A and B farm A produces milk with 3 ½ percent fat and farm B produces milk with 4 ¾ percent fat.
(a) Determine
(i) The total mass fat in 50 kg milk from farm A and 30 kg of milk from farm B
(ii) The percentage of fat in a mixture of 50 kg of milk from A and 30 kg of milk from B
(b) Determine the range of values of mass of milk from farm B that must in a 50 kg mixture so that the mixture may have at least 4 percent fat.
(i) 50(0.035) + 30(0.0475) = 3.175
(ii) 3.175/(50+30) = 0.03968 = 3.97%
.035x + .0475(50-x) >= 0.04*50
x <= 30
That is, with 30kg from A and 20kg from B, the result is exactly 4% fat
Since A's content is less than 4%, using any more of A will reduce the result below the desired 4% mix. Having less than 30kg from A will mean more than 20kg from B, raising the fat content above 4%.
(a)
(i) To determine the total mass of fat in 50 kg of milk from farm A and 30 kg of milk from farm B, we need to calculate the mass of fat in each type of milk separately and then add them together.
Mass of Fat in Farm A milk:
In 50 kg of milk,
Percentage of fat = 3 ½ % = 3.5%
Mass of fat in 50 kg of milk = (3.5/100) * 50 kg = 1.75 kg
Mass of Fat in Farm B milk:
In 30 kg of milk,
Percentage of fat = 4 ¾ % = 4.75%
Mass of fat in 30 kg of milk = (4.75/100) * 30 kg = 1.425 kg
Total mass of fat = Mass of fat in Farm A milk + Mass of fat in Farm B milk
Total mass of fat = 1.75 kg + 1.425 kg = 3.175 kg
(ii) To determine the percentage of fat in a mixture of 50 kg of milk from farm A and 30 kg of milk from farm B, we need to calculate the total mass of fat in the mixture and divide it by the total mass of the mixture.
Total mass of the mixture = 50 kg (milk from farm A) + 30 kg (milk from farm B) = 80 kg
Percentage of fat in the mixture = (Total mass of fat in the mixture / Total mass of the mixture) * 100
Percentage of fat in the mixture = (3.175 kg / 80 kg) * 100 = 3.97%
(b) To determine the range of values for the mass of milk from farm B in a 50 kg mixture so that the mixture may have at least 4% fat, we need to calculate the maximum and minimum values for the mass of milk from farm B.
Let x be the mass of milk from farm B in the mixture.
Since we want the mixture to have at least 4% fat, we can set up the following inequality:
(1.425 kg + x * (4.75/100)) / (50 kg + x) ≥ 4/100
Simplifying the inequality:
1.425 kg + 0.0475x ≥ (0.04 * (50 kg + x))
1.425 kg + 0.0475x ≥ 2 kg + 0.04x
0.0475x - 0.04x ≥ 2 kg - 1.425 kg
0.0075x ≥ 0.575 kg
x ≥ 0.575 kg / 0.0075
x ≥ 76.67 kg
Therefore, the range of values for the mass of milk from farm B in a 50 kg mixture is x ≥ 76.67 kg.
To solve this problem, we will follow these steps:
(a)(i) To determine the total mass of fat in 50 kg of milk from farm A and 30 kg of milk from farm B, we need to calculate the fat mass from each farm individually.
Farm A:
50 kg of milk contains 3 ½ percent fat.
To find the mass of fat, we calculate 3 ½ percent of 50 kg:
Mass of fat from Farm A = (3 ½ / 100) * 50 kg
Farm B:
30 kg of milk contains 4 ¾ percent fat.
To find the mass of fat, we calculate 4 ¾ percent of 30 kg:
Mass of fat from Farm B = (4 ¾ / 100) * 30 kg
To find the total mass of fat, we add the mass of fat from Farm A and Farm B:
Total mass of fat = Mass of fat from Farm A + Mass of fat from Farm B
(a)(ii) To determine the percentage of fat in a mixture of 50 kg of milk from Farm A and 30 kg of milk from Farm B, we need to calculate the total mass of fat in the mixture and then determine its percentage.
Total mass of milk in the mixture = 50 kg + 30 kg
Total mass of fat in the mixture = Mass of fat from Farm A + Mass of fat from Farm B
To find the percentage of fat, we divide the total mass of fat in the mixture by the total mass of milk in the mixture and multiply by 100.
Percentage of fat in the mixture = (Total mass of fat in the mixture / Total mass of milk in the mixture) * 100
(b) To determine the range of values of mass of milk from Farm B that must be in a 50 kg mixture so that the mixture may have at least 4 percent fat, we need to find the minimum and maximum mass of milk from Farm B.
We can set up two equations based on the given conditions:
Equation 1: (Mass of milk from Farm B / Total mass of milk in the mixture) * 100 = 4 percent
Equation 2: Mass of milk from Farm A + Mass of milk from Farm B = 50 kg
Solve these two equations simultaneously to find the range of values for the mass of milk from Farm B.