If a stone is thossed from the top of a 230 meter building, the height of the ston as a function of time is given by h(t)=9.8t^2-10t+230 t is seconds and h in meters. How many seconds will the stone hit the ground?Round to the nearest hundredreths place include units.
To find the time it takes for the stone to hit the ground, we need to determine when the height of the stone, represented by the function h(t), becomes zero.
The given function for the height of the stone as a function of time is h(t) = 9.8t^2 - 10t + 230. We set h(t) equal to zero and solve for t:
0 = 9.8t^2 - 10t + 230
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Comparing the equation to the quadratic formula, we have:
a = 9.8
b = -10
c = 230
Plugging these values into the quadratic formula:
t = (-(-10) ± √((-10)^2 - 4 * 9.8 * 230)) / (2 * 9.8)
t = (10 ± √(100 + 9016)) / 19.6
t = (10 ± √9116) / 19.6
Calculating the square root:
√9116 ≈ 95.5
t ≈ (10 ± 95.5) / 19.6
Now we can determine the two possible values of t:
t₁ ≈ (10 + 95.5) / 19.6 ≈ 5.8 seconds
t₂ ≈ (10 - 95.5) / 19.6 ≈ -4.8 seconds
Since time cannot be negative in this context, we discard the negative solution: t₂ ≈ -4.8 seconds.
Therefore, the stone will hit the ground approximately 5.8 seconds after it was thrown.