QuestionA ball is thrown from 7 feet high. Its height is modeled by $f(x)=-0.1 x^{2}+0.7 x+7$ . Find the max height and distance.

Studdy Solution

STEP 1

Assumptions1. The height of the ball, $f(x)$ , is given by the equation $f(x)=-0.1 x^{}+0.7 x+7$ . $x$ is the ball's horizontal distance, in feet, from where it was thrown3. We are looking for the maximum height of the ball and the horizontal distance at which this occurs

STEP 2

The maximum height of the ball corresponds to the vertex of the parabola described by the equation $f(x)=-0.1 x^{2}+0.7 x+7$ . For a parabola in the form $f(x)=ax^2+bx+c$ , the x-coordinate of the vertex is given by $-\frac{b}{2a}$ .

STEP 3

Plug in the values for $a$ and $b$ from the equation $f(x)=-0.1 x^{2}+0.7 x+7$ into the formula $-\frac{b}{2a}$ to find the x-coordinate of the vertex.
$x = -\frac{0.7}{2(-0.1)}$

STEP 4

Calculate the x-coordinate of the vertex.
$x = -\frac{0.7}{-0.2} =3.$

STEP 5

The x-coordinate of the vertex is the horizontal distance from where the ball was thrown to where it reaches its maximum height. So, the ball reaches its maximum height3.5 feet from where it was thrown.

STEP 6

To find the maximum height of the ball, plug the x-coordinate of the vertex into the equation $f(x)=-0.1 x^{2}+0. x+$ .
$f(3.5) = -0.1(3.5)^2 +0.(3.5) +$

STEP 7

Calculate the maximum height of the ball.
$f(3.5) = -0.1(12.25) +2.45 +7 =.775$ The maximum height of the ball is. feet (rounded to the nearest tenth), which occurs3.5 feet from the point of release.

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