Question

Kun-cha has 150 feet of fencing to make a corral for her horses. The
barn will be one side of the partitioned rectangular enclosure, as shown
in the diagram above. The graph illustrates the function that represents
the area that could be enclosed.
a. Write a function, A(x), that represents the area that can be enclosed
by the corral.
b. What information does the graph provide about the function?
c. Which ordered pair indicates the maximum area possible for the
corral? Explain what each coordinate tells about the problem.
d. What values of x will give a total area of 1000 ft2
? 2000 ft2
?

Answer 1

Critique the reasoning of others. Tim is the punter for the Bitterroot
Springs Mustangs football team. He wrote a function h(t) = 16t
2 + 8t + 1
that he thinks will give the height of a football in terms of t, the number
of seconds after he kicks the ball. Use two different methods to determine
the values of t for which h(t) = 0. Show your work. Is Tim’s function
correct? Why or why not?

Answer 2

a. The function A(x) that represents the area enclosed by the corral can be expressed as A(x) = x(150-2x), where x is the length of the side perpendicular to the barn.

b. The graph provides information about the maximum and minimum values of the function, as well as the shape of the graph, which indicates the behavior of the area as the length of the side changes.

c. The ordered pair that indicates the maximum area possible for the corral is (25, 1875). The x-coordinate (25) represents the length of the side perpendicular to the barn, and the y-coordinate (1875) represents the maximum area enclosed by the corral.

d. To find the values of x that will give a total area of 1000 ft^2 and 2000 ft^2, we can set up the equation x(150-2x) = 1000 and x(150-2x) = 2000 and solve for x. The solutions will give us the values of x for the corresponding total areas.