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# Points of intersection? @Asinus

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In a chemistry class, the students in lab derived a function to model the results of their experiment on the effect of heat on a chemical where x represents the number of minutes heat was applied at a constant temperature set by the lab instructions.  Their function was f(x)=16x/x^2+2 The teacher said the function should have been f(x)=12x/x^2+1
a) Was there ever any time at which these two functions were the same.  If so, when?
b) For what values of x is their derived function greater than the actual function?

Nov 10, 2017
edited by Julius  Nov 10, 2017

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a)   Was there ever a time when the first function  =  the second function?

Was there ever a time when         16x/x^2+2   =   12x/x^2+1       ?

16x / x2 + 2   =   12x / x2 + 1       I am guessing that this is supposed to be...

16x / (x2 + 2)   =   12x / (x2 + 1)        Cross-multiply.

16x(x2 + 1)  =  12x(x2 + 2)                 Distribute.

16x3 + 16x   =   12x3 + 24x               Subtract  12x3  from both sides.

4x3  + 16x  =  24x                             Subtract  24x  from both sides.

4x3 - 8x  =  0               Factor  4x  out of both terms.

4x(x2 - 2)  =  0             Set each factor equal to zero and solve for  x .

4x  =  0          or          x2 - 2  =  0

x  =  0            or          x2  =  2

x  =  ± √2

Yes, they are the same when  x = 0 ,  x = √2 ,  and  x = -√2 .

b)   For what values of  x  is  16x/x^2+2  greater than  12x/x^2+1    ?

16x / (x2 + 2)   >   12x / (x2 + 1)

Look at the graph here.

We can see  16x / (x2 + 2)  is greater than  12x / (x2 + 1)  for  x > √2  and for  -√2 < x < 0  .

You might just need to say positive possibilities:   x > √2

Nov 10, 2017

#1
+2

a)   Was there ever a time when the first function  =  the second function?

Was there ever a time when         16x/x^2+2   =   12x/x^2+1       ?

16x / x2 + 2   =   12x / x2 + 1       I am guessing that this is supposed to be...

16x / (x2 + 2)   =   12x / (x2 + 1)        Cross-multiply.

16x(x2 + 1)  =  12x(x2 + 2)                 Distribute.

16x3 + 16x   =   12x3 + 24x               Subtract  12x3  from both sides.

4x3  + 16x  =  24x                             Subtract  24x  from both sides.

4x3 - 8x  =  0               Factor  4x  out of both terms.

4x(x2 - 2)  =  0             Set each factor equal to zero and solve for  x .

4x  =  0          or          x2 - 2  =  0

x  =  0            or          x2  =  2

x  =  ± √2

Yes, they are the same when  x = 0 ,  x = √2 ,  and  x = -√2 .

b)   For what values of  x  is  16x/x^2+2  greater than  12x/x^2+1    ?

16x / (x2 + 2)   >   12x / (x2 + 1)

Look at the graph here.

We can see  16x / (x2 + 2)  is greater than  12x / (x2 + 1)  for  x > √2  and for  -√2 < x < 0  .

You might just need to say positive possibilities:   x > √2

hectictar Nov 10, 2017