A firework is set off from the roof of a building. Its height, h, above the ground in feet after t seconds is given by the function h(t)=-t^2+6t+16. How tall is the building? What is the maximum height of the firework? How long would it take the firework to reach ground? Explain how the key features of the function are related to your answers.
Help?
(a) what is h(0) ?
(b) as with all parabolas, the vertex is at t = -b/2a = -6/-2 = 3
(c) solve h(t) = 0
think of what you know about parabolas.
To find the height of the building, we need to determine the value of h when t is 0. In the given function, h(t) = -t^2 + 6t + 16.
When t is 0, the equation becomes h(0) = -(0)^2 + 6(0) + 16. Simplifying this expression, we get h(0) = 16. Therefore, the height of the building is 16 feet.
To find the maximum height of the firework, we need to identify the vertex of the quadratic function. The vertex is the highest or lowest point on a parabolic graph.
The vertex of the quadratic function h(t) = -t^2 + 6t + 16 can be found using the formula t = -b / (2a), where a, b, and c are coefficients of the quadratic. In this case, a = -1, b = 6, and c = 16.
Substituting these values into the formula, we get t = -6 / (2 * -1) = -6 / -2 = 3. Therefore, the firework reaches its maximum height at t = 3 seconds.
To find the maximum height, we substitute t = 3 into the function h(t). h(3) = -(3)^2 + 6(3) + 16 = -9 + 18 + 16 = 25. So, the maximum height of the firework is 25 feet.
To determine the time it takes for the firework to reach the ground, we need to find when the height becomes zero. This can be done by setting the function h(t) equal to zero and solving for t.
Setting -t^2 + 6t + 16 = 0, we have to solve this quadratic equation. By factoring, completing the square, or using the quadratic formula, we find that the roots of this equation are t = -2 and t = 8.
Since time cannot be negative in this context, we discard t = -2, leaving us with t = 8 as the solution. Therefore, it would take the firework 8 seconds to reach the ground.
In summary, the height of the building is 16 feet, the maximum height of the firework is 25 feet, and it takes 8 seconds for the firework to reach the ground. The height of the building is given by the y-intercept of the function, the maximum height is determined by the vertex of the parabolic graph, and the time it takes to reach the ground can be found by solving the equation for h(t) = 0.