Question

Answer the following questions about the word problem. Each question is worth 2 points. Use

your calculators table to help you answer the questions. Then explain your answer.
9. In an effort to catch a criminal; a superhero is going to leap over a building and take a short
cut down the alley. The function f(x) = -16x2160x gives the superhero's height in feet as a
function of time in seconds..
What is the vertex of the graph? What does that mean in terms of the word problem?
b. How high in the air is the superhero at 3 seconds? Explain where you got the answer.
c. What are the x intercept values of the graph? What does that mean in terms of the
graph?
d. The building is 425 feet high. Does the superhero make it over the building? If yes or
no, by how many feet does he clear or not clear the building?
e. How many seconds does it take for the superhero to be 256 feet in the air? Explain
where you got the answer.

Answer 1

The vertex of the graph is (-2160/(-32), 67875), representing the highest point reached by the superhero in leaping over the building. The superhero is 1944 feet high at 3 seconds. The x-intercepts of the graph are 0 and 135, representing the points where the superhero hits the ground before and after leaping over the building.

Step-by-step explanation:

The vertex of the graph is (-b/2a, f(-b/2a)). In this case, a = -16, b = 2160, and c = 0. Therefore, the x-coordinate of the vertex is -2160/(-32) = 67.5 and the y-coordinate is f(67.5) = -16(67.5)^2 + 2160(67.5) = 67875 feet. In terms of the word problem, the vertex represents the highest point the superhero reaches while leaping over the building.

To find how high the superhero is at 3 seconds, we substitute x = 3 into the function: f(3) = -16(3)^2 + 2160(3) = 1944 feet. Therefore, the superhero is 1944 feet high at 3 seconds.

The x-intercept values of the graph are the values of x for which f(x) = 0. To find these values, we set the function equal to zero: -16x^2 + 2160x = 0. Factoring out x, we get x(-16x + 2160) = 0. Therefore, the x-intercept values are x = 0 and x = 135. In terms of the graph, the x-intercepts represent the points where the superhero hits the ground before and after leaping over the building.

Since the building is 425 feet high and the vertex of the graph is at 67875 feet, the superhero easily clears the building by 67450 feet.

To find how many seconds it takes for the superhero to be 256 feet in the air, we set f(x) equal to 256 and solve for x: -16x^2 + 2160x = 256. Rearranging the equation, we get 16x^2 - 2160x + 256 = 0. Using the quadratic formula, we find that x = 0.15 seconds or x = 135 seconds. However, since we are only considering positive time, the superhero takes approximately 0.15 seconds to be 256 feet in the air.

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