Find two nonnegative numbers whose sum is $50$ and the product of the square of one $(x)$ and the cube of the other $(y)$ is a maximum. The number $y$ is:

a$)40$

b$)30$

c$)25$

d$)20$

e$)10$

A rectangular sheet of cardboard $3$ m by $8$ m is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut to obtain the largest possible volume?

a$)\frac{2}{3}$

b$)\frac{3}{2}$

c$)\frac{1}{3}$

d$)0\mathrm{.}35$

e$)\frac{3}{4}$

A divided field is to be constructed with $5000$ feet of fence as shown. For what value of $x$ will the area be a maximum $?$

$0$

$24.$ Find the radius of the right circular cone with least volume that can be circumseribed about the sphere of radius $\sqrt[\phantom{2}]{2}.$

a$)3\sqrt[\phantom{2}]{2}$

b$)4\sqrt[\phantom{2}]{2}$

c$)4$

d$)8\sqrt[\phantom{2}]{2}$

e$)2$

$25.$ A piece of wire $40$ cm long is cut into two pieces. One piece is bent into a square and the other piece is shaped into a circle. To minimize the total area enclosed, the length of wire for the square should be:

a$)0$

b$)40$

c$)\frac{40r}{4}$

d$)\frac{100}{4x}$

e$)$ Not enough information

$26.f(x)=\frac{1}{y^2}$ has an absolute maximum on $3,5$ of

a$)\frac{1}{5}$

b$)\frac{1}{9}$

c$)9$

d$)\frac{1}{25}$

e$)\frac{1}{3}$