A ball is launched into the air from below a cliff. after t seconds its height above the cliff top is h metres, and is given by the equation

h =  − 4.9t 2 + 19.6t − 14.6.

Calculate, to the nearest metre, the maximum height the ball achieves above the cliff top.

Well, let's solve this problem step by step. First, we need to find the maximum height the ball achieves above the cliff top. This occurs when the ball reaches its peak, which is the highest point on its trajectory.

To find this, we'll use a little mathematics. The equation you gave us is quadratic, where h represents the height and t represents time.

Now, the maximum value of a quadratic equation occurs at its vertex. The t-coordinate of the vertex can be found using the formula t = -b/(2a), where a, b, and c are the coefficients of the quadratic equation.

In our case, a = -4.9, b = 19.6, and c = -14.6. Substituting these values into the formula, we get:

t = -19.6 / (2 * -4.9)
t = -19.6 / -9.8
t = 2 seconds

So, the ball reaches its peak at 2 seconds.

Now, let's find the maximum height by substituting t = 2 into the equation:

h = -4.9(2)^2 + 19.6(2) - 14.6
h = -4.9(4) + 39.2 - 14.6
h = -19.6 + 39.2 - 14.6
h = 5 meters

Therefore, the maximum height the ball achieves above the cliff top is approximately 5 meters.

To find the maximum height the ball achieves above the cliff top, we need to determine the vertex of the quadratic equation. The vertex represents the highest or lowest point of the parabolic curve.

The equation for the height of the ball above the cliff top is given by: h = -4.9t^2 + 19.6t - 14.6.

We can write this equation in the form: h = -a(t - b)^2 + c, where a, b, and c are constants.

Comparing this with the given equation, we can see that a = -4.9, b = (19.6 / (2 * -4.9)) = -2, and c = -14.6.

Now, we can determine the vertex of the equation using the formula:

t = -b / (2a)

Substituting the values, we get:

t = -(-2) / (2 * -4.9)
t = 0.204

To find the maximum height, we substitute this value of t into the equation:

h = -4.9(0.204)^2 + 19.6(0.204) - 14.6
h ≈ 21 metres

Therefore, the maximum height the ball achieves above the cliff top is approximately 21 meters.

To find the maximum height the ball achieves above the cliff top, we need to determine the vertex of the parabolic equation given.

The equation for the height of the ball above the cliff top is h = -4.9t^2 + 19.6t - 14.6.

The vertex of a parabolic equation in the form y = ax^2 + bx + c can be found using the formula x = -b / (2a). In this case, a = -4.9 and b = 19.6.

Substituting these values into the formula, we get x = -19.6 / (2 * -4.9) = -19.6 / -9.8 = 2.

So, the ball reaches its maximum height after 2 seconds.

To find the maximum height, we need to substitute the value of t = 2 into the equation.

h = -4.9 * (2)^2 + 19.6 * (2) - 14.6 = -19.6 + 39.2 - 14.6 = 4.6.

Therefore, the maximum height the ball achieves above the cliff top is approximately 5 meters (rounded to the nearest meter).