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The population of birds in a park is inversely proportional to the square of the...

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torrip996 | Student, Grade 10 | Salutatorian

Posted September 21, 2013 at 5:32 PM via web

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The population of birds in a park is inversely proportional to the square of the temperature and directly proportional to the number of the day of the week starting with Sunday = 1. On Tuesday, the temperature is 20 C and there are 150 birds in the park. If temperature on Friday is 35 C what is the number of birds.

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted September 21, 2013 at 5:38 PM (Answer #1)

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The population of birds in a park is inversely proportional to the square of the temperature and directly proportional to the number of the day of the week starting with Sunday = 1.

On Tuesday, the temperature is 20 C and there are 150 birds in the park. The temperature is 20C and the day of the week is 3. On Friday the day of the week is 6 and the temperature is 35 C.

As the population is inversely proportional to the square of the temperature and directly proportional to the number of the day of the week, the population on Friday is `150/(35^2/20^2)*(6/3)` = `150*(400/1225)*(6/3) ~~ 97`

As the number of birds is a whole number, there are 97 birds in the park on Friday.

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steveschoen | College Teacher | (Level 3) Assistant Educator

Posted September 21, 2013 at 6:28 PM (Answer #2)

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hi, torrip,

I believe i will be able to assist with this.  First, one must consider what directly and inversely proportional mean.  Directly proportional means an equation would look like y = kx, where k is the constant of proportion (something like slope, but not exactly).  So, with the population of birds being directly porportional to the day of the week, we could have specifically;

P = kd

P = population

k = constant

d = day of the week.

Inversely proportional means the equation would look like y = k/x.  So, for this, with the population being inversely proportional to the square of the temperature, we would have;

P = k/(t^2)

t = temperature

when we "combine' these equations together, we get:

P = k*d/(t^2)

From here, there can be a couple of ways to solve this problem.  Let's do this.  with the initial given conditions:

t = 20, d = Tuesday, so 3, and p = 150

we can plug them into the equation to solve for k, the constant of variation:

P = k*d/(t^2)

150 = k*3/(20^2)

solving thise for k, we get k = 20,000.  For Friday, we are given:

d = 6 for Friday and t = 35

Then remember k = 20,000, we can plug those into the formula to find the population for Friday:

P = k*d/(t^2)

P = 20,000*6/(35^2) = approximately 97.96

So, there are going to be 97 birds, almost 98.

Good luck, torrip.  i hope this helps.

Till then,

Steve

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