Engineering Board Exam Reviewer · Optimization (Maxima and Minima) · hard
A rectangular box with a square base and no top is to have a total surface area of 108 square inches. Find the maximum possible volume.
- 108 cu in
- 144 cu in
- 96 cu in
- 120 cu in
Answer
A. 108 cu in
Solution
Let x = side of the base, h = height. S=x2+4xh=108 h=108-x2/4x. V=x2h=108x-x3/4. V'=108-3x2/4=0 x=6, h=3. Max volume =x2h=36(3)=108 cu in.
More Optimization (Maxima and Minima) questions
- A closed cylindrical can is to have a volume of 16π cubic inches. Find the radius that minimizes the total surface area.
- Find the point on the line y=2x+3 that is closest to the origin.
- A right circular cylinder is inscribed in a sphere of radius 6. Find the height of the cylinder that maximizes its volume.
- A wire 100 cm long is cut into two pieces. One piece is bent into a square and the other into a circle. How many centimeters of wire…
- An open rectangular box with a square base is to have a volume of 32 cubic feet. Find the dimensions that minimize the amount of material…
- A Norman window (a rectangle surmounted by a semicircle) is to have a perimeter of 20 ft. Find the radius of the semicircle that maximizes…
- A cone-shaped paper cup is to hold 100 cm3 of water. Find the height that minimizes the amount of paper needed to make the cup (i.e…
- A page is to contain 24 in2 of print. The margins at the top and bottom are 1.5 in each, and at the sides 1 in each. Find the dimensions…
- Find the maximum area of a rectangle that can be inscribed in a semicircle of radius 5, with one side on the diameter.
- A closed box with a square base is to have a surface area of 150 square inches. Find the dimensions that maximize the volume.