A miniature rocket is launched so that its height in meters after t seconds is given by the equation h=-4.9t^2+29.4t+7.
(a)draw a graph of the equation
(b)how high is the rocket after 5 seconds?
(c)what is the maximum height the rocket will attain?
(d)how long after the launch will the rocket hit the ground?
a)
http://www.wolframalpha.com/input/?i=h%3D-4.9t%5E2%2B29.4t%2B7
consider only the first quadrant
b) just plug t = 5 and evaluate
c) find the vertex of the parabola , and use the properties of the vertex.
d) set
4.9t^2 - 29.4t - 7 = 0
and solve using the quadratic equation formula
use only the positive result
To answer the given questions, we need to use the equation of the rocket's height over time, which is represented by h = -4.9t^2 + 29.4t + 7. Let's go through each question step by step:
(a) To draw a graph of the equation, we need to plot points on a coordinate plane. The x-axis represents time (t) in seconds, and the y-axis represents height (h) in meters. We can select a range of values for t, calculate the corresponding h values using the equation, and then plot the points on the graph. For simplicity, let's choose t values between 0 and 6 seconds:
When t = 0, h = -4.9(0)^2 + 29.4(0) + 7 = 7.
When t = 1, h = -4.9(1)^2 + 29.4(1) + 7 = 32.5.
When t = 2, h = -4.9(2)^2 + 29.4(2) + 7 = 38.2.
When t = 3, h = -4.9(3)^2 + 29.4(3) + 7 = 32.9.
When t = 4, h = -4.9(4)^2 + 29.4(4) + 7 = 16.6.
When t = 5, h = -4.9(5)^2 + 29.4(5) + 7 = 10.5.
When t = 6, h = -4.9(6)^2 + 29.4(6) + 7 = 14.6.
Plotting these points and connecting them smoothly will give you the graph of the equation.
(b) To find how high the rocket is after 5 seconds, we substitute t = 5 into the equation h = -4.9t^2 + 29.4t + 7:
h = -4.9(5)^2 + 29.4(5) + 7 = 10.5 meters. Therefore, the rocket will be 10.5 meters high after 5 seconds.
(c) To find the maximum height the rocket will attain, we need to determine the vertex of the quadratic equation. The vertex represents the highest point on the parabolic graph. The equation for the height of the rocket can be rewritten as h = -4.9t^2 + 29.4t + 7 in the form of h = at^2 + bt + c, where a = -4.9, b = 29.4, and c = 7.
The x-coordinate of the vertex, t = -b/(2a), can be calculated as:
t = -29.4/(2*(-4.9)) = -29.4/(-9.8) = 3.
Substituting t = 3 back into the equation, we can find the maximum height as follows:
h = -4.9(3)^2 + 29.4(3) + 7 = 44.1 meters. Therefore, the maximum height the rocket will attain is 44.1 meters.
(d) To determine how long after the launch the rocket will hit the ground, we need to find the time (t) when the height (h) is zero. This represents the point where the rocket lands and touches the ground.
Setting h = 0 in the equation -4.9t^2 + 29.4t + 7 = 0, we can solve for t using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = -4.9, b = 29.4, and c = 7. Substituting these values into the quadratic formula, we get:
t = (-29.4 ± √(29.4^2 - 4(-4.9)(7))) / (2*(-4.9)).
Simplifying the equation gives us two possible solutions:
t = (-29.4 ± √(861.24 + 137.2)) / (-9.8).
t = (-29.4 ± √998.44) / (-9.8),
t = (-29.4 ± 31.60) / (-9.8).
This results in two solutions:
t1 = (31.60 - 29.4) / (-9.8) ≈ 0.225 seconds, and
t2 = (-31.60 - 29.4) / (-9.8) ≈ 6.225 seconds.
The negative value is discarded since time cannot be negative. Therefore, the rocket will hit the ground approximately 6.225 seconds after the launch.
So, to summarize:
(a) Plot the points obtained from substituting different values of t into the equation on a graph to get the graph of h = -4.9t^2 + 29.4t + 7.
(b) The height of the rocket after 5 seconds is approximately 10.5 meters.
(c) The maximum height the rocket will attain is approximately 44.1 meters.
(d) The rocket will hit the ground approximately 6.225 seconds after the launch.