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Avani launches a toy rocket from a platform. The height of the rocket in feet is given by h(t)=-16t^(2)+16 t+32 where t represents the time in seconds after launch. What is the rocket's greatest height? Answer: feet

Avani launches a toy rocket from a platform. The height of the rocket in feet is given by h(t)=−16t2+16t+32 h(t)=-16 t^{2}+16 t+32 where t t represents the time in seconds after launch. What is the rocket's greatest height?\newlineAnswer: feet

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Q. Avani launches a toy rocket from a platform. The height of the rocket in feet is given by h(t)=−16t2+16t+32 h(t)=-16 t^{2}+16 t+32 where t t represents the time in seconds after launch. What is the rocket's greatest height?\newlineAnswer: feet
  1. Given Equation Analysis: The given equation for the height of the rocket is h(t)=−16t2+16t+32h(t) = -16t^2 + 16t + 32. We need to find the greatest height, which corresponds to the maximum value of the quadratic function. The maximum value occurs at the vertex of the parabola represented by the equation.
  2. Calculate Time of Vertex: To find the time at which the rocket reaches its greatest height, we need to calculate the tt-coordinate of the vertex of the parabola. The formula for the tt-coordinate of the vertex is given by t=−b2at = -\frac{b}{2a}, where aa and bb are the coefficients from the quadratic equation in the form h(t)=at2+bt+ch(t) = at^2 + bt + c.
  3. Find Time Value: In our equation, a=−16a = -16 and b=16b = 16. Plugging these values into the vertex formula gives us t=−16/(2∗(−16))=−16/(−32)=1/2t = -16/(2*(-16)) = -16/(-32) = 1/2.
  4. Calculate Maximum Height: Now that we have the time at which the rocket reaches its greatest height, we can find the actual height by plugging t=12t = \frac{1}{2} back into the original equation. So, h(12)=−16∗(12)2+16∗(12)+32h(\frac{1}{2}) = -16*(\frac{1}{2})^2 + 16*(\frac{1}{2}) + 32.
  5. Final Height Calculation: Calculating h(12)h(\frac{1}{2}) gives us h(12)=−16∗(14)+16∗(12)+32=−4+8+32=36h(\frac{1}{2}) = -16*(\frac{1}{4}) + 16*(\frac{1}{2}) + 32 = -4 + 8 + 32 = 36 feet.

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