An artist makes three types of ceramic statues (large, medium, and small) at a monthly cost of $820 for 180 statues. The manufacturing costs for the three types are $6, $5, and $4. If the statues sell for $22, $14, and $11, respectively, how many of each type should be made to produce $2,380 in monthly revenue?

Question

An artist makes three types of ceramic statues (large, medium, and small) at a monthly cost of $820 for 180 statues. The manufacturing costs for the three types are $6, $5, and $4. If the statues sell for $22, $14, and $11, respectively, how many of each type should be made to produce $2,380 in monthly revenue?

Answer 1

number of large --- x

number of medium -- y
number of small --- 180-x-y

6x + 5y + 4(180-x-y) = 820
6x+5y+720-4x-4y=820
2x+y=100 or y = 100-2x

22x+14y+11(180-x-y) = 2380
11x + 3y = 400

sub in y = 100-2x
11x + 3(100-2x) = 400
5x = 100
x=20
then y = 60

So 20 large, 60 medium and 100 small ones

btw, it asked for $2380 in revenue, not $2380 in profit

Answer 2

To find out how many statues of each type should be made to produce $2,380 in monthly revenue, we need to set up a system of equations based on the given information.

Let's assume the artist makes x large statues, y medium statues, and z small statues.

The total monthly cost of manufacturing the statues is $820. The cost of manufacturing each type of statue is $6, $5, and $4 respectively. So we can create the equation:

6x + 5y + 4z = 820 ---(1)

The revenue generated by selling the statues is $2,380. The selling price of each type of statue is $22, $14, and $11 respectively. So we can create the equation:

22x + 14y + 11z = 2,380 ---(2)

Now we have a system of two equations with three variables. We can solve this system using algebraic methods.

To solve the system of equations, we can use substitution, elimination, or matrix methods. In this case, let's solve it using the elimination method.

Multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of x in both equations equal:

12x + 10y + 8z = 1640 ---(3)
66x + 42y + 33z = 7140 ---(4)

Next, we can multiply equation (1) by 3 and equation (2) by -2 to make the coefficients of y in both equations equal:

18x + 15y + 12z = 2460 ---(5)
-28x - 28y - 22z = -4760 ---(6)

Now we can eliminate y by subtracting equation (6) from equation (5):

18x + 15y + 12z - (-28x - 28y - 22z) = 2460 - (-4760)

Combine like terms:

46x + 43z = 7220 ---(7)

Now we have two equations with two variables: equation (3) and equation (7). We can solve this system of equations.

Multiply equation (3) by 43 and equation (7) by 12 to make the coefficients of z in both equations equal:

516x + 430y + 344z = 70520 ---(8)
552x + 516z = 86640 ---(9)

Now subtract equation (8) from equation (9) to eliminate z:

552x + 516z - (516x + 430y + 344z) = 86640 - 70520

Combine like terms:

36x + 172z = 16120 ---(10)

Now we have one equation with one variable: equation (10). We can solve for x.

To isolate x, we can subtract 172z from both sides:

36x = 16120 - 172z

Finally, divide both sides by 36 to solve for x:

x = (16120 - 172z) / 36

Now we have the value of x in terms of z. We can substitute this expression into equation (7) to solve for z.

46 ((16120 - 172z) / 36) + 43z = 7220

Simplify this equation to isolate z. Then, solve for z:

(736120 - 7984z + 46×43z) / 36 = 7220

Multiply both sides by 36 to eliminate the denominator:

736120 - 7984z + 46×43z = 36×7220

Combine like terms:

736120 - 7984z + 1978z = 259920

Combine like terms again:

-6006z = -476200

Divide both sides by -6006 to solve for z:

z = -476200 / -6006

Simplify:

z ≈ 79.3

Since we cannot have a fractional number of statues, we need to round z to the nearest whole number.

z ≈ 79

Now we have the value of z. We can substitute this value into equation (7) to solve for x.

46x + 43×79 = 7220

Simplify:

46x + 3397 = 7220

Subtract 3397 from both sides:

46x = 3823

Divide both sides by 46 to solve for x:

x ≈ 83

Now we have the value of x. We can substitute both z and x into equation (1) to solve for y:

6×83 + 5y + 4×79 = 820

Simplify:

498 + 5y + 316 = 820

Combine like terms:

5y + 814 = 820

Subtract 814 from both sides:

5y = 6

Divide both sides by 5 to solve for y:

y = 6/5

Since we cannot have a fractional number of statues, we need to round y to the nearest whole number.

y ≈ 1

So, the number of each type of statue should be approximately:
x ≈ 83 (large statues)
y ≈ 1 (medium statues)
z ≈ 79 (small statues)

Therefore, the artist should make approximately 83 large statues, 1 medium statue, and 79 small statues to produce $2,380 in monthly revenue.