A rock is thrown upward at a speed of 85 ft/sec from the top of a 100-foot-high cliff overlooking the ocean. The rock's height above the ocean can be modeled by the function f(x) = - 16x ^ 2 + 85x + 100 where (x) represents the height of the rock as a function of time , . Match each question below with the characteristic of the function that is necessary to answer it.
It seems that there is some information missing in your question. Could you please provide the questions that need to be matched with the characteristics of the function f(x)?
1) What is the initial height of the rock?
2) How long does it take for the rock to reach its maximum height?
3) What is the maximum height the rock reaches?
4) When does the rock hit the ocean?
5) What is the rock's height after 2 seconds?
Characteristics of the function needed to answer each question:
1) Constant term (100 in this case)
2) Maximum value (vertex of the parabola)
3) Maximum value (vertex of the parabola)
4) The root(s) of the function
5) Substituting the time value in the function
To match each question with the characteristic of the function necessary to answer it, we first need to understand the given function and its meaning:
The given function is f(x) = -16x^2 + 85x + 100, where x represents the height of the rock as a function of time. This function is a quadratic function, in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants.
Now, let's go through each question and determine the characteristic of the function that is necessary to answer it:
1. At what time will the rock reach its maximum height?
This question asks about the time when the rock reaches its peak height. To answer this, we need to find the vertex of the parabola represented by the function. The x-coordinate of the vertex can be determined using the formula x = -b / (2a) in the quadratic equation.
2. How high above the ocean will the rock be after 2 seconds?
This question asks for the height of the rock at a specific time (2 seconds in this case). To answer this, we need to substitute the given time (x = 2) into the function and calculate the resulting height f(x).
3. When will the rock hit the ocean?
This question asks about the time when the rock reaches the ocean height (0 feet). To answer this, we need to find the x-intercept(s) of the function, which represents the time(s) when the rock hits the ocean. The x-intercepts can be found by setting the function f(x) equal to zero and solving the quadratic equation.
4. How long will the rock take to hit the ocean after being thrown?
This question asks for the time difference between when the rock was thrown from the cliff and when it hits the ocean. To answer this, we need to find the time it takes for the rock to reach the ocean by subtracting the initial time (when the rock was thrown) from the time it hits the ocean.
So, to match each question with the characteristic of the function necessary to answer it:
1. At what time will the rock reach its maximum height? (x-coordinate of the vertex)
2. How high above the ocean will the rock be after 2 seconds? (substituting x = 2 into the function)
3. When will the rock hit the ocean? (x-intercepts of the function)
4. How long will the rock take to hit the ocean after being thrown? (time difference)