A model rocket is launched with an initial velocity of 250 ft/s. The height, h, in feet, of the rocket t seconds after the launch is given by
h = −16t2 + 250t.
How many seconds after launch will the rocket be 730 ft above the ground? Round to the nearest hundredth of a second.
730=-16t^2+250t
16t^2-250t+730=0
Using the quadratic equation...
t= (250+-sqrt (250^2-4*16*730))/32
solve for t
To find how many seconds after the launch the rocket will be 730 ft above the ground, we need to set up an equation and solve for t.
In this case, the height of the rocket is given by the equation h = -16t^2 + 250t, where h represents the height in feet and t represents the time in seconds.
We want to find the time (t) when the height (h) is 730 ft. So, we can set up the equation as follows:
-16t^2 + 250t = 730
To solve this quadratic equation, we can rearrange it into the standard quadratic form:
-16t^2 + 250t - 730 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the values of t.
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -16, b = 250, and c = -730. Plugging these values into the quadratic formula, we get:
t = (-250 ± √(250^2 - 4*(-16)*(-730))) / (2*(-16))
Simplifying this expression, we have:
t = (-250 ± √(62500 - 46720)) / (-32)
t = (-250 ± √(15780)) / (-32)
The square root of 15780 is approximately 125.65. Plugging this value back into the equation, we have:
t ≈ (-250 ± 125.65) / (-32)
To find the two possible values of t, we evaluate the expression with both the plus and minus signs:
t1 ≈ (-250 + 125.65) / (-32) ≈ -3.8 (rounded to the nearest hundredth)
t2 ≈ (-250 - 125.65) / (-32) ≈ 11.72 (rounded to the nearest hundredth)
Therefore, there are two possible times when the rocket is 730 ft above the ground: approximately -3.8 seconds and approximately 11.72 seconds after the launch. However, since we cannot have a negative time in this context, we discard the negative value.
Thus, the rocket will be approximately 730 ft above the ground after approximately 11.72 seconds (rounded to the nearest hundredth of a second).