Math Problem Statement

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 the maximum height reached by the rocket, to the nearest tenth of a foot. y, equals, minus, 16, x, squared, plus, 254, x, plus, 90 y=−16x 2 +254x+90

Solution

The height of the rocket is modeled by the quadratic equation:

y=16x2+254x+90y = -16x^2 + 254x + 90

Here, yy is the height in feet, and xx is the time in seconds. To find the maximum height, we need to determine the vertex of the parabola represented by this equation. For a quadratic equation of the form:

y=ax2+bx+cy = ax^2 + bx + c

the time xx at which the maximum height occurs can be found using the formula:

x=b2ax = \frac{-b}{2a}

For our equation:

  • a=16a = -16
  • b=254b = 254

Let's calculate the time xx at which the rocket reaches its maximum height.

x=2542(16)=25432=7.9375secondsx = \frac{-254}{2(-16)} = \frac{-254}{-32} = 7.9375 \, \text{seconds}

Now, substitute x=7.9375x = 7.9375 back into the original equation to find the maximum height yy.

y=16(7.9375)2+254(7.9375)+90y = -16(7.9375)^2 + 254(7.9375) + 90

Let me calculate that.The rocket reaches its maximum height at approximately 7.97.9 seconds, and the maximum height is 1098.11098.1 feet (to the nearest tenth).

Would you like more details or have any further questions?

Here are five related questions to explore:

  1. How would the maximum height change if the initial velocity were increased?
  2. What would happen to the maximum height if the acceleration due to gravity was lower?
  3. How long does the rocket stay in the air before hitting the ground?
  4. What is the rocket's velocity at the maximum height?
  5. How would you graph the rocket's trajectory over time?

Tip: The vertex formula b2a\frac{-b}{2a} is a useful tool for quickly finding maximums or minimums of quadratic equations in physics.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Maxima and Minima
Parabolas

Formulas

y = ax^2 + bx + c
Vertex formula: x = -b / 2a

Theorems

Quadratic formula
Properties of parabolas

Suitable Grade Level

Grades 9-11