is thrown from 7 feet high, modeled by f(x)=−0.4x2+2.7x+7f(x)=-0.4 x^{2}+2.7 x+7f(x)=−0.4x2+2.7x+7. Find its max height and distance from release.

Studdy Solution

STEP 1

STEP 2

STEP 3

Now, plug in the given values for aaa and bbb to calculate the x-coordinate of the vertex.x=−b2a=−2.72(−0.)x = -\frac{b}{2a} = -\frac{2.7}{2(-0.)}x=−2ab​=−2(−0.)2.7​

STEP 4

Calculate the x-coordinate of the vertex.x=−2.7−0.8=3.375x = -\frac{2.7}{-0.8} =3.375x=−−0.82.7​=3.375

STEP 5

Now that we have the x-coordinate of the vertex, we can find the maximum height of the ball by plugging this value into the equation for f(x)f(x)f(x).f(x)=−0.4(3.375)2+2.7(3.375)+7f(x) = -0.4(3.375)^2 +2.7(3.375) +7f(x)=−0.4(3.375)2+2.7(3.375)+7

STEP 6

Calculate the maximum height of the ball.f(3.375)=−0.4(3.375)2+2.(3.375)+=11.35625f(3.375) = -0.4(3.375)^2 +2.(3.375) + =11.35625f(3.375)=−0.4(3.375)2+2.(3.375)+=11.35625The maximum height of the ball is approximately11.4 feet, which occurs3.4 feet from the point of release.

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QuestionA ball is thrown from 7 feet high, modeled by $f(x)=-0.4 x^{2}+2.7 x+7$ . Find its max height and distance from release.

Now, plug in the given values for $a$ and $b$ to calculate the x-coordinate of the vertex.
$x = -\frac{b}{2a} = -\frac{2.7}{2(-0.)}$

Calculate the x-coordinate of the vertex.
$x = -\frac{2.7}{-0.8} =3.375$

Now that we have the x-coordinate of the vertex, we can find the maximum height of the ball by plugging this value into the equation for $f(x)$ .
$f(x) = -0.4(3.375)^2 +2.7(3.375) +7$

Calculate the maximum height of the ball.
$f(3.375) = -0.4(3.375)^2 +2.(3.375) + =11.35625$ The maximum height of the ball is approximately11.4 feet, which occurs3.4 feet from the point of release.