Let’s reanalyze the problem and recalculate the volume to ensure accuracy.
We need to find the maximum volume of the open-top box defined by the volume function:
\[
V(x) = x(10 - 2x)(12 - 2x).
\]
### Step 1: Expand and Simplify
First, we'll simplify the volume function:
1. **Expand the terms:**
- \((10 - 2x)(12 - 2x) = 120 - 20x - 24x + 4x^2 = 120 - 44x + 4x^2.\)
2. **Thus, we have:**
\[
V(x) = x(120 - 44x + 4x^2) = 120x - 44x^2 + 4x^3.
\]
### Step 2: Find the Derivative
Next, find the derivative of \( V(x) \):
\[
V'(x) = 120 - 88x + 12x^2.
\]
### Step 3: Set the Derivative to Zero
To find critical points, set the derivative equal to zero:
\[
12x^2 - 88x + 120 = 0.
\]
### Step 4: Use the Quadratic Formula
Divide by 4:
\[
3x^2 - 22x + 30 = 0.
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- \( a = 3 \)
- \( b = -22 \)
- \( c = 30 \)
\[
x = \frac{22 \pm \sqrt{(-22)^2 - 4 \cdot 3 \cdot 30}}{2 \cdot 3}
= \frac{22 \pm \sqrt{484 - 360}}{6}
= \frac{22 \pm \sqrt{124}}{6}
= \frac{22 \pm 2\sqrt{31}}{6}
= \frac{11 \pm \sqrt{31}}{3}.
\]
### Step 5: Evaluate the Critical Points
Calculating the discriminant:
- \(\sqrt{31} \approx 5.57\).
So:
\[
x_1 = \frac{11 + \sqrt{31}}{3} \approx \frac{11 + 5.57}{3} \approx 5.19 \quad (\text{not feasible}).
\]
\[
x_2 = \frac{11 - \sqrt{31}}{3} \approx \frac{11 - 5.57}{3} \approx \frac{5.43}{3} \approx 1.81.
\]
### Step 6: Calculate Volume at \( x = 1.81 \)
Now substitute \( x = 1.81 \) back into the volume formula:
\[
V(1.81) = 1.81(10 - 2 \cdot 1.81)(12 - 2 \cdot 1.81).
\]
Calculating the dimensions:
- \( 10 - 2 \cdot 1.81 = 10 - 3.62 = 6.38 \)
- \( 12 - 2 \cdot 1.81 = 12 - 3.62 = 8.38 \)
So the volume is:
\[
V(1.81) = 1.81 \cdot 6.38 \cdot 8.38.
\]
Now calculating this:
1. Calculate:
- \( 6.38 \cdot 8.38 \approx 53.441 \)
2. Then:
- \( V(1.81) = 1.81 \cdot 53.441 \approx 96.749 \).
### Conclusion
Thus, when rounded to one decimal place, the maximum volume is approximately:
\[
\boxed{96.8 \text{ cubic inches}}.
\]
Thank you for your patience, and I apologize for the earlier miscalculation!