13.8.33 Question Help
If an object is propelled upward from a height of s feet at an initial velocity of v feet per second, then its height h after t seconds is given by the equation h=−16t squared + vt +s, where h is in feet. If the object is propelled from a height of 4 feet with an initial velocity of 64 feet per second, its height h is given by the equation h=−16t squared + 64t + 4.
After how many seconds is the height 6464 feet?
just solve
-16t^2 + 64t + 4 = 6464
If an object is propelled upward from a height of s feet at an initial velocity of v feet per second, then its height h after t seconds is given by the equation h=-16t^2 + vt + s, where h is in feet. If the object is propelled from a height of 8 feet with an initial velocity lf 64 feet per second, its height h is given by the equation h=16t^2 + 64t +8. After how many seconds is the height 64 feet?
To find the number of seconds when the height is 6464 feet, we need to solve the equation:
h = -16t^2 + 64t + 4
Setting h equal to 6464, we have:
6464 = -16t^2 + 64t + 4
Rearranging the equation, we get:
-16t^2 + 64t - 6456 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula. Since factoring may not be possible in this case, we will use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = -16, b = 64, and c = -6456.
Substituting these values into the formula, we get:
t = (-64 ± √(64^2 - 4(-16)(-6456))) / 2(-16)
Simplifying further:
t = (-64 ± √(4096 - 41472))/(-32)
t = (-64 ± √(-37376))/(-32)
Since the term inside the square root is negative, the quadratic equation has no real solutions. This means that the object will never reach a height of 6464 feet.
To find out after how many seconds the height is 6464 feet, we need to solve the equation h = -16t^2 + 64t + 4 for t when h = 6464.
Substituting the value of h into the equation, we get:
6464 = -16t^2 + 64t + 4
Now, we have a quadratic equation. To solve it, we can set it equal to zero:
-16t^2 + 64t + 4 - 6464 = 0
Simplifying further:
-16t^2 + 64t - 6460 = 0
Now, we can solve this equation for t. There are different methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula.
In this case, let's use the quadratic formula:
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac))/(2a)
In our equation, a = -16, b = 64, and c = -6460. Substituting these values into the quadratic formula, we get:
t = (-64 ± √(64^2 - 4(-16)(-6460)))/(2(-16))
Simplifying further:
t = (-64 ± √(4096 - 412480))/(-32)
t = (-64 ± √(-408384))/(-32)
Since the expression inside the square root is negative, it means there are no real solutions to this equation. Therefore, there is no value of t for which the height is exactly 6464 feet in this scenario.