How to Find the Remaining Length of a Cone With Only a Part of It

I took three measurements for a certain plastic cup in my kitchen. One was of the circle on the bottom of the cup, and the other was the top(the larger opening) and the height in between the two. Assuming that the cup is a cone it should collapse to a single point. How can I go about figuring out the length of the remaining section? My maths level: just finished calculus II.

NOTE: This mini-project is just for the fun of maths, so there isn't the need to rush, also I'm perfectly fine being given a long answer that uses higher mathematics and I'll research it at long as it isn't too high above my maths level.

Measurements: Bottom Circle:(.75in diameter), Top Circle(3.5in diameter), and Height(5.5in).

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Draw a 2D picture of a cross-section of the cup. Let $x$ be the length of the remaining section of the tip of the cone. Then, by similar triangles in your picture, you will see that $$\frac{x}{0.75} = \frac{x+5.5}{3.5}$$ Solve to get $x$.