A professional stunt performer at a theme park dives off a tower 21metres high into the water. His height above the ground at time tseconds is given by the equation h=-4.9t^2+21
a. How long does it take to reach the halfway mark?
To find the time it takes for the stunt performer to reach the halfway mark, we need to set the height equation equal to half the height of the tower, which is 21 ÷ 2 = 10.5 meters.
So, the equation becomes:
-4.9t^2 + 21 = 10.5
To solve for t, we first need to isolate the t term. Let's start by subtracting 10.5 from both sides:
-4.9t^2 + 21 - 10.5 = 0
Simplifying, we have:
-4.9t^2 + 10.5 = 0
Next, we can divide the entire equation by -4.9 to make the coefficient of the t^2 term 1:
t^2 - (10.5 / 4.9) = 0
Now, we can solve for t by factoring or using the quadratic formula. In this case, let's use the quadratic formula:
t = [-b ± √(b^2 - 4ac)] / (2a)
For our equation, a = 1, b = 0, and c = -(10.5 / 4.9).
Plugging in these values, we have:
t = [-(0) ± √((0)^2 - 4(1)(-(10.5 / 4.9)))] / (2(1))
Simplifying further:
t = ± √(0 + (42 / 4.9)) / 2
t = ± √(42 / 4.9) / 2
t = ± √(8.571) / 2
Taking the positive square root:
t = √8.571 / 2
t ≈ 1.849 seconds
Therefore, it takes approximately 1.849 seconds for the stunt performer to reach the halfway mark.
To find the time it takes for the stunt performer to reach the halfway mark, we need to set the height equation equal to half of the total height and solve for t.
Given height equation: h = -4.9t^2 + 21
Halfway mark height: h/2 = (21/2)
Now, substitute the value of the halfway mark height into the equation:
(21/2) = -4.9t^2 + 21
Next, rearrange the equation to isolate the term with t^2:
-4.9t^2 = (21/2) - 21
-4.9t^2 = (21 - 42)/2
-4.9t^2 = -21/2
Now, divide both sides of the equation by -4.9 to solve for t^2:
t^2 = (-21/2) / (-4.9)
t^2 = (21/2) / (4.9)
Finally, take the square root of both sides to find the value of t:
t = √(21/2) / √(4.9)
t ≈ 2.073 seconds
Therefore, it takes approximately 2.073 seconds for the stunt performer to reach the halfway mark.
Solve the equation
10.5 = -4.9 t^2 + 21
which can be rewritten
t^2 = 10.5/4.9 = 2.14 s^2
t = sqrt2.14 = ___ s