Question

A ball is thrown from an initial height of 3 meters with an initial upward velocity of 13 m/s. The ball's height h(in meters) after 1 seconds is given by the following.h - 3+131-512Find all values of 1 for which the ball's height is 8 meters.Round your answer(s) to the nearest hundredth.(If there is more than one answer, use the "or" button.)secondsinitialХ5?heightground

Answer 1

In order to find all values of t for which the height is 8 meters, we can use h = 8 in the given equation, and then solve it for the variable t:

h = 3 + 13t - 5t²

8 = 3 + 13t - 5t²

Now, we can rewrite this equation in the standard form for a quadratic equation:

8 = 3 + 13t - 5t²

5t² - 13t - 3 + 8 = 0

5t² - 13t + 5 = 0

Then, we can use Bhaskara's Formula to find the values of t for which this equation is true:

t = [-b ± √ (b²-4ac)]/(2a)

In this case, we have:

a = 5

b = -13

c = 5

So, we find:

t = {-(-13) ± √ [(-13)²-4*5*5]}/(2*5)

= [13 ± √(169 - 100)]/10

= (13 ± √69)/10

Thus, we have two values of t:

t₁ = (13 + √69)/10 ≅ (13 + 8.3066)/10 ≅ 2.13

t₂ = (13 - √69)/10 ≅ (13 - 8.3066)/10 ≅ 0.47

Therefore, values of t for which the ball's height is 8 meters are

2.13

and

0.47