An artist makes three types of ceramic statues (large, medium, and small) at a monthly cost of $650 for 180 statues. The manufacturing costs for the three types are $5, $4, and $3. If the statues sell for $20, $12, and $9, respectively, how many of each type should be made to produce $2,100 in monthly revenue?
x+y+z = 180
5x+4y+3z = 650
20x+12y+9z = 2100
To solve this problem, we can set up a system of equations based on the given information.
Let's represent the number of large, medium, and small statues made as L, M, and S, respectively.
Based on the manufacturing costs, we can write the equation for the total manufacturing cost as:
5L + 4M + 3S = 650
Based on the selling prices, we can write the equation for the total revenue as:
20L + 12M + 9S = 2100
We can solve this system of equations to find the values of L, M, and S.
Using a method such as substitution or elimination, let's solve this system of equations.
Step 1: Solve the first equation for L in terms of M and S:
5L = 650 - 4M - 3S
L = (650 - 4M - 3S) / 5
Step 2: Substitute the value of L in the second equation:
20((650 - 4M - 3S) / 5) + 12M + 9S = 2100
Multiply through by 5 to eliminate the denominator:
1300 - 8M - 6S + 12M + 9S = 10500
Combine like terms:
-2M + 3S = 800 ----(Equation 3)
Step 3: We now have two equations with two variables (Equation 1 and Equation 3). Solve this system of equations.
Using the substitution method:
From Equation 3, solve for M in terms of S:
-2M = 800 - 3S
M = (800 - 3S) / -2
Substitute the value of M in Equation 1:
5L + 4((800 - 3S) / -2) + 3S = 650
Multiply through by -2 to eliminate the denominator:
-10L - 8(800 - 3S) + 6S = -1300
Expand and simplify:
-10L - 6400 + 24S + 6S = -1300
Combine like terms:
-10L + 30S = 5100 ----(Equation 4)
Step 4: We now have two equations with two variables (Equation 3 and Equation 4). We can solve this system of equations.
Multiply Equation 3 by 5 and Equation 4 by 3:
-10M + 15S = 4000 ----(Equation 5)
-30L + 90S = 15300 ----(Equation 6)
Multiply Equation 6 by 2 and add it to Equation 5 to eliminate S:
(-30L + 90S) * 2 + (-10M + 15S) = 0
-60L + 180S - 10M + 15S = 0
-10M - 60L + 195S = 0
Now, substitute the value of M from Equation 3 in the above equation:
-10((800 - 3S) / -2) - 60L + 195S = 0
Expand and simplify:
4000 + 30S - 60L + 195S = 0
Combine like terms:
225S - 60L = -4000 ----(Equation 7)
Step 5: We now have two equations with two variables (Equation 7 and Equation 4). We can solve this system of equations.
Multiply Equation 7 by 10 and Equation 4 by 225:
2250S - 600L = -40000 ----(Equation 8)
-2250L + 6750S = 1147500 ----(Equation 9)
Add Equation 8 and Equation 9 to eliminate L:
(2250S - 600L) + (-2250L + 6750S) = -40000 + 1147500
Combine like terms:
9000S = 1107500
Divide both sides by 9000:
S = 123.0556
Since we cannot have a fraction of a statue, we can round S to the nearest whole number:
S ≈ 123
Now, substitute the value of S in Equation 3 to find M:
-2M + 3(123) = 800
-2M + 369 = 800
-2M = 431
M ≈ -215.5
Again, since we cannot have a negative number of statues, we can round M up to zero:
M ≈ 0
Finally, substitute the values of M and S in Equation 1 to find L:
5L + 4(0) + 3(123) = 650
5L + 369 = 650
5L = 281
L ≈ 56.2
Again, since we cannot have a fraction of a statue, we can round L to the nearest whole number:
L ≈ 56
Therefore, to produce $2,100 in monthly revenue, the artist should make approximately 56 large, 0 medium, and 123 small statues.