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An object is thrown straight up with a speed of v(t)=−32t+96v(t) = -32t + 96, where tt is the time in seconds, from a height of 64 ft64\,\text{ft} above the ground. What is the maximum height above the ground that the object reaches?

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Q. An object is thrown straight up with a speed of v(t)=−32t+96v(t) = -32t + 96, where tt is the time in seconds, from a height of 64 ft64\,\text{ft} above the ground. What is the maximum height above the ground that the object reaches?
  1. Identify velocity function: Identify the velocity function and initial height. v(t)=−32t+96v(t) = -32t + 96, initial height h0=64h_0 = 64 ft.
  2. Find height function: The height function h(t)h(t) is the integral of the velocity function.h(t)=∫v(t) dt=∫(−32t+96) dt.h(t) = \int v(t) \, dt = \int (-32t + 96) \, dt.
  3. Integrate velocity function: Integrate the velocity function to find the height function. h(t)=−16t2+96t+Ch(t) = -16t^2 + 96t + C.
  4. Determine constant of integration: Determine the constant of integration CC using the initial height. \newlineh(0)=64 fth(0) = 64 \, \text{ft}, so C=64C = 64.
  5. Write complete height function: Write the complete height function. h(t)=−16t2+96t+64h(t) = -16t^2 + 96t + 64.
  6. Find time of maximum height: Find the time at which the object reaches maximum height by finding the vertex of the parabola.\newlineThe vertex occurs at t=−b2at = -\frac{b}{2a}, where a=−16a = -16 and b=96b = 96.
  7. Calculate time of maximum height: Calculate the time at which the object reaches maximum height. t=−962(−16)t = -\frac{96}{2(-16)}, t=9632t = \frac{96}{32}, t=3t = 3 seconds.
  8. Substitute time into height function: Substitute t=3t = 3 into the height function to find the maximum height.h(3)=−16(3)2+96(3)+64.h(3) = -16(3)^2 + 96(3) + 64.
  9. Perform calculation for maximum height: Perform the calculation to find the maximum height. h(3)=−16(9)+288+64h(3) = -16(9) + 288 + 64, h(3)=−144+288+64h(3) = -144 + 288 + 64.
  10. Add values for maximum height: Add the values to get the maximum height.\newlineh(3)=144+64h(3) = 144 + 64, h(3)=208h(3) = 208 feet.

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