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Problem 28 is a Pre-Algebra word problem that is considered a Medium-level ACT math question.
Try to work this problem in one minute. The correct answer is J .
In this problem we've got a tracking device attached to a bear, and a graph that shows
the distance between the bear and its den. We want to know which activity best matches
the graph. Let's label some points where things are happening
so we can talk about them: O, P, Q, and R. Now the horizontal axis is TIME and the vertical
axis is DISTANCE from the den. If the hours and distances were labeled on the axis, then
at a particular time, like 10:00 am, you could see how far the bear was from its den. It
looks like toward the middle of the day, the distance to the den is larger, and in the
afternoon the distance to the den gets smaller again.
We can see that the bear must have started out at its den, since the point O, first thing
in the morning, shows that the distance from the den, the y-value, is zero. So the bear
started at it's den. It begins to move away from its den, getting farther and farther
away, until it gets to the point represented by point P on this graph. At this point the
distance to the den is shown by the height up to that level of the graph.
Between P and Q, the bear's distance from its den doesn't change, that is, the y-value
stays the same. This means the bear is either staying in one place, or moving in a way that
causes the distance to remain unchanged, like, for example, walking in a circle with the
den at the center point. But that's pretty unlikely and it's not one of the answer choices,
so we'll assume the bear stays in one spot between P and Q.
After staying in one place for a while the bear begins to move again at point Q. This
time the bear is getting steadily closer to its den, since the y-value is getting smaller
and smaller. The bear reaches the den when the line reaches the horizontal axis at point
R. Now let's look at the answer choices.
Choice F is wrong because the bear DOES return to its den, at point R. Choice G could explain
the flat part of the graph between P and Q, but if the bear lost its collar, it's not
going to pick it up and carry it back to its den, so the flat part of the graph would just
continued forever. H doesn't describe the graph very well since it suggests that the
bear is running during P to Q, but staying the same distance from the den -- not likely.
J looks good. The bear walks directly to the berry patch, stops awhile, then walks directly
back to its den. K is wrong because if the bear had climbed a tree and stayed there for
the entire day, you'd see a long flat region like PQ, but you wouldn't see the bear moving
closer to the den. The best answer is J.
In problems like this, the x-axis is often used for TIME, and the y-axis could be a distance,
a depth, a population, or some quantity that varies with time. Parts of the graph with
positive slope, describe where the y-value is increasing as time increases. Flat regions
show where the y-value is staying the same as time goes on. And regions with negative
slope show where the y-value is decreasing with time. We saw all three of these cases
in this problem. The bear's distance to its den started at zero, grew steadily larger
for a while (that's a line segment with positive slope), stayed the same for a while (that's
a line segment with zero slope), and then grew steadily smaller (that's a line segment
with negative slope). The segments of the graph in problems like
this don't necessarily have to be line segments, they can be curved. Here's an example similar
to a problem that showed up on an old ACT exam. The graph represents the number of fish
in a pond as a function of time. The fish population increased rapidly at first, but
leveled off because the pond became overcrowded with too many fish and not enough to eat.