So I am confused on how to tackle this problem. Since the function starts with x = t I put the Z function with respect to x by substituting y for 5x^2 and then replaced x with t. I don't know what I'm doing wrong.
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You're making it too hard. First of all, as they tell you, r(t)=< t, some stuff, some other stuff >. So right away you know that x(t)=t in your parameterization. Then since y = 5x^2, you know that y=5t^2. You almost had the right notion since 5t^2 is what you used for z(t). Then the paraboloid gives you z in terms of x and y and you alread have expressions for those in terms of t, so you just need to substitute those. Where do you think you should put that?