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Question

A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation, find out the time at which the rocket will reach its max, to the nearest 100th of a second.

y=−16x2+218x+121

Answer 1

x = -b/2a

x = -218/(2(16)) = -218/32 = 6.81 seconds

Answer 2

Equations in the form y = ax² + bx + c always graph as a u-shaped curve, or parabola, and its turning point is called the vertex. For the rocket in this problem, its path will follow an upside-down u-shaped curve, so to find the time for it to get to the max height, we need to find the x-coordinate of the vertex. Thankfully, there is a handy formula that can be used to find the x-coordinate of the vertex. The formula is: x = -b/2a.

The "a" value in the formula is the coefficient of the x² term and the "b" value is the coefficient of the x term. So for your equation, a = -16 and b = 218. Inserting these into the formula, we have x = -218/(2)(-16), or x = -218/-32, so x = 6.8125. Rounding this to the nearest 100th, we have x = 6.81 seconds, the time it will take for the rocket to reach its maximum height.

Answer 3

Hi there.

For this problem, first note that the equation for the rocket is a parabola in standard form:

y = ax² + bx + c  

where a = -16, b=218, and c=121.

Since a < 0, you know that this parabola is concave down, meaning it looks like a frowny face, so there will be a vertex that is the maximum point of this parabola.

To start, figure out what the question is asking us to solve. It asks for the time at which the rocket will be at it's max. This means that the question is asking us to find the x value of the vertex.

All parabolas have an axis of symmetry. That is the line that acts as the mirror for the parabola. It is a vertical line, and the parabola on either side of the axis of symmetry is the mirror image of the other side. This means that the axis of symmetry runs through the vertex, and is the line x = the x value of the vertex.

The formula for the axis of symmetry is x = -b / 2a. Since we know that is the x-value of the vertex, we can use this to solve our problem.

Plugging in the values for b and a from our parabola, we get:

x = -b / 2a

x = -(218) / 2(-16)

x = -218 / -32

x = 6.8125

Rounding that to the nearest 100th of a second, we get 6.81 seconds is the time at which the rocket will reach it's max.

I hope this helps. Let me know if you have any follow-up questions to this one. Good luck!