To solve this problem, we'll use the concept of tangent line approximation, which is essentially linear approximation. Here’s the step-by-step approach:

Step 1: Define the Surface Area Function

The formula for the surface area $S$ of a cone with height $h$ and base radius $r$ is given by: $S = \pi r^2 + \pi r \sqrt{r^2 + h^2}$

Step 2: Set Up the Original Values and Changes

Given:

• Initial height $h = 8$ cm
• Initial radius $r = 6$ cm
• Increase in height $\Delta h = 0.32 \times h = 0.32 \times 8 = 2.56$ cm

Thus, the new height $h' = h + \Delta h = 8 + 2.56 = 10.56$ cm.

Step 3: Find the Derivative of $S$ with Respect to $r$

Since we're using tangent line approximation, we need the partial derivatives of $S$ with respect to $r$ and $h$.

The derivative of $S$ with respect to $r$, denoted $\frac{\partial S}{\partial r}$, is: $\frac{\partial S}{\partial r} = 2\pi r + \pi \sqrt{r^2 + h^2} + \frac{\pi r^2}{\sqrt{r^2 + h^2}}$

Step 4: Find the Derivative of $S$ with Respect to $h$

The derivative of $S$ with respect to $h$, denoted $\frac{\partial S}{\partial h}$, is: $\frac{\partial S}{\partial h} = \frac{\pi r h}{\sqrt{r^2 + h^2}}$

Step 5: Apply Linear Approximation

For the surface area $S$ to remain the same, the total differential $dS$ must be zero: $dS = \frac{\partial S}{\partial r} \, dr + \frac{\partial S}{\partial h} \, dh = 0$

Thus, $\frac{\partial S}{\partial r} \, dr = - \frac{\partial S}{\partial h} \, dh$

Solving for $dr$ (the change in $r$): $dr = -\frac{\frac{\partial S}{\partial h} \, dh}{\frac{\partial S}{\partial r}}$

Step 6: Substitute Values

Substitute $r = 6$, $h = 8$, and $dh = 2.56$ into the derivatives and calculate $dr$. Then, find the new radius $r' = r + dr$.

I'll perform the calculations for $dr$ and find the approximate value of the new radius.The estimated new value of the radius $r$ to keep the surface area constant after increasing the height is approximately $r \approx 5.52$ cm.

Would you like further details or have any questions?

Here are some additional related questions for practice:

1. How would the result change if the height increased by 50% instead of 32%?
2. Can you derive a general formula for the radius change $dr$ in terms of $dh$, $r$, and $h$?
3. What is the volume of the cone before and after the height increase, using the new radius?
4. How does the approximation accuracy vary if we increase or decrease both $r$ and $h$ proportionally?
5. What happens to the surface area if only the height changes without adjusting the radius?

Tip: Tangent line approximations work best for small changes in variables. For larger changes, other methods may provide more accurate results.