Question

The density of trees (measured in trees per square mile) in a particular forest can be modeled by the following function:

p(x) = 500/((x+2)^3)



The forest is rectangular shaped. It borders a straight road for 8 miles while extending 3 miles away from the road. The x in the density function is measured in miles away from the road.


(a) Write a definite integral giving the total number of trees in the forest. Express your answer using the form below.

Answer 1

What part of this problem are you having trouble with? (writing the integral, solving it?)

Answer 2

in writing the integral

Answer 3

This problem statement is sort of wonky. There are a few ways to interpret it. I'd ask your teacher for clarification.

Answer 4

I know that the integral is 0 to 3

Answer 5

the integral is Density X Area

Answer 6

so you can't help me? thank you so much

Answer 7

Sorry, phone rang. I think it's a strangely worded problem. Here is one way to interpret it:

This gives us the number of trees in a 3 mile slice of one mile wide.
\[\huge 1\text{ mile} \times \int_{x=0}^{x=3}\frac{500}{(x+2)^2}dx\]

The forest is 8 miles wide...

That should be enough hint for now and the answer I get is a whole number, so I'm fairly certain this is how it's supposed to be interpreted.

Answer 8

why 500/((x+2)^2) and not 500/((x+2)^3)?

Answer 9

I think you are wrong?

Answer 10

@mathteacher1729 you are wrong

Answer 11

@Ldaniel because I pressed the 2 button instead of the 3 button. Sorry. :( Everything else about the interpretation seems to check out though.

Answer 12

no it's wrong it not 500/((x+2)^3)

Answer 13

it should be 500/((x+2)^3) X the Area. but i don't know how to find the area of of the forest

Answer 14

never-mind I got it . the total number of trees is its 500/((x+2)^3) X 8 miles X deltal x

Answer 15

The area of the forest is the area of the rectangle. 3 miles x 8 miles = 24 square miles.

Answer 16

\[\int\limits_{0}^{3}\frac{ 500 }{ (x+2)^3 }\times 8 dx\]

Answer 17

that's the right integral.

Answer 18

Bingo.

Answer 19

you need to practice your calculus mathteacher :)

Answer 20

thanks anyways :) you helped me a lot

Answer 21

Glad it was helpful, sorry for the typo where I wrote 2 instead of 3. The idea of my post was still correct. I gave you a 1 x 3 rectangle. You need an 8 x 3 rectangle. Multiply by 8 and you get your integral. :) nighty night.

Answer 22

yeah thanks