The height above ground level in meters of missile launched vertically, is given by h(t) = -16t cubed + 100t. Find at what time is the missile 72m above ground level? (t is time in second)
sorry. The equation of a trajectory is
h(t) = -16t^2 + 100t
so just solve
-16t^2 + 100t = 72
thanks
To find the time when the missile is 72 meters above ground level, we need to solve the equation h(t) = 72.
Let's substitute h(t) with its given formula:
-16t^3 + 100t = 72
Now, we need to solve this equation for t. Rearranging the equation, we get:
-16t^3 + 100t - 72 = 0
To solve this cubic equation, we can use various methods such as factoring, the rational root theorem, or numerical methods like Newton's method. Let's use factoring for simplicity:
Since 8 is a common factor among all three terms, we can divide the equation by 8:
-2t^3 + 12.5t - 9 = 0
Now, let's try to find a solution by trial and error, substituting different values for t:
If we substitute t = 1, we get -2(1)^3 + 12.5(1) - 9 = -2 + 12.5 - 9 = 1.5
If we substitute t = 2, we get -2(2)^3 + 12.5(2) - 9 = -2(8) + 25 - 9 = -16 + 25 - 9 = 0
Therefore, t = 2 is a solution to the equation.
Now, we can use polynomial long division to factorize the equation:
(t - 2)(-2t^2 - 4t + 4.5) = 0
Now, we can solve the quadratic equation -2t^2 - 4t + 4.5 = 0 to find the remaining solutions.
Using the quadratic formula, we have:
t = (-b ± √(b^2 - 4ac)) / 2a
For our quadratic equation, a = -2, b = -4, and c = 4.5. Substituting these values into the formula, we get:
t = (-(-4) ± √((-4)^2 - 4(-2)(4.5))) / (2(-2))
Simplifying this equation, we get:
t = (4 ± √(16 + 36)) / (-4)
t = (4 ± √52) / (-4)
t = (4 ± 2√13) / (-4)
This gives us two possible solutions:
t ≈ (4 + 2√13) / (-4)
t ≈ (4 - 2√13) / (-4)
Simplifying these solutions,
t ≈ -1 + 0.5√13
t ≈ -1 - 0.5√13
Therefore, the missile is approximately 72 meters above ground level at three different times:
t = 2 seconds,
t ≈ -1 + 0.5√13 seconds,
t ≈ -1 - 0.5√13 seconds.
To find at what time the missile is 72m above ground level, we need to solve the equation h(t) = 72.
Given that h(t) = -16t^3 + 100t, we can set up the equation:
-16t^3 + 100t = 72
To solve this equation, follow these steps:
Step 1: Move all terms to one side of the equation to set it equal to zero:
-16t^3 + 100t - 72 = 0
Step 2: Simplify the equation:
-16t^3 + 100t - 72 = 0
Step 3: Factor out the common factor (-8):
-8(2t^3 - 12.5t + 9) = 0
Step 4: Solve the equation inside the parentheses. Since this is a cubic equation, we may need a numerical method, such as Newton's method or the bisection method, to find the exact solutions. However, in this case, we can observe that t = 1 is a solution by trial and error.
Step 5: Divide the equation by (t - 1) to get the quadratic equation:
2t^2 + 2t - 9 = 0
Step 6: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. The quadratic equation can be factored as follows:
(2t + 3)(t - 3) = 0
Setting each factor equal to zero, we get two possible solutions:
2t + 3 = 0 -> 2t = -3 -> t = -3/2 (discard this solution since time cannot be negative)
t - 3 = 0 -> t = 3
Therefore, the missile is 72m above ground level at time t = 3 seconds.