Question

A model rocket is fired from the side of a hill and its flight is given by the quadratic equation shown here:
ℎ(𝑡𝑡) = −4.9𝑡𝑡2 + 50𝑡𝑡 + 25
a) What is the initial height of the rocket? (1 mark)
b) What is the maximum height of the rocket? (1 mark)
c) How long after launch does the rocket reach the maximum height? (1 mark)
d) How long does it take for the rocket to reach the ground? (3 marks)
e) State the domain and range for this situation.

Answer 1

a) The initial height of the rocket is given by the constant term in the equation, which is 25.

b) The maximum height of the rocket can be found by determining the vertex of the quadratic equation. The formula for finding the x-coordinate of the vertex of a quadratic equation in the form 𝑓(𝑥) = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 is given by 𝑥 = -𝑏/(2𝑎). In this case, 𝑎 = -4.9 and 𝑏 = 50.
So, 𝑡 = -50/(2*(-4.9)) ≈ 5.102.
To find the maximum height, substitute this value into the equation: ℎ(𝑡) = −4.9(5.102)^2 + 50(5.102) + 25 ≈ 125.375.
Therefore, the maximum height of the rocket is approximately 125.375.

c) The time at which the rocket reaches the maximum height is given by the x-coordinate of the vertex, which we found to be 𝑡 = 5.102. Therefore, the rocket reaches its maximum height approximately 5.102 seconds after launch.

d) To find the time it takes for the rocket to reach the ground, we need to find the time at which the height ℎ(𝑡) equals zero. To do this, set ℎ(𝑡) = 0 in the quadratic equation and solve for 𝑡.
-4.9𝑡^2 + 50𝑡 + 25 = 0
Using the quadratic formula, we get 𝑡 = (-50 ± sqrt(50^2 - 4*(-4.9)*25))/(2*(-4.9)).
Calculating this, we find that 𝑡 ≈ -1.02 and 𝑡 ≈ 5.13.
Since the time cannot be negative in this context, we discard the 𝑡 ≈ -1.02 solution.
Therefore, it takes approximately 5.13 seconds for the rocket to reach the ground.

e) The domain of this situation is the set of all real numbers, as time (𝑡) can be any real number.
The range is the set of all real numbers greater than or equal to the initial height of the rocket, which is 25.

Answer 2

. Explain how the coordinates of the vertex help you find the maximum or minimum value of a quadratic
and where that maximum or minimum occurs.

Answer 3

The coordinates of the vertex of a quadratic equation in the form 𝑦 = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 0, help us find the maximum or minimum value of the quadratic and where that maximum or minimum occurs.

The x-coordinate of the vertex, denoted as 𝑥 = -𝑏/(2𝑎), gives us the value of 𝑥 at which the quadratic reaches its maximum or minimum. If 𝑎 is positive, the vertex represents the minimum point, and if 𝑎 is negative, the vertex represents the maximum point.

The y-coordinate of the vertex, denoted as 𝑦 = 𝑓(𝑥), gives us the value of 𝑦 at the maximum or minimum point. In other words, it gives us the maximum or minimum value of the quadratic.

By calculating the coordinates of the vertex, we can determine the maximum or minimum value and the corresponding 𝑥-value of the quadratic. This information is useful in various real-life situations, such as determining the maximum height of a projectile or optimizing the production of a product to maximize profits.