To find an expression for $y$ in terms of $x$, we need to use the given information that the curved surface area of the cone is equal to the surface area of the sphere.

Step 1: Write down the formulas

1. Surface area of the sphere with radius $x$: $\text{Surface area of sphere} = 4\pi x^2$

2. Curved surface area of the cone with radius $x$ and slant height $l$: $\text{Curved surface area of cone} = \pi x l$

We are given that these two areas are equal: $4\pi x^2 = \pi x l$

Step 2: Simplify the equation

Cancel $\pi$ and $x$ (where $x > 0$): $4x = l$

Thus, the slant height $l$ is: $l = 4x$

Step 3: Relate the slant height $l$ to the height $y$ of the cone

The slant height $l$, radius $x$, and height $y$ of the cone form a right triangle. Using the Pythagorean theorem: $l^2 = x^2 + y^2$

Substitute $l = 4x$ into this equation: $(4x)^2 = x^2 + y^2$

Simplify: $16x^2 = x^2 + y^2$

Step 4: Solve for $y^2$

Rearrange to isolate $y^2$: $y^2 = 16x^2 - x^2$

Simplify: $y^2 = 15x^2$

Take the square root of both sides: $y = \sqrt{15}x$

Final Answer:

$y = \sqrt{15}x$

Let me know if you need further clarifications or additional steps explained!

Related Questions:

1. How is the Pythagorean theorem applied to find the height of the cone?
2. What would happen if the sphere had a different radius?
3. How do you calculate the surface area of a full cone (including its base)?
4. What assumptions were made when solving for $y$ in terms of $x$?
5. How does changing the height of the cone affect the slant height?

Tip:

Always check if terms in equations can be simplified or canceled, like $\pi$ in this problem, to make solving easier.