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QuestionA ball is thrown from 5 feet high. Its height is modeled by f(x)=−0.1x2+1.4x+5f(x)=-0.1 x^{2}+1.4 x+5. Find its max height and distance from the throw point.

Studdy Solution

STEP 1

Assumptions1. The height of the ball, f(x)f(x), is modeled by the quadratic equation f(x)=−0.1x+1.4x+5f(x)=-0.1 x^{}+1.4 x+5 . xx is the ball's horizontal distance, in feet, from where it was thrown

STEP 2

The maximum height of the ball can be found by finding the vertex of the parabola represented by the equation f(x)=−0.1x2+1.4x+5f(x)=-0.1 x^{2}+1.4 x+5. The x-coordinate of the vertex of a parabola given by the equation 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 given values for aa and bb to calculate the x-coordinate of the vertex.
xvertex=−1.2(−0.1)x_{vertex} = -\frac{1.}{2(-0.1)}

STEP 4

Calculate the x-coordinate of the vertex.
xvertex=−1.4−0.2=7x_{vertex} = -\frac{1.4}{-0.2} =7

STEP 5

Now that we have the x-coordinate of the vertex, we can find the y-coordinate (which represents the maximum height of the ball) by plugging xvertexx_{vertex} into the equation f(x)f(x).
f(xvertex)=−0.1xvertex2+1.4xvertex+5f(x_{vertex}) = -0.1 x_{vertex}^{2}+1.4 x_{vertex}+5

STEP 6

Plug in the value for xvertexx_{vertex} to calculate the maximum height.
f()=−0.1()2+1.4()+5f() = -0.1 ()^{2}+1.4 ()+5

STEP 7

Calculate the maximum height of the ball.
f(7)=−0.1(49)+9.+5=4.1+9.+5=18.9f(7) = -0.1 (49)+9.+5 =4.1+9.+5 =18.9The maximum height of the ball is18.9 feet, which occurs7 feet from the point of release.

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