solve for t. h(t)=16t^2+60t+30
To solve for t in the equation h(t) = 16t^2 + 60t + 30, we can set h(t) equal to a specific value and solve for t.
Let's solve for h(t) = 0:
0 = 16t^2 + 60t + 30
To simplify the equation, we can divide all terms by 2:
0 = 8t^2 + 30t + 15
Now, let's solve this quadratic equation. We can either factor it or use the quadratic formula.
Since the equation doesn't factor easily, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 8, b = 30, and c = 15.
t = (-30 ± √(30^2 - 4(8)(15))) / (2(8))
Simplifying further:
t = (-30 ± √(900 - 480)) / 16
t = (-30 ± √420) / 16
Now, we have two possible solutions for t:
t = (-30 + √420) / 16
and
t = (-30 - √420) / 16
These are the solutions for t in the equation h(t) = 16t^2 + 60t + 30.
To solve for t, we start by setting the equation h(t) equal to zero:
16t^2 + 60t + 30 = 0
Next, we attempt to factor the equation:
2(8t^2 + 30t + 15) = 0
Now, we can try to factor the quadratic equation:
2(2t + 3)(4t + 5) = 0
Setting each factor equal to zero gives two possible solutions:
2t + 3 = 0 or 4t + 5 = 0
Solving these equations for t, we find:
2t = -3 or 4t = -5
t = -3/2 or t = -5/4
So, the possible values of t that solve the equation are t = -3/2 or t = -5/4.
To solve for t in the equation h(t) = 16t^2 + 60t + 30, we need to set h(t) to zero and then solve for t.
Step 1: Set h(t) to zero.
0 = 16t^2 + 60t + 30
Step 2: Rearrange the equation.
16t^2 + 60t + 30 = 0
Step 3: Simplify the equation if possible. In this case, we cannot simplify further.
Step 4: Solve the quadratic equation. There are different methods to solve quadratic equations. The most common methods are factoring, completing the square, and using the quadratic formula. Let's use the quadratic formula in this case.
The quadratic formula is:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
In our equation, a = 16, b = 60, and c = 30.
Plugging in the values, we get:
t = (-60 ± sqrt(60^2 - 4*16*30)) / (2*16)
Step 5: Simplify the equation further.
t = (-60 ± sqrt(3600 - 1920)) / 32
t = (-60 ± sqrt(1680)) / 32
Step 6: Calculate the square root of 1680.
t = (-60 ± sqrt(1680)) / 32
t ≈ (-60 ± 40.99) / 32
So, t ≈ (-60 + 40.99) / 32 or t ≈ (-60 - 40.99) / 32
This gives us two possible solutions:
t ≈ (-19.01) / 32 ≈ -0.593
t ≈ (-100.99) / 32 ≈ -3.156
Therefore, the solutions for t are approximately -0.593 and -3.156.