The problem is asking us to find the height $h$, in meters, of an open-top cylinder that maximizes the volume, given a surface area constraint of 61 square meters.

Analyzing the Problem:

1. Surface Area Constraint: The surface area $S$ of an open-top cylinder is given by: $S = \pi r^2 + 2\pi rh$ where:

   • $r$ is the radius of the base of the cylinder.
   • $h$ is the height of the cylinder.

   The equation consists of the area of the base ($\pi r^2$) and the lateral surface area ($2\pi rh$).

2. Volume of the Cylinder: The volume $V$ of the cylinder is given by: $V = \pi r^2 h$

3. Given:

   • Total surface area $S$ is 61 square meters.
   • We need to find $h$ that maximizes $V$.

Step-by-Step Solution:

1. Express $h$ in Terms of $r$: Using the surface area constraint: $\pi r^2 + 2\pi rh = 61$ Solve for $h$: $h = \frac{61 - \pi r^2}{2\pi r}$

2. Substitute $h$ into the Volume Formula: Substitute $h$ in the volume formula $V = \pi r^2 h$: $V = \pi r^2 \left( \frac{61 - \pi r^2}{2\pi r} \right)$ Simplify: $V = \frac{r(61 - \pi r^2)}{2}$ This is the volume equation in terms of $r$.

3. Maximize the Volume: Take the derivative of $V$ with respect to $r$ and set it to zero to find the critical points: $\frac{dV}{dr} = \frac{61}{2} - \frac{3\pi r^2}{2} = 0$ Solve for $r$: $\frac{3\pi r^2}{2} = \frac{61}{2}$ $r^2 = \frac{61}{3\pi}$ $r = \sqrt{\frac{61}{3\pi}}$

4. Find $h$ using $r$: Substitute $r$ back into the equation for $h$: $h = \frac{61 - \pi \left( \frac{61}{3\pi} \right)}{2\pi \sqrt{\frac{61}{3\pi}}}$ Simplifying this will give the height $h$.

Let me calculate $h$ numerically to provide the exact height rounded to the nearest tenth.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you on how to compute the height step-by-step:

1. Find the radius $r$:

   • Use the formula $r = \sqrt{\frac{61}{3\pi}}$.

2. Substitute $r$ into $h$:

   • Use $h = \frac{61 - \pi r^2}{2\pi r}$.

3. Evaluate and round $h$:

   • Calculate $h$ and round it to the nearest tenth.

If you want to proceed with manual calculations or upload a new image for further help, let me know!

Related Questions

1. How can we verify that the found height gives the maximum volume?
2. What would the volume of the cylinder be using this height?
3. How does the ratio between height and radius affect the maximum volume?
4. Can this problem be solved using optimization techniques like Lagrange multipliers?
5. What are the practical applications of this type of problem in real-world scenarios?

Tip: When solving optimization problems, always check the second derivative to confirm whether it is a maximum or minimum.