Height of an object
Initial explanation
We know that if we replace t by some number in the equation:
f(t) = -5t² + 20t + 60
the result will be the height of a launched object.
When the object hits the ground its height is 0.
Then
-5t² + 20t + 60 = 0
We want to solve the highlighted equation fot t, the values of t that make it result in 0 will be the seconds when the object is in the ground.
Solving the equation for t
In order to solve the equation
-5t² + 20t + 60 = 0
we want to factor the left side:
Step 1- Common factor
The common factor of -5t², 20t and 60 is -5:
(-5) · 1 = -5
(-5) · (-4) = 20
(-5) · (-12) = 60
Then
-5t² + 20t + 60 = -5(t² -4t - 12)
Then, replacing the equation
-5t² + 20t + 60 = 0
↓
-5(t² -4t - 12) = 0
↓ taking -5 to the right side
(t² -4t - 12) = 0/(-5) = 0
t² - 4t - 12 = 0
Step 2- Factoring a trinomial
We continue the factoring.
We want to factor t² - 4t - 12.
It should look something like:
t² - 4t - 12 = (t + _ )(t + _ )
To complete it we will use two numbers whose:
- product is -12 (third term t² - 4t - 12)
- sum is -4 (second term t² - 4t - 12)
The pair of numbers whose product is -12 are:
(-1) · 12 = -12
1 · (-12) = -12
(-2) · 6 = -12
2 · (-6) = -12
(-3) · 4 = -12
3 · (-4) = -12
We add them, the pairs whose result is -4 is the pair we will use:
-1 + 12 = 11
1 - 12= -11
-2 + 6 = 4
2 - 6 = -4
-3 + 4 = 1
3 - 4 = -1
Then, we will use 2 and -6 (their product is the third term and their sum is the second term):
t² - 4t - 12 = (t + _ )(t + _ )
↓
t² - 4t - 12 = (t + 2)(t - 6)
Then, using the equation we had:
t² - 4t - 12 = 0
↓
(t + 2)(t - 6) = 0
Step 3- finding the possible t values
When a product of two numbers is 0, it means that one of them is 0:
In this case
We have that
(t + 2)(t - 6) = 0
then
t + 2 = 0 or t - 6 = 0
This means that t could have two possible values:
t + 2 = 0 → t = -2
or
t - 6 = 0 → t = 6
Since t means the seconds the object takes to hit the ground, it cannot have a negative value, because it would mean that it happened in the past. so t cannot be -2
Answer: It takes to the object 6 seconds to hit the ground t = 6