The height (in feet) attained by a rocket t seconds into flight is giving by the function
h(t)= -1/3t^3 + 16t^2 + 33t + 10
When is the rocket rising?
When is the rocket decreasing?
when does it reach its maximum height above the ground?
what is the maximum height?
what is the velocity and when does maximum velocity occur?
To determine when the rocket is rising, decreasing, and when it reaches its maximum height, we need to consider the derivative of the function h(t), which gives us the velocity of the rocket. To find the maximum height, we need to find the vertex of the parabola formed by the function h(t). Let's break down each question step by step:
1. When is the rocket rising?
The rocket is rising whenever the velocity is positive. To determine this, we need to find the intervals where the derivative of h(t) is greater than 0.
Derivative of h(t) = h'(t) = -t^2 + 32t + 33
Now, we set h'(t) > 0 and solve for t:
-t^2 + 32t + 33 > 0
We can solve this inequality by factoring or using the quadratic formula. After solving, we find the interval(s) where t satisfies the inequality. These intervals indicate when the rocket is rising.
2. When is the rocket decreasing?
The rocket is decreasing whenever the velocity is negative. To determine this, we find the intervals where the derivative of h(t) is less than 0.
Using the derivative h'(t) = -t^2 + 32t + 33, we set h'(t) < 0 and solve for t.
3. When does the rocket reach its maximum height above the ground?
To find when the rocket reaches its maximum height, we need to find the vertex of the parabola formed by the function h(t). The vertex gives us the maximum or minimum point on the graph of the function.
The vertex is found using the formula: t = -b/2a, where the function is in the form at^2 + bt + c.
For h(t) = -1/3t^3 + 16t^2 + 33t + 10, the coefficients are a = -1/3, b = 16, and c = 10.
4. What is the maximum height?
The maximum height is given by the value of h(t) at the t-value we found in step 3. Substituting the t-value into the function h(t) will give us the maximum height.
5. What is the velocity, and when does the maximum velocity occur?
We already have the expression for velocity, which is the derivative h'(t) = -t^2 + 32t + 33. To find the maximum velocity, we look for the highest point on the graph of h'(t), similar to finding the maximum height.