Question

LaToya throws a ball from the top of a bridge. Her throw is modeled by the equation y=-0.5x² + 4x + 9, and the bridge is modeled by the equation y=-0.1x+6. About how far does the ball travel

horizontally before its first bounce?
About feet
(Round to two decimal places as needed)

Answer 1

To find out how far the ball travels horizontally, we set up an equation where the trajectory of the ball intersects with the bridge model, then solve the resulting quadratic equation for the positive value of the horizontal distance x.

Step-by-step explanation:

To determine how far the ball travels horizontally before its first bounce, we need to find the point where the ball's trajectory (given by the equation y = -0.5x² + 4x + 9) intersects with the model of the top of the bridge (y = -0.1x + 6). This is the point where the ball hits the ground (or in this case, the bridge).

We solve for x by setting the two equations equal to each other:

-0.5x² + 4x + 9 = -0.1x + 6

Next, we simplify and solve the quadratic equation:

-0.5x² + 4x + 9 - (-0.1x + 6) = 0

-0.5x² + 4.1x + 3 = 0

Using the quadratic formula, x = [-b ± sqrt(b² - 4ac)] / (2a), where a = -0.5, b = 4.1, and c = 3, we find the positive root to get the horizontal distance (since time cannot be negative).

The ball will travel horizontally approximately x feet before landing. Please note that this calculation was done for your understanding, and you will need a calculator to obtain the numerical value of x.

Answer 2

See below.

Step-by-step explanation:

Model the system of equations, and solve for x to determine the required horizontal distance.

`\left \{ {{y=-0.5x^2+3x+10} \atop {y=-0.2+7}} \right.`

Transform the quadratic equation into standard form and substitute for y into linear equation. Since the quadratic equation already is in the standard form, substitute for y:

`y=-0.2x+7` ---> Rewrite the linear equation.

`-0.5x^2+3x+10=-0.2x+7` ---> Substitute
`y=-0.5x^2+3x+10`.

`-0.5x^2+3.2x+3=0` ----> Simplify.

Solve the equation by using the Quadratic Formula:
Identify
`a=-0.5,b=3.2,c=3.`

`x=(-b\pm√(b^2-4ac) )/(2a)` -----------------> Write the Quadratic Formula.

`=(-3.2\pm√(3.2^2-4*(-0.5)*3) )/(2*(-0.5))` -------------------> Substitute for a, b, and c.

`=(-3.2\pm√(16.24) )/(-1)` ----------------------> Simplify.

`=3.2\pm4.03` -------------------->
`√(16.24)=(\text{Estimated})4.03` and simplify.

`x=7.23`, and

`x = -0.83`

The distance does not make sense for negative numbers, so only consider
`x=7.23.`

Thank you,

Eddie Echevarria