A goldsmith uses a platinum alloy and a silver alloy to make jewelry; the densit
ID: 2943426 • Letter: A
Question
A goldsmith uses a platinum alloy and a silver alloy to make jewelry; the densities of these alloys are exactly 20 and 10 grams per cubic centimeter, respectively.
a. King Hiero of Syyracuse orders a crown from this goldsmith, with a total mass of 5 kilograms (or 5,000 grams), with the stipulation that the platinum alloy must make up at least 90% of the mass. The goldsmith delivers a beautiful piece, but the king's friend Archimedes has doubts about its purity. While taking a bath, he comes up with a method to check the composition of the crown (famously shouting "Eureka!" in the process, and running to the king's palace naked). Submerging the crown in water, he finds its volume to be 370 cubic centimeters. How much of each alloy went into this piece (by mass)? Is this goldsmith a crook?
b. Find the matrix A that transforms the vector
mass of platinum alloy
mass of silver alloy
Explanation / Answer
A goldsmith uses a platinum alloy and a silver alloy to make jewelry; the densities of these alloys are exactly 20 and 10 grams per cubic centimeter, respectively.
a. King Hiero of Syyracuse orders a crown from this goldsmith, with a total mass of 5 kilograms (or 5,000 grams), with the stipulation that the platinum alloy must make up at least 90% of the mass. The goldsmith delivers a beautiful piece, but the king's friend Archimedes has doubts about its purity. While taking a bath, he comes up with a method to check the composition of the crown (famously shouting "Eureka!" in the process, and running to the king's palace naked). Submerging the crown in water, he finds its volume to be 370 cubic centimeters. How much of each alloy went into this piece (by mass)? Is this goldsmith a crook?
b. Find the matrix A that transforms the vector
mass of platinum alloy
mass of silver alloy
----------------------------------------
volume OF CROWN FOUND = 370 CC
LET THE CROWN CONTAIN
P GMS OF PLATINUM ALLOY AND S GMS OF SILVER ALLOY
CROWN MASS = 5000 GMS....SO.....
P+S=5000...............S=5000-P....................1
VOLUME OF PLATINUM ALLOY IN THE CROWN = P/20 CC.
VOLUME OF SILVER ALLOY IN THE CROWN = S/10 CC
TOTAL VOLUME = V = [P/20]+[S/10] = [P+2S]/20 CC ........................2.
SUBSTITUTING FOR S FROM EQN.1
V=[P+2(5000-P)]/20 = [10000-P]/20
BUT volume OF CROWN FOUND = 370 CC....SO
V=370=[10000-P]/20
10000-P=370*20=7400
P=10000-7400=2600 GMS.
THAT IS
PLATINUM ALLOY CONTENT = [2600/5000][100]=52%
SILVER ALLOY CONTENT = 100-52=48%
BUT platinum alloy must make up at least 90% of the mass.
WHERE AS IT IS ACTUALLY 52% ONLY..
SO this goldsmith IS a SUPER crook!!!
b. Find the matrix A that transforms the vector
mass of platinum alloy
mass of silver alloy IN TO WHAT?????
ANY WAY ASSUMING THAT YOU WANT TO KNOW THE TOTAL VOLUME
OF AN ALLOY WITH P GMS.OF PLATINUM ALLOY AND S GMS OF
SILVER ALLOY ..
T[P,S]=M*[P,S]=[V]
WHERE V IS THE VOLUME OF THE RESULTANT MASS...
WE HAVE FROM THE ABOVE CALCULATION THAT
V= [(P/20)+(S/10)] ..SO ....
T[P,S]'=M*[P,S]'=[V]=[(P/20)+(S/10)]
M*[(2 X 1)..MATRIX]=[(1 X 1 ) MATRIX]..SO
M SHALL BE A [1,2] MATRIX FOR COMPATABILITY
EQN. EDITOR NOT WORKING ...
WE HAVE
[X , Y ] * [P] = [PX+SY] = [P/20+S/10]
[S]
HENCE
X=1/20
S=1/10
SO THE REQD. TRANSFORM MATRIX IS
[(1/20) , (1/10) ] = [0.05 , 0.1]
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