Let's go step by step to solve the problem. We are working with the surface area formula for a cylinder:

$A(r) = 2\pi r^2 + 16\pi r$

where:

• $r$ is the radius of the top (and bottom) of the cylinder,
• $h = 8$ inches is the height of the cylinder.

Part a: Domain of $A(r)$

The surface area $A(r)$ is defined in terms of the radius $r$, and the function involves both $r^2$ and $r$. Since the radius of a cylinder must be a positive real number (you can't have a negative or zero radius for a physical cylinder), the domain of $A(r)$ is:

$r > 0$

Thus, the domain of the function $A(r)$ is $(0, \infty)$.

Part b: Finding the inverse function $r(A)$

We want to solve for $r$ as a function of $A$. Start with the surface area formula:

$A = 2\pi r^2 + 16\pi r$

Rearrange the equation to set it equal to zero:

$2\pi r^2 + 16\pi r - A = 0$

This is a quadratic equation in terms of $r$. We can solve it using the quadratic formula, which is given by:

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For our equation $2\pi r^2 + 16\pi r - A = 0$, the coefficients are:

• $a = 2\pi$,
• $b = 16\pi$,
• $c = -A$.

Plugging these into the quadratic formula:

$r = \frac{-16\pi \pm \sqrt{(16\pi)^2 - 4(2\pi)(-A)}}{2(2\pi)}$

Simplify:

$r = \frac{-16\pi \pm \sqrt{256\pi^2 + 8\pi A}}{4\pi}$

Now, divide everything by $4\pi$:

$r = \frac{-16\pi}{4\pi} \pm \frac{\sqrt{256\pi^2 + 8\pi A}}{4\pi}$

This simplifies to:

$r = -4 \pm \frac{\sqrt{256\pi^2 + 8\pi A}}{4\pi}$

Since $r$ must be positive, we only take the positive root:

$r = \frac{-4 + \sqrt{256\pi^2 + 8\pi A}}{4\pi}$

Thus, the inverse function $r(A)$ is:

$r(A) = \frac{-4 + \sqrt{256\pi^2 + 8\pi A}}{4\pi}$

Part c: Calculate the radius when the surface area is 250 square inches

To find $r(250)$, substitute $A = 250$ into the inverse function we found earlier:

$r(250) = \frac{-4 + \sqrt{256\pi^2 + 8\pi \times 250}}{4\pi}$

Let's compute this step by step.

The radius of the cylinder when the surface area is 250 square inches is approximately $7.15$ inches.

Summary:

• Part a: The domain of $A(r)$ is $r > 0$.
• Part b: The inverse function is $r(A) = \frac{-4 + \sqrt{256\pi^2 + 8\pi A}}{4\pi}$.
• Part c: When the surface area is 250 square inches, the radius is approximately $7.15$ inches.

Would you like more details or have any questions?

Here are some related questions:

1. What is the volume of a cylinder given its radius and height?
2. How would the surface area change if the height were doubled?
3. Can the quadratic formula be used for non-circular cylinders?
4. How does the inverse function behave for large values of $A$?
5. Can we express the surface area in terms of the volume?

Tip: When solving quadratic equations, always check if the discriminant (the part under the square root) is positive to ensure real solutions.