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)=−16x2+85x+100
where f(x) represents the height of the rock as a function of time, x.
Match each question below with the characteristic of the function that is necessary to answer it.
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
After how many seconds does the rock reach its maximum height?
After how many seconds would the rock hit the ocean?
What is the maximum height of the rock?
After how many seconds does the rock reach its maximum height? - Time(s)
After how many seconds would the rock hit the ocean? - Time(s)
What is the maximum height of the rock? - Height(ft)
After how many seconds does the rock reach its maximum height?
To answer these questions, we need to analyze the quadratic function f(x) = -16x^2 + 85x + 100.
1. After how many seconds does the rock reach its maximum height?
To find the time when the rock reaches its maximum height, we need to determine the x-coordinate of the vertex of the quadratic function. The vertex can be found using the formula x = -b / (2a), where a is the coefficient of the x^2 term and b is the coefficient of the x term. In this case, a = -16 and b = 85. Substituting these values into the formula, we can calculate the time when the rock reaches its maximum height.
2. After how many seconds would the rock hit the ocean?
To determine when the rock hits the ocean, we need to find the time when the height (f(x)) of the rock is equal to zero. This can be done by setting f(x) = 0 and solving for x. The solutions (if any) will represent the time when the rock hits the ocean.
3. What is the maximum height of the rock?
The maximum height of the rock can be found by substituting the x-coordinate of the vertex (which we found in the first question) into the quadratic function f(x). By evaluating f(x) at the x-coordinate of the vertex, we can determine the maximum height of the rock.
So, to answer each question:
1. After how many seconds does the rock reach its maximum height?
We can use the formula x = -b / (2a) to find the time. Substitute a = -16 and b = 85 into the formula and calculate the value of x.
2. After how many seconds would the rock hit the ocean?
Set f(x) = 0 and solve for x to find the time when the rock hits the ocean.
3. What is the maximum height of the rock?
Substitute the x-coordinate of the vertex (which we found in the first question) into the quadratic function f(x) to determine the maximum height.