Previous Problem Problem List Next Problem HW22 Optimization: Problem 1 (1 point) If 1100 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box Volume $=$ (include units) Preview My Answers Submit Answers You have attempted this problem 0 times. You have 10 attempts remaining Email Instructor
Solution
Step 1 ：Let the side length of the square base be \( x \) and the height of the box be \( h \).
Step 2 ：The volume of the box is \( V = x^2h \).
Step 3 ：The surface area of the material is \( A = x^2 + 4xh = 1100 \) square centimeters.
Step 4 ：Solve for \( h \) in terms of \( x \) using the surface area equation: \( h = \frac{1100 - x^2}{4x} \).
Step 5 ：Substitute \( h \) into the volume equation to get \( V(x) = x^2 \left( \frac{1100 - x^2}{4x} \right) \).
Step 6 ：Simplify the volume function to \( V(x) = \frac{1100x - x^3}{4} \).
Step 7 ：Take the derivative of \( V(x) \) with respect to \( x \) and set it equal to zero to find the critical points.
Step 8 ：Solve \( \frac{dV}{dx} = 0 \) to find the optimal value of \( x \) that maximizes the volume.
Step 9 ：The optimal side length of the base is \( x_{\text{optimal}} = 19.148541073664777 \) centimeters.
Step 10 ：The optimal height of the box is \( h_{\text{optimal}} = 9.574272159025412 \) centimeters.
Step 11 ：The maximum volume of the box is \( V_{\text{max}} = 3510.566061773223 \) cubic centimeters.
Step 12 ：The largest possible volume of the box with a square base and an open top, using 1100 square centimeters of material, is \(\boxed{3510.566061773223} \) cubic centimeters.
