+956 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?
+9466 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
+9466 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
