There is an unspoken assumption here that going from a chicken leg to a turkey leg, all proportions remain the same. Think of a birds leg as being like a cylinder- doubling only the length gives a very skinny cylinder as compared to the original. To keep the same proportions you also have to double the radius (thickness). Since the volume of a cylinder (the amount of meat on the leg) is given by \(\displaystyle \pi r^2h\) doubling both length (h) and radius (r) gives \(\displaystyle \pi (2r)^2(2h)= 8\pi r^2h\). The turkey leg will feed 8 times as many people as a chicken leg.
Now, lets assume that a pan consists of a circular base, of radius r, with a rectangle, of width h and length \(\displaystyle 2\pi r\), bent around it to form the sides. Further we need to assume that both pans are made from the same material so that the weight is proportional to the volume of base and sides. The volume of the pan is given by \(\displaystyle \pi r^2h\) again. This problem is the opposite of the previous problem- here we are given the new volume and asked to find the new dimensions so lets take the "constant of proportionality" to be "c". The original pan had radius r and height h so the new pan has radius cr and height ch so volume \(\displaystyle \pi(cr)^2(ch)= c^3(\pi r^2h)\). We have \(\displaystyle \pi r^2h= 2\) quarts for the first pan, \(\displaystyle c^3(\pi r^2 h)= 1\). Dividing the first of those two equations by the second, \(\displaystyle \frac{1}{c^3}= 2\) so \(\displaystyle c^3= \frac{1}{2}\) or \(\displaystyle c= \frac{1}{2^{1/3}}\). Now we are told that "A company's brand of kitchen pans all have constant thickness." The weight of the base is proportional to its volume which is the area of the base times its thickness. Since the thickness is constant, the weight is proportional to the area, \(\displaystyle \pi r^2\). The area of the smaller pan is then \(\displaystyle \pi (cr)^2= c^2(\pi r^2)\). The weight of the side is also proportional to its volume which, because the thickness is constant, is proportional to the area of the rectangle \(\displaystyle 2\pi rh\). The weight of the side of the smaller pan is \(\displaystyle 2\pi (cr)(ch)= c^2(2\pi rh)\). That is, the weight of the smaller pan, adding base and side together, is \(\displaystyle c^2(\pi r^2)+ c^2(2\pi rh)= c^2)(\pi r^2h+ 2\pi rh)\). That is, the weight of the smaller pan is \(\displaystyle c^2\) times the weight of the larger pan. The weight of the larger pan is 3.75 pounds and we have found, above, that \(\displaystyle c= \frac{1}{2^{1/3}}\) so the weight of the smaller pan is \(\displaystyle \frac{1}{2^{2/3}}(3.75)\) pounds.
All of that can be done more quickly by making the general observation that, if all dimensions of an object are multiplied by "c" then surface area is multiplied by \(\displaystyle c^2\) and volume by \(\displaystyle c^3\).
That is, by the way, why slender animals, like a gazelle, cannot be as large as an elephant. Weight is proportional to volume so to "length" cubed. Muscle strength is proportional to cross section area so to "lengths" squared. If a gazelle were made, in exactly the same proportions, but legs 4 times as long, its strength would be multiplied by \(\displaystyle 4^2= 16\) but its weight would be multiplied by \(\displaystyle 4^3= 64\). It strength would not be enough to support that weight. Large animals cannot have a gazelle's slender legs- they have to be thick, like an elephant's.
