Solve h(t)=-16t^3+100t-50=0 please answer me!
surely by now you have learned to use the Quadratic Formula ...
If not, learn it and use it now.
post your work if you get stuck
To solve the equation h(t) = -16t^3 + 100t - 50 = 0, you can use the method of solving a cubic equation. Here are the steps:
Step 1: Start with the given equation:
-16t^3 + 100t - 50 = 0
Step 2: Divide the entire equation by the coefficient of the highest power term to make the coefficient of the highest power term equal to 1:
(-16t^3 + 100t - 50) ÷ (-16) = 0
This simplifies to:
t^3 - (100/16)t + (50/16) = 0
Step 3: The equation now has the form: t^3 + pt + q = 0, where p = (-100/16) and q = (50/16).
Step 4: Next, identify an initial guess for one of the roots of the cubic equation. You can use various methods like factoring, graphing, or approximation to find a reasonable initial guess. Let's assume the initial guess is t = 1.
Step 5: Now, use a numerical method like Newton's method or the cubic formula to refine the initial guess and find the exact roots of the equation. These methods involve iterative calculations. Since the calculations are complex, it may be more appropriate to use numerical software or calculators that can solve cubic equations.
There are several online cubic equation solvers or computer algebra systems that can help you find the roots. Input the equation in the proper format and let the software or tool solve it for you.
To solve the equation h(t) = -16t^3 + 100t - 50 = 0, you can use the method of factoring or the quadratic formula. However, notice that this equation is cubic (degree 3), so we will use a different method to solve it.
One way to solve cubic equations is by using the Rational Root Theorem. According to this theorem, if the equation has a rational root, it will be a divisor of the constant term (in this case, -50) divided by a divisor of the leading coefficient (in this case, -16).
The factors of -50 are ±1, ±2, ±5, ±10, ±25, and ±50. The factors of -16 are ±1, ±2, ±4, ±8, and ±16.
Now, we will check these possible rational roots by substituting them into the equation until we find one that makes h(t) equal to zero.
By trying different values for t from this list of possible roots, we find that t = 1 is a root of the equation.
To find the other roots, we divide the original equation by (t - 1) to obtain a quadratic equation, which can be solved using the quadratic formula.
Dividing the equation h(t) = -16t^3 + 100t - 50 by (t - 1), we have:
(-16t^3 + 100t - 50) / (t - 1) = 0
Simplifying further, we have:
-16t^2 - 16t + 116 = 0
Now, we can find the remaining roots by using the quadratic formula, which states that for any quadratic equation of the form ax^2 + bx + c = 0, the roots can be found using the following formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation -16t^2 - 16t + 116 = 0, we have a = -16, b = -16, and c = 116.
Using the quadratic formula, we can find the remaining roots as follows:
t = (-(-16) ± √((-16)^2 - 4(-16)(116))) / (2(-16))
Simplifying further, we have:
t = (16 ± √(256 + 7424)) / (-32)
t = (16 ± √7680) / (-32)
Now, we can simplify the expression inside the square root:
t = (16 ± √(256 * 30)) / (-32)
t = (16 ± 16√30) / (-32)
t = (1 ± √30) / (-2)
Therefore, the solutions to the equation h(t) = -16t^3 + 100t - 50 = 0 are:
t = 1, t = (1 + √30) / (-2), and t = (1 - √30) / (-2).