3. A soccer ball is thrown from the top of a cliff. Its height, h, in meters, above the sea after t seconds can be modelled by h = -5t^2 + 21t + 120. How long will the ball take to fall 20 m below its initial height?
The equation for h(t) tells you that the initial height (when t=0) is 120 m.
just solve in the usual way:
-5t^2 + 21t + 120 = 120-20
To find the time it takes for the ball to fall 20 meters below its initial height, we need to set up the equation:
h = -5t^2 + 21t + 120
We want to find the time when the height is 20 meters below the initial height, so we substitute h with -20 in the equation:
-20 = -5t^2 + 21t + 120
Next, we rearrange the equation to put it in standard quadratic form:
0 = -5t^2 + 21t + 140
Now we can solve this quadratic equation for t. There are multiple ways to do this, but one common method is using factoring or the quadratic formula. In this case, let's use factoring.
To factor the quadratic equation, we need to find two numbers whose product is -5 * 140 = -700, and whose sum is 21.
After finding the numbers, we can rewrite the quadratic equation as:
0 = -5t^2 + 35t - 14t + 140
Now we can factor by grouping:
0 = (-5t^2 + 35t) + (-14t + 140)
0 = 5t(t - 7) - 14(t - 7)
We can see that we have a common binomial factor, (t - 7), so we can now factor that out:
0 = (5t - 14)(t - 7)
Now we can set each factor equal to zero and solve for t:
5t - 14 = 0
t - 7 = 0
Solving these equations, we get:
5t = 14
t = 14/5 = 2.8
t = 7
Since time cannot be negative, we discard the solution t = -7. Thus, the only valid solution is t = 2.8.
Therefore, the ball will take approximately 2.8 seconds to fall 20 meters below its initial height.
To find the time it takes for the soccer ball to fall 20 meters below its initial height, we can set up an equation and solve for t.
The height of the soccer ball after time t can be represented by the equation h = -5t^2 + 21t + 120.
We want to find the time it takes for the ball to reach a height of 20 meters below its initial height. This means we need to solve for t when h = -20.
Substituting -20 for h in the equation, we get:
-20 = -5t^2 + 21t + 120
To solve this quadratic equation, we need to rearrange it into the standard form, which is ax^2 + bx + c = 0.
So, let's rearrange the equation:
-5t^2 + 21t + 120 + 20 = 0
-5t^2 + 21t + 140 = 0
Now, we can solve this quadratic equation by factoring, using the quadratic formula, or by completing the square.
Alternatively, we can use the quadratic formula to find the values of t.
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a = -5, b = 21, and c = 140.
Using these values, we can substitute them into the quadratic formula and solve for t:
t = [-(21) ± √((21)^2 - 4(-5)(140))] / 2(-5)
t = [-21 ± √(441 + 2800)] / -10
t = [-21 ± √3241] / -10
Now, we need to find the values of t that make sense in this context.
Since time cannot be negative, we need to discard the negative value obtained from the ± sign.
Calculating the square root, we get:
√3241 ≈ 56.97
So, one possible value of t is:
t = (-21 + 56.97) / -10 ≈ -1.596
However, this value of t is not meaningful in this context since time cannot be negative. Therefore, we need to discard it.
Therefore, the ball will not fall 20 meters below its initial height.