High School

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If a toy rocket is launched vertically upward from ground level with an initial velocity of 120 feet per second, then its height \( h \) after \( t \) seconds is given by the equation \( h(t) = -16t^2 + 120t \).

How long will it take the rocket to return to the ground?

Answer :

Answer:

[tex]Time = 7.5\ seconds[/tex]

Step-by-step explanation:

Given

[tex]Equation:\ h(t) = -16t^2 + 120t[/tex]

[tex]Initial\ Velocity = 160ft/s[/tex]

Required:

Determine the time taken to return to the ground

From the equation given; height (h) is a function of time (t)

When the rocket returns to the ground level, h(t) = 0

Substitute 0 for h(t) in the given equation

[tex]h(t) = -16t^2 + 120t[/tex]

becomes

[tex]0 = -16t^2 + 120t[/tex]

Solve for t in the above equation

[tex]-16t^2 + 120t = 0[/tex]

Factorize the above expression

[tex]-4t(4t - 30) = 0[/tex]

Split the expression to 2

[tex]-4t = 0\ or\ 4t - 30 = 0[/tex]

Solving the first expression

[tex]-4t = 0[/tex]

Divide both sides by -4

[tex]\frac{-4t}{-4} = \frac{0}{-4}[/tex]

[tex]t = \frac{0}{-4}[/tex]

[tex]t =0[/tex]

Solving the second expression

[tex]4t - 30 = 0[/tex]

Add 30 to both sides

[tex]4t - 30+30 = 0+30[/tex]

[tex]4t = 30[/tex]

Divide both sides by 4

[tex]\frac{4t}{4} = \frac{30}{4}[/tex]

[tex]t = \frac{30}{4}[/tex]

[tex]t = 7.5[/tex]

Hence, the values of t are:

[tex]t =0[/tex] and [tex]t = 7.5[/tex]

[tex]t =0[/tex] shows the time before the launching the rocket

while

[tex]t = 7.5[/tex] shows the time after the rocket returns to the floor

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Rewritten by : Barada