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Al released his balloon from the $10$ -yard line, and it landed at the $16$ -yard line. If the ball reached a height of $27$ yards, what equation represents the path of his toss?

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Mean, variance, and standard deviation of binomial distributions

Full solution

Q. Al released his balloon from the

10

-yard line, and it landed at the

16

-yard line. If the ball reached a height of

27

yards, what equation represents the path of his toss?

Identify Parabolic Path: To find the equation that represents the path of Al's balloon toss, we need to determine the type of path it would take. Since the balloon is being released and lands at a different point, we can assume it follows a parabolic path, similar to the path of a projectile. We will use the vertex form of a parabolic equation, which is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.
Determine Vertex Coordinates: The vertex of the parabola is the highest point of the toss. Since the balloon reached a height of $27$ yards, this is the $k$ value in our vertex form equation. We need to find the $h$ value, which is the $x$ -coordinate of the vertex. The balloon was released from the $10$ -yard line and landed at the $16$ -yard line, so the vertex is halfway between these two points. The midpoint is $(10 + 16) / 2 = 13$ yards.
Substitute Vertex into Equation: Now that we have the vertex $(h, k) = (13, 27)$ , we can substitute these values into the vertex form equation to get $y = a(x - 13)^2 + 27$ . However, we still need to find the value of $a$ , which determines the shape of the parabola.
Find Value of $a$ : To find the value of $a$ , we need another point on the parabola. We know that the balloon started at the $10$ -yard line, which is $3$ yards away from the vertex horizontally. Since the balloon starts at ground level, the $y$ -coordinate at this point is $0$ . So, we have the point $(10, 0)$ to use in our equation.
Use Additional Point for $a$ : Substitute the point $(10, 0)$ into the equation $y = a(x - 13)^2 + 27$ to solve for $a$ . This gives us $0 = a(10 - 13)^2 + 27$ . Simplifying, we get $0 = a(3)^2 + 27$ , which is $0 = 9a + 27$ .
Solve for a: To solve for a, we subtract $27$ from both sides of the equation, which gives us $-27 = 9a$ . Then, we divide both sides by $9$ to get $a = -27 / 9$ , which simplifies to $a = -3$ .
Write Final Equation: Now that we have the value of $a$ , we can write the final equation of the parabola. The equation representing the path of Al's balloon toss is $y = -3(x - 13)^2 + 27$ .

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