the height, h, in meters, of a gold ball t seconds after is is hit can be modelled by the function:
h (t) = 4.9t^2 + 32t + 0.2.
When is the height of the ball 15 m?
thanks so much Reiny! :D for correcting my mistake and for the answer!
To find when the height of the ball is 15 meters, we need to solve the equation:
15 = 4.9t^2 + 32t + 0.2
First, let's rearrange the equation to standard form:
4.9t^2 + 32t + 0.2 - 15 = 0
Combine like terms:
4.9t^2 + 32t - 14.8 = 0
Now, we can solve this quadratic equation. There are multiple ways to solve it, but one common method is by using the quadratic formula:
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our case:
a = 4.9, b = 32, and c = -14.8
Plugging these values into the quadratic formula, we get:
t = (-32 ± √(32^2 - 4 * 4.9 * -14.8)) / (2 * 4.9)
Simplifying further:
t = (-32 ± √(1024 + 287.84)) / 9.8
t = (-32 ± √(1311.84)) / 9.8
Now we can calculate the two possible values for t:
t1 = (-32 + √(1311.84)) / 9.8
t2 = (-32 - √(1311.84)) / 9.8
Using a calculator, we find:
t1 ≈ 0.757
t2 ≈ -6.626
Since time cannot be negative, we disregard t2. Therefore, the height of the ball is 15 meters approximately 0.757 seconds after it is hit.
To find when the height of the ball is 15 meters, we can solve the equation h(t) = 15.
The given function is h(t) = 4.9t^2 + 32t + 0.2, so we substitute 15 for h(t):
15 = 4.9t^2 + 32t + 0.2.
To solve this quadratic equation, we need to manipulate it into the standard form ax^2 + bx + c = 0.
Rearrange the equation to set it equal to zero:
4.9t^2 + 32t + 0.2 - 15 = 0.
Combining like terms, we have:
4.9t^2 + 32t - 14.8 = 0.
Now, there are a few methods you can use to solve this quadratic equation: factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the solutions.
The quadratic formula is:
t = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = 4.9, b = 32, and c = -14.8. Substituting these values into the formula, we get:
t = (-32 ± √(32^2 - 4 * 4.9 * -14.8)) / (2 * 4.9).
Simplifying further:
t = (-32 ± √(1024 + 294.4)) / 9.8.
t = (-32 ± √(1318.4)) / 9.8.
Now, we can calculate the two possible values of t by evaluating the expression:
t ≈ (-32 + √(1318.4)) / 9.8 and t ≈ (-32 - √(1318.4)) / 9.8.
Using a calculator, we find:
t ≈ 0.987 seconds and t ≈ -6.958 seconds.
Since time cannot be negative in this context, we can ignore the negative solution. Therefore, the height of the ball is approximately 15 meters after 0.987 seconds.
You have a typo, there should be a negative sign in front of 4.9t^2
so you want h(t) to be 5, ...
15 = -4.9t^2 + 32t + .2
4.9t^2 - 32t + 14.8 = 0
solve using the quadratic formula, you should get two positve answers, one will be the time on its upwards path, the other will be the time on its downwards path.
(should be appr. t=0.5 and t=6)