A ball is thrown up on the surface of a moon. Its height above the lunar surface (in feet) after t seconds is given by the formula
h=308t−(14/6t^2)
Find the time that the ball reaches its maximum height.
Answer =
Find the maximal height attained by the ball
Answer =
To find the time that the ball reaches its maximum height, we need to find the value of t that gives the maximum value for the equation h = 308t - (14/6)t^2.
To determine the maximum value, we can find the vertex of the quadratic equation. The vertex of a quadratic equation in the form ax^2 + bx + c is given by the formula x = -b/2a.
In our equation, a = -(14/6), and b = 308. Plugging these values into the formula, we get:
t = -308 / 2(-14/6)
= -308 / (-28/6)
To simplify, we can divide 308 by 28 and multiply by 6:
t = -11 * 6
= -66
So, the time the ball reaches its maximum height is t = -66 seconds.
To find the maximal height attained by the ball, we substitute the value of t = -66 into the equation for height:
h = 308(-66) - (14/6)(-66)^2
= -20328 - (14/6)(4356)
= -20328 - 10184
= -30512
Therefore, the maximal height attained by the ball is -30512 feet.
To find the time that the ball reaches its maximum height, we need to find the vertex of the equation.
The equation for the height of the ball above the lunar surface is given by: h = 308t - (14/6)t^2
This equation is in the form h = at^2 + bt + c, where a = -(14/6), b = 308, and c = 0.
The vertex of the quadratic equation in this form is given by the formula: t = -b / (2a)
Plugging in the values, we get: t = -(308) / (2*(-14/6))
Simplifying further, t = -308 / (-28/6)
To divide fractions, we multiply by the reciprocal, so t = -308 * (-6/28)
Multiplying, t = 308 * 6 / 28
Simplifying, t = 66 seconds.
Therefore, the ball reaches its maximum height after 66 seconds.
To find the maximal height attained by the ball, we substitute the value of t into the height equation.
Plugging t = 66 into the height equation: h = 308(66) - (14/6)(66)^2
Calculating, h = 20328 - (14/6)(4356)
Simplifying, h = 20328 - (14/6)(4356) = 20328 - 10164 = 10164 feet.
Therefore, the maximal height attained by the ball is 10164 feet.
as usual for quadratics, the vertex is at t = -b/2a = 308/(14/3) = 66
Odd, since g is 1/6 of earth's, I expected the equation to be -16/6 t^2