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McGraw Hill Integrated II, 2012

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McGraw Hill Integrated II, 2012 View details

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Exercise 12 Page 713

By finding the second zero of the equation, we can evaluate when the rocket would hit the ground if it did not explode.

See solution.

Practice makes perfect

We know that a rocket followed a path modeled by a quadratic equation, where t is the time in seconds. d=80t-16t^2 We are given that the rocket failed to explode, so we can determine after how many seconds the rocket would hit the ground. To do this, we will substitute 0 for d and solve the quadratic equation. 0=80t-16t^2 ⇒ -16t^2+80t=0Let's recall the Quadratic Formula. ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a To solve our equation, we first need to identify the values of a, b, and c. -16t^2+80t=0 ⇕ -16t^2+ 80t+ 0=0 We see that a= -16, b= 80, and c= 0. Let's substitute these values into the Quadratic Formula.

t=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

t=- 80±sqrt(80^2-4( -16)( 0))/2( -16)

▼

Solve for t and Simplify

Zero Property of Multiplication

t=-80±sqrt(80^2)/2(-16)

SqrtPowToNumber

sqrt(a^2)=a

t=-80± 80/2(-16)

a(- b)=- a * b

t=-80± 80/-32

The solutions for this equation are t= -80± 80-32. Let's separate them into the positive and negative cases.

t=-80± 80/-32
t_1=-80+80/-32	t_2=-80-80/-32
t_1=0/-32	t_2=-160/-32
t_1=0	t_2=5

Using the Quadratic Formula, we found that the solutions of the given equation are t_1=0 and t_2=5. Therefore, the rocket would hit the ground after 5 seconds.

Alternative Solution

Graphing Method

Let's draw the path of the rocket using the given equation. To do this, we will substitute some values for t and evaluate corresponding values of d to find the points that lie on this graph.

t	80t-16t^2	d
0	80( 0)-16( 0)^2	0
1	80( 1)-16( 1)^2	64
2	80( 2)-16( 2)^2	96
3	80( 3)-16( 3)^2	96
4	80( 4)-16( 4)^2	64
5	80( 5)-16( 5)^2	0

Now let's plot the points and connect them with a smooth curve. Remember that both time and distance are non-negative, so our graph will be drawn only in the first quadrant.

From the graph, we can see that the rocket would hit the ground after 5 seconds.

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