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Algebra 1

8.4.3 Applying the Zero Product Property to Solve a Real-World Projectile Problem

Algebra 18.4.3 Applying the Zero Product Property to Solve a Real-World Projectile Problem

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Activity

We have seen quadratic functions modeling the height of a projectile as a function of time.

Here are two ways to define the same function that approximates the height of a projectile in meters, tt seconds after launch:

h(t)=5t2+27t+18h(t)=(5t3)(t6)h(t)=5t2+27t+18h(t)=(5t3)(t6)

1.

Which way of defining the function allows us to use the zero product property to find out when the height of the object is 0 meters (when the object is on the ground)?

2.

Without graphing, determine at what time(s) the height of the object is 0 meters (when the object is on the ground). Show your reasoning.

3.

Using the function in standard form, determine the height of the object before it started its motion, at t=0t=0 seconds. Be prepared to show your reasoning.

Video: Applying the Zero Product Property

Watch the following video to learn more about applying the zero product property to solve a real-world projectile problem.

Self Check

The equivalent functions below model the height of a projectile in meters, t seconds after launch.

h ( t ) = 8 t 2 + 27 t + 20 h ( t ) = ( 8 t 5 ) ( t 4 )

Without graphing, determine at what time the height of the object is 0 meters (when the object is on the ground).

  1. 8 seconds
  2. 4 seconds
  3. 5 8 seconds
  4. 0 seconds

Additional Resources

Applying the Zero Product Property to Solve a Real-World Projectile Problem

We have previously looked at functions representing the motion of projectiles. We analyzed two equivalent expressions that define the same quadratic function, one in quadratic form and the other in factored form.

Now we’re going to look at a similar problem, but with the additional knowledge of how the function relates to the zero product property.

Let’s look at an example.

The equivalent functions below model the height, in feet, of a baseball tt seconds after being hit.

h(t)=15t2+44t+3h(t)=(15t1)(t3)h(t)=15t2+44t+3h(t)=(15t1)(t3)

Using the zero product property, the quadratic expression in factored form allows you to more easily find out when the height of the baseball is 0 feet.

Without graphing, we can determine at what time the baseball reaches the ground.

Let’s apply the zero product property to solve (15t1)(t3)=0(15t1)(t3)=0.

Set each factor equal to 0 and solve.

15t1=0t3=015t1+1=0+1t3+3=0+315t=1t=315t15=115t=11515t1=0t3=015t1+1=0+1t3+3=0+315t=1t=315t15=115t=115

So, t=115t=115 and t=3t=3. These values of tt represent the roots of the function. Because the object was launched at t=0t=0, the negative solution doesn’t make sense in this context of time. So the baseball reaches the ground in 3 seconds.

There is one benefit of knowing the function in quadratic form. By substituting 0 in for tt, we can easily determine at what height the baseball was hit, when t=0t=0 seconds.

h(t)=15t2+44t+3h(t)=15t2+44t+3

h(0)=15(0)2+44(0)+3h(0)=15(0)2+44(0)+3

h(0)=3h(0)=3

So, the baseball was hit at a height of 3 feet. Note that this point, (0,3)(0,3), is the yy-intercept of the function when graphed.

Try it

Try It: Applying the Zero Product Property to Solve a Real-World Projectile Problem

The functions below model the height of a projectile in feet, tt seconds after launch.

  • h(t)=7t2+34t+48h(t)=7t2+34t+48

  • h(t)=(7t8)(t6)h(t)=(7t8)(t6)

1. Without graphing, determine at what time the height of the object is 0 feet.

2. Determine the height of the object before it is launched.

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