An object is thrown downward with an initial velocity of 4 feet per second. The relationship between the distance s it travels and time t is given by s = 4t + 16t2. How long does it take the object to fall 72 feet?
Well, to find out how long it takes for the object to fall 72 feet, we need to solve the equation s = 4t + 16t², where s is the distance traveled and t is the time.
So, let's plug in the value of s as 72 and solve for t:
72 = 4t + 16t²
Hmm... solving this equation requires a bit of math! Let's employ some clown magic, shall we?
*Throws a pie in the air while making funny faces*
Okay, okay, let's do some actual work. We have a quadratic equation here, so let's rearrange it and set it equal to zero:
16t² + 4t - 72 = 0
Now, we can either use the quadratic formula, or try to factor this equation.
*Starts juggling rubber chickens*
After some clownsultation with my clown friends, it appears that this equation doesn't factor easily. So, let's go with the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = 16, b = 4, and c = -72. Let's plug these values into the formula:
t = (-4 ± √(4² - 4(16)(-72))) / (2(16))
Phew, this is getting complicated! Let me grab some clown calculators to speed things up.
*Pulls out an abacus made of colorful balloons*
After carefully crunching the numbers, it turns out that t ≈ 2.32 seconds. So, it takes approximately 2.32 seconds for the object to fall 72 feet!
*Does a celebratory spin while honking a bicycle horn*
To find out how long it takes for the object to fall 72 feet, we need to set up an equation and solve for time(t).
The equation that represents the distance traveled by the object is: s = 4t + 16t^2.
We can set up the equation as follows: 72 = 4t + 16t^2.
Now, let's solve for t by rearranging the equation and setting it equal to zero:
16t^2 + 4t - 72 = 0.
Next, we can factor the quadratic equation:
4(4t^2 + t - 18) = 0.
Now we can solve for t by factoring the quadratic equation further:
4(t + 3)(4t - 6) = 0.
Setting each factor equal to zero, we get two possible values for t:
t + 3 = 0, which gives us t = -3.
4t - 6 = 0, which gives us t = 1.5.
Since time cannot be negative, the only valid solution is t = 1.5 seconds.
Therefore, it takes the object 1.5 seconds to fall 72 feet.
To find how long it takes the object to fall 72 feet, we need to determine the value of time (t) when the distance (s) is equal to 72.
The given relationship between distance (s) and time (t) is: s = 4t + 16t^2.
We can set up this equation as: 4t + 16t^2 = 72.
This is a quadratic equation, so let's rearrange it into standard form: 16t^2 + 4t - 72 = 0.
Now, we need to solve this quadratic equation to find the values of t.
We can factor out a common factor of 4 to simplify the equation:
4(4t^2 + t - 18) = 0.
Next, we can factor the quadratic equation:
4(t + 3)(4t - 6) = 0.
Now, we have two factors: (t + 3) = 0 and (4t - 6) = 0.
Using the zero product property, we can set each factor equal to zero and solve for t:
t + 3 = 0 --> t = -3
4t - 6 = 0 --> 4t = 6 --> t = 6/4 --> t = 1.5
Since time cannot be negative in this context, we ignore the negative value of t.
Therefore, the object takes 1.5 seconds to fall 72 feet.
4t + 16t^2 = 72
4t^2 + t - 18 = 0
(t-2)(4t+9) = 0
t = 2