A rocket is shot into the air at a velocity of 48 ft/ sec. The equation is h(t) = -16t^2+48t+4
1) How far off the ground did the rocket start from?
2) Using the quadratic function, how high the rocket will be at its maximum point?
3) How long will it take the rocket to reach that maximum point?
4) Rewrite the function into vertex form.
5) Predict when the rocket will reach the ground.
6) Sketch the function. ( Label at least 5 points.)
1) To find how far off the ground the rocket started from, we need to find the height when time (t) is equal to 0. We can substitute t=0 into the equation h(t) = -16t^2+48t+4:
h(0) = -16(0)^2 + 48(0) + 4
h(0) = 4
Therefore, the rocket started 4 feet off the ground.
2) To find the maximum height of the rocket, we need to find the vertex of the quadratic function. The vertex of a quadratic function in the form h(t) = at^2 + bt + c is given by the formula:
t = -b/2a
In this case, a = -16 and b = 48. Substituting these values:
t = -48 / 2(-16)
t = -48 / -32
t = 1.5
Now, we substitute t = 1.5 into the equation to find the maximum height:
h(1.5) = -16(1.5)^2 + 48(1.5) + 4
h(1.5) = 54
Therefore, the rocket will reach a maximum height of 54 feet.
3) The time it takes for the rocket to reach its maximum height is given by the value of t we found in the previous step, which is 1.5 seconds.
4) To rewrite the function into vertex form, we complete the square. The vertex form of a quadratic function is given by h(t) = a(t - h)^2 + k, where (h, k) represents the coordinates of the vertex. Let's complete the square:
h(t) = -16t^2 + 48t + 4
h(t) = -16(t^2 - 3t) + 4
Now, we need to find the value that completes the square inside the parentheses. We take half of the coefficient of 't' (-3) and square it, which is 9. But since we multiply by -16, we need to subtract 9* (-16) from the equation.
h(t) = -16(t^2 - 3t + 9) - 9(-16) + 4
h(t) = -16(t - 1.5)^2 + 144 + 4
h(t) = -16(t - 1.5)^2 + 148
Therefore, the function rewritten in vertex form is h(t) = -16(t - 1.5)^2 + 148.
5) To determine when the rocket will reach the ground, we set the height function equal to zero and solve for t:
0 = -16t^2 + 48t + 4
This is a quadratic equation, and we can solve it by factoring, completing the square, or using the quadratic formula.
Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values, we have:
t = (-48 ± √(48^2 - 4(-16)(4))) / (2(-16))
t = (-48 ± √(2304 + 256)) / -32
t = (-48 ± √(2560)) / -32
After simplifying and taking the positive root:
t = (-48 + 16√10) / -32
This will give us the time it takes for the rocket to reach the ground. You can calculate the exact value using a calculator for the square root and the arithmetic operations.
6) To sketch the function, we can choose specific values of t and calculate the corresponding height h(t) using the given equation h(t) = -16t^2 + 48t + 4. Here are five points that can be used to sketch the function:
- t = 0, h(t) = 4 (starting point)
- t = 0.5, calculate h(t)
- t = 1, calculate h(t)
- t = 1.5 (maximum point), h(t) = 54 (highest point)
- t = 2, calculate h(t)
By plotting these points and connecting them smoothly, you can sketch the function. Remember, this is a downward-facing quadratic function, so it will be a parabola opening downwards.