A rocket is launched from atop a 36-foot cliff with an initial velocity of 107 feet per second. The height of the rocket, [tex]$t$[/tex] seconds after launch is given by the equation [tex]$h=-16 t^2+107 t+36$[/tex]. Use the quadratic formula to find out how long after the rocket is launched it will hit the ground ([tex]$h=0$[/tex]). Round your answer to the nearest tenth of a second.
A. 1.5 sec
B. 0.3 sec
C. 7.0 sec
D. 6.3 sec
Answer (2)
Set up the quadratic equation − 16 t 2 + 107 t + 36 = 0 .
Apply the quadratic formula: t = 2 a − b ± b 2 − 4 a c .
Calculate the discriminant and the two possible values for t .
Discard the negative value and round the positive value to the nearest tenth: 7.0
Explanation
Problem Setup We are given the height of a rocket as a function of time: h = − 16 t 2 + 107 t + 36 . We want to find the time t when the rocket hits the ground, which means h = 0 . So we need to solve the quadratic equation − 16 t 2 + 107 t + 36 = 0 for t .
Applying the Quadratic Formula To solve the quadratic equation, we will use the quadratic formula: t = 2 a − b ± b 2 − 4 a c where a = − 16 , b = 107 , and c = 36 .
Substituting Values Substitute the values of a , b , and c into the quadratic formula: t = 2 ( − 16 ) − 107 ± 10 7 2 − 4 ( − 16 ) ( 36 )
Simplifying the Discriminant Simplify the expression under the square root: 10 7 2 − 4 ( − 16 ) ( 36 ) = 11449 + 2304 = 13753
Calculating the Roots Now we have: t = − 32 − 107 ± 13753
Finding the Two Possible Times We calculate the two possible values for t : t 1 = − 32 − 107 + 13753 ≈ − 32 − 107 + 117.27 ≈ − 0.32 t 2 = − 32 − 107 − 13753 ≈ − 32 − 107 − 117.27 ≈ 7.01
Choosing the Correct Root Since time cannot be negative, we discard the negative solution t 1 ≈ − 0.32 . Therefore, the time when the rocket hits the ground is approximately t 2 ≈ 7.01 seconds. Rounding to the nearest tenth of a second, we get t ≈ 7.0 seconds.
Final Answer The rocket will hit the ground approximately 7.0 seconds after launch.
Examples
Understanding quadratic equations is crucial in various real-world scenarios, such as physics, engineering, and economics. For instance, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch, ensuring structural integrity and stability. Similarly, in economics, quadratic functions can model cost and revenue curves, helping businesses determine optimal pricing strategies to maximize profits. In sports, quadratic equations can describe the trajectory of a ball, enabling athletes to optimize their performance. These examples highlight the practical significance of quadratic equations in solving real-world problems.
The rocket will hit the ground approximately 7.0 seconds after launch, based on the quadratic equation derived from its height above the ground. This was calculated using the quadratic formula, resulting in a time of about 7.01 seconds, rounded to the nearest tenth. Therefore, the correct option is C. 7.0 sec.
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