a) If 10 liters of paint A is mixed with 40 liters of paint B, the percentage of acrylic is 20%.
b) If 10 liters of paint A is mixed with (50-10z) liters of paint B, there are (11 - z) liters of acrylic in the mixture.
c) To make 50 liters of paint that is 30% acrylic, the customer should use 20 liters of Paint A and 30 liters of Paint B.
d) If PC sells the paint to the customer for $3,000, PC makes a profit of $1,400.
e) The total amount of water in the paint mixture is 19 liters.
The percentage of acrylic in the mixture = The total amount of acrylic in each type of paint divided by the total volume of the mixture.
In 10 liters of Paint A, the amount of acrylic is 60% of 10 liters, which = 6 liters (10 x 60%).
In 40 liters of Paint B, the amount of acrylic is 10% of 40 liters, which = 4 liters (40 x 10%).
Thus, the total amount of acrylic in the mixture is 6 liters (from Paint A) + 4 liters (from Paint B) = 10 liters.
The total volume of the mixture is 10 liters (of Paint A) + 40 liters (of Paint B) = 50 liters.
Therefore, the percentage of acrylic in the mixture is (10 liters / 50 liters) * 100% = 20%.
Thus, we can conclude that the mixture is 20% acrylic.
b) Let’s calculate the amount of acrylic in the mixture:
In 10 liters of Paint A, the amount of acrylic is 60% of 10 liters, which is 6 liters.
In ((50 - 10z)) liters of Paint B, the amount of acrylic is 10% of ((50 - 10z)) liters, which is (0.1 \times (50 - 10z)) liters.
So, the total amount of acrylic in the mixture is the sum of the acrylic from Paint A and Paint B:
Total Acrylic = 6 + 0.1 × (50−10z) = 6 + 5 − z
Simplifying this gives:
Total Acrylic = 11−z liters
Thus, there are (11 - z) liters of acrylic in the mixture.
c) Let the amount of Paint A used = x liters
Let the amount of Paint B used = y liters.
The total amount of paint used is 50 liters, so (x + y = 50).
The total amount of acrylic in the mixture is 30% of 50 liters, which is 15 liters. Since Paint A is 60% acrylic and Paint B is 10% acrylic, we have (0.6x + 0.1y = 15).
We have formed a system of two equations with two unknowns and can solve this system to find the values of (x) and (y).
Multiplying the second equation by 10 gives us (6x + y = 150). Subtracting the first equation from this gives us (5x = 100), so (x = 20).
Substituting (x = 20) into the first equation gives us (20 + y = 50), so (y = 30).
Thus, to make 50 liters of paint that is 30% acrylic, the customer should use 20 liters of Paint A and 30 liters of Paint B.
d) Computation of Profit:
The total revenue = $3,000
From the previous parts of the problem, we know that 20 liters of Paint A and 30 liters of Paint B are used. Given that Paint A costs $50 per liter and Paint B costs $20 per liter, we can calculate the total cost:
Cost of Paint A = 20 liters * $50/liter = $1,000
Cost of Paint B = 30 liters * $20/liter = $600
So, the total cost is $1000 (for Paint A) + $600 (for Paint B) = $1,600.
The profit:
Profit = Total Revenue - Total Cost
= $3000 - $1600 = $1400.
So, PC makes a profit of $1400.
e) In part (a), we mixed 10 liters of Paint A and 40 liters of Paint B.
Paint A is 60% acrylic, 10% latex, and the rest is water. So, in 10 liters of Paint A, there is 30% of water, which is 3 liters.
Paint B is 10% acrylic, 50% latex, and the rest is water. So, in 40 liters of Paint B, there is 40% of water, which is 16 liters.
Thus, the total amount of water in the paint mixture is 3 liters (from Paint A) + 16 liters (from Paint B) = 19 liters.
Complete Question:
e) How many liters of water is in the paint used in part (a)?