Math

QuestionA ball is thrown from 8 feet high. Its height is given by f(x)=0.1x2+0.7x+8f(x)=-0.1 x^{2}+0.7 x+8. Find its max height and distance from release.

Studdy Solution

STEP 1

Assumptions1. The height of the ball is given by the function f(x)=0.1x+0.7x+8f(x)=-0.1 x^{}+0.7 x+8 . xx represents the ball's horizontal distance, in feet, from where it was thrown3. We are looking for the maximum height of the ball and the distance from the point of release where this occurs

STEP 2

The maximum height of the ball occurs at the vertex of the parabola represented by the function f(x)f(x). The xx-coordinate of the vertex of a parabola given by the function f(x)=ax2+bx+cf(x)=ax^{2}+bx+c is given by the formula b2a-\frac{b}{2a}.
xvertex=b2ax_{vertex} = -\frac{b}{2a}

STEP 3

Now, plug in the values for aa and bb from the function f(x)=0.1x2+0.7x+8f(x)=-0.1 x^{2}+0.7 x+8 to find the xx-coordinate of the vertex.
xvertex=0.72(0.1)x_{vertex} = -\frac{0.7}{2(-0.1)}

STEP 4

Calculate the xx-coordinate of the vertex.
xvertex=0.70.2=3.x_{vertex} = -\frac{0.7}{-0.2} =3.

STEP 5

The yy-coordinate of the vertex (which is the maximum height of the ball) is found by substituting the xx-coordinate of the vertex into the function f(x)f(x).
f(xvertex)=0.1(xvertex)2+0.7xvertex+8f(x_{vertex}) = -0.1 (x_{vertex})^{2}+0.7 x_{vertex}+8

STEP 6

Substitute xvertex=3.5x_{vertex} =3.5 into the function to find the maximum height.
f(3.5)=0.1(3.5)2+0.3.5+8f(3.5) = -0.1 (3.5)^{2}+0. \cdot3.5+8

STEP 7

Calculate the maximum height of the ball.
f(3.5)=0.112.25+2.45+=9.725f(3.5) = -0.1 \cdot12.25 +2.45 + =9.725The maximum height of the ball is9.7 feet (rounded to the nearest tenth), which occurs3.5 feet from the point of release.

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