A team of researchers finds that there is a relationship between the overall health of a forest they're studying and the population of a certain beetle which feeds on the bark of old-growth trees. Due to the warming climate, the beetles are now able to reproduce one more time per year than they previously did, exacerbating the stress caused by the drought of the past few seasons. They use the function f (x) = 13x"+67 to model .6x5+39 the relationship between the two populations, where x is in tens of thousands of beetles and f is in thousands of trees. They note that there is a beneficial contribution to the ecosystem as a whole for a smaller population of the beetles, but that the number of healthy trees sharply declines as the beetle population rises too much. 1. Graph f (x) using technology. a. What appears to be the maximum number of healthy trees, to the nearest thousand? What is the estimated beetle population corresponding to this, to the nearest ten thousand? b. What is the slope of the tangent line to the graph of f (x) at the point corresponding to the maximum number of healthy trees (and the optimal population of beetles)? How do you know?
A team of researchers finds that there is a relationship between the overall health of a forest they're studying and the population of a certain beetle which feeds on the bark of old-growth trees. Due to the warming climate, the beetles are now able to reproduce one more time per year than they previously did, exacerbating the stress caused by the drought of the past few seasons. They use the function f (x) = 13x"+67 to model .6x5+39 the relationship between the two populations, where x is in tens of thousands of beetles and f is in thousands of trees. They note that there is a beneficial contribution to the ecosystem as a whole for a smaller population of the beetles, but that the number of healthy trees sharply declines as the beetle population rises too much. 1. Graph f (x) using technology. a. What appears to be the maximum number of healthy trees, to the nearest thousand? What is the estimated beetle population corresponding to this, to the nearest ten thousand? b. What is the slope of the tangent line to the graph of f (x) at the point corresponding to the maximum number of healthy trees (and the optimal population of beetles)? How do you know?
Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
Related questions
Question
![A team of researchers finds that there is a
relationship between the overall health of a forest
they're studying and the population of a certain
beetle which feeds on the bark of old-growth
trees. Due to the warming climate, the beetles are
now able to reproduce one more time per year
than they previously did, exacerbating the stress
caused by the drought of the past few seasons.
13x³+67
They use the function f (x)
to model
.6x5+39
the relationship between the two populations,
where x is in tens of thousands of beetles and f is
in thousands of trees. They note that there is a
beneficial contribution to the ecosystem as a
whole for a smaller population of the beetles, but
that the number of healthy trees sharply declines
as the beetle population rises too much.
1. Graph f (x) using technology.
a. What appears to be the maximum
number of healthy trees, to the nearest
thousand? What is the estimated beetle
population corresponding to this, to the
nearest ten thousand?
b. What is the slope of the tangent line to
the graph of f (x) at the point
corresponding to the maximum number
of healthy trees (and the optimal
population of beetles)? How do you
know?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F034edb7e-6612-4ac8-8429-66e4e199f366%2F629c106f-37ea-4864-9834-4f6386427447%2Fwr6bl4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A team of researchers finds that there is a
relationship between the overall health of a forest
they're studying and the population of a certain
beetle which feeds on the bark of old-growth
trees. Due to the warming climate, the beetles are
now able to reproduce one more time per year
than they previously did, exacerbating the stress
caused by the drought of the past few seasons.
13x³+67
They use the function f (x)
to model
.6x5+39
the relationship between the two populations,
where x is in tens of thousands of beetles and f is
in thousands of trees. They note that there is a
beneficial contribution to the ecosystem as a
whole for a smaller population of the beetles, but
that the number of healthy trees sharply declines
as the beetle population rises too much.
1. Graph f (x) using technology.
a. What appears to be the maximum
number of healthy trees, to the nearest
thousand? What is the estimated beetle
population corresponding to this, to the
nearest ten thousand?
b. What is the slope of the tangent line to
the graph of f (x) at the point
corresponding to the maximum number
of healthy trees (and the optimal
population of beetles)? How do you
know?
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