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 would the rock hit the ocean? After how many seconds does the rock reach its maximum height? What is the maximum height of the rock?
After how many seconds would the rock hit the ocean? - The rock hits the ocean when its height above the ocean is 0.
After how many seconds does the rock reach its maximum height? - The rock reaches its maximum height at the vertex of the parabolic function.
What is the maximum height of the rock? - The maximum height of the rock is the y-coordinate of the vertex of the parabolic function.
After how many seconds would the rock hit the ocean?
Characteristic of the function needed: Finding the zeros of the function (the x-intercepts).
Required step: Set f(x) = 0 and solve for x.
After how many seconds does the rock reach its maximum height?
Characteristic of the function needed: Finding the x-value of the vertex of the parabola.
Required step: Divide the x-coordinate of the vertex formula by -b/2a.
What is the maximum height of the rock?
Characteristic of the function needed: Finding the y-coordinate of the vertex of the parabola.
Required step: Substitute the x-coordinate of the vertex into the function and solve for f(x).
To answer each question, we need to analyze the function f(x) = -16x^2 + 85x + 100:
1. "After how many seconds would the rock hit the ocean?" This question is asking for the time or value of x when the rock's height equals zero, which represents the point where it hits the ocean.
2. "After how many seconds does the rock reach its maximum height?" Here, we need to determine the time or value of x when the rock reaches its maximum height. This corresponds to the vertex of the parabolic function.
3. "What is the maximum height of the rock?" To find the maximum height, we need to evaluate the height function at the x-value that represents the vertex. The vertex is the highest point of the parabola and gives us the maximum height.
Now, let's solve each question individually:
1. To find when the rock hits the ocean, we set f(x) = 0 and solve for x:
-16x^2 + 85x + 100 = 0
We can use the quadratic formula to solve this equation, which is: x = (-b ± sqrt(b^2 - 4ac))/(2a)
Plugging in the values a = -16, b = 85, and c = 100, we can calculate the two possible solutions of x.
The positive solution represents the time it takes for the rock to hit the ocean.
2. To determine when the rock reaches its maximum height, we need to find the x-coordinate of the vertex of the parabola represented by f(x).
The formula to find the x-coordinate of the vertex of a quadratic function in the form ax^2 + bx + c is x = -b/(2a).
For our function, a = -16 and b = 85. Substituting these values into the formula will give us the x-value of the vertex.
3. The maximum height of the rock can be found by evaluating the height function at the x-value of the vertex, which we found in the previous step.
Substituting this x-value into the height function will give us the maximum height of the rock.