QuestionJuly 19, 2025

An object is thrown upward at a speed of 119 feet per second by a machine from a height of 15 feet off the ground. The height of the object after t seconds can be found using the equation h(t)=-16t^2+119t+15 Give all numerical answers to 2 decimal places. When will the height be 141 feet? At square units: square When will the object reach the ground? square units: square When will the object reach its maximum height? square units: square What is its maximum height? square units: square

An object is thrown upward at a speed of 119 feet per second by a machine from a height of 15 feet off the ground. The height of the object after t seconds can be found using the equation h(t)=-16t^2+119t+15 Give all numerical answers to 2 decimal places. When will the height be 141 feet? At square units: square When will the object reach the ground? square units: square When will the object reach its maximum height? square units: square What is its maximum height? square units: square
An object is thrown upward at a speed of 119 feet per second by a machine from a height of 15 feet off the ground. The height of the object after t seconds can be found using the equation h(t)=-16t^2+119t+15 Give all numerical answers to 2 decimal places. When will the height be 141 feet? At square units: square When will the object reach the ground? square units: square When will the object reach its maximum height? square units: square What is its maximum height? square units: square

Solution
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Answer

When will the height be 141 feet? At 0.56 units: 14.06 ### When will the object reach the ground? 7.58 units: ### When will the object reach its maximum height? 3.72 units: ### What is its maximum height? 236.19 units: Explanation 1. Solve for when height is 141 feet Set h(t) = 141: -16t^2 + 119t + 15 = 141. Simplify to -16t^2 + 119t - 126 = 0. Use the quadratic formula t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} with a = -16, b = 119, c = -126. 2. Calculate discriminant and roots Discriminant \Delta = 119^2 - 4(-16)(-126) = 14161 - 8064 = 6097. Roots are t = \frac{-119 \pm \sqrt{6097}}{-32}. 3. Evaluate roots t_1 = \frac{-119 + \sqrt{6097}}{-32} \approx 0.56, t_2 = \frac{-119 - \sqrt{6097}}{-32} \approx 14.06. 4. Solve for when object reaches the ground Set h(t) = 0: -16t^2 + 119t + 15 = 0. Use the quadratic formula again with a = -16, b = 119, c = 15. 5. Calculate discriminant and roots for ground Discriminant \Delta = 119^2 - 4(-16)(15) = 14161 + 960 = 15121. Roots are t = \frac{-119 \pm \sqrt{15121}}{-32}. 6. Evaluate roots for ground t_1 = \frac{-119 + \sqrt{15121}}{-32} \approx -0.13, t_2 = \frac{-119 - \sqrt{15121}}{-32} \approx 7.58. Only positive root is valid. 7. Find time of maximum height Maximum height occurs at vertex t = \frac{-b}{2a} = \frac{-119}{2(-16)} = 3.72. 8. Calculate maximum height Substitute t = 3.72 into h(t): h(3.72) = -16(3.72)^2 + 119(3.72) + 15 \approx 236.19.

Explanation

1. Solve for when height is 141 feet<br /> Set $h(t) = 141$: $-16t^2 + 119t + 15 = 141$. Simplify to $-16t^2 + 119t - 126 = 0$. Use the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a = -16$, $b = 119$, $c = -126$.<br /><br />2. Calculate discriminant and roots<br /> Discriminant $\Delta = 119^2 - 4(-16)(-126) = 14161 - 8064 = 6097$. Roots are $t = \frac{-119 \pm \sqrt{6097}}{-32}$.<br /><br />3. Evaluate roots<br /> $t_1 = \frac{-119 + \sqrt{6097}}{-32} \approx 0.56$, $t_2 = \frac{-119 - \sqrt{6097}}{-32} \approx 14.06$.<br /><br />4. Solve for when object reaches the ground<br /> Set $h(t) = 0$: $-16t^2 + 119t + 15 = 0$. Use the quadratic formula again with $a = -16$, $b = 119$, $c = 15$.<br /><br />5. Calculate discriminant and roots for ground<br /> Discriminant $\Delta = 119^2 - 4(-16)(15) = 14161 + 960 = 15121$. Roots are $t = \frac{-119 \pm \sqrt{15121}}{-32}$.<br /><br />6. Evaluate roots for ground<br /> $t_1 = \frac{-119 + \sqrt{15121}}{-32} \approx -0.13$, $t_2 = \frac{-119 - \sqrt{15121}}{-32} \approx 7.58$. Only positive root is valid.<br /><br />7. Find time of maximum height<br /> Maximum height occurs at vertex $t = \frac{-b}{2a} = \frac{-119}{2(-16)} = 3.72$.<br /><br />8. Calculate maximum height<br /> Substitute $t = 3.72$ into $h(t)$: $h(3.72) = -16(3.72)^2 + 119(3.72) + 15 \approx 236.19$.
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