At what point do the curves r1(t) = t^4 + 35t^2 and r2(s) = 7s + 3s^2 intersect?

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To find the intersection points of the curves r1(t) = t^4 + 35t^2 and r2(s) = 7s + 3s^2, we need to set the two equations equal to each other, since the intersection points occur where the values are the same for corresponding parameters.

First, we can rewrite the curves as:

  • r1(t): y = t^4 + 35t^2
  • r2(s): y = 7s + 3s^2

To find the intersection points, we’ll set:

t^4 + 35t^2 = 7s + 3s^2

Next, we can rearrange this equation to isolate one of the variables. However, we first need to express either s or t in terms of the other before solving for the intersection points:

  1. Make a substitution. Assume s is a function of t or solve for t in terms of s.
  2. After simplification, you may find a quadratic equation in one variable, which can be solved using the quadratic formula or by factoring.

For example, if we express s in terms of t and substitute back:

7s + 3s^2 = t^4 + 35t^2

Assuming s = kt (where k is some constant), substitute s back into the equation and solve for t.

Once you identify the t values of interest, plug them back into either curve function to find the corresponding s values, thus obtaining the intersection points:

(t, r1(t)) and (s, r2(s))

Note that these equations will yield specific points. Depending on how the parameters are defined (and the nature of the polynomial functions), you may find multiple intersection points or possibly none. Make sure to check the roots and consider their physical meanings, graphing the curves if necessary for insight.

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