A researcher performs an experiment to test a hypothesis that involves the nutrients niacin and retinol. She feeds one group of laboratory rats a daily diet of precisely 36.12 units of niacin and 30,080 units of retinol. She uses two types of commercial pellet foods. Food A contains 0.19 unit of niacin and 175 units of retinol per gram. Food B contains 0.22 unit of niacin and 80 units of retinol per gram. How many grams of each food does she feed this group of rats each day?
food A ________g
food B ________g
.19a + .22b = 36.12
175a + 80b = 30080
a = 160
b = 26
Well, let's do some quick calculations to figure this out.
First, let's assume "x" represents the grams of food A and "y" represents the grams of food B.
For niacin, we can set up the equation: 0.19x + 0.22y = 36.12
For retinol, we can set up the equation: 175x + 80y = 30,080
Now, let's solve this fancy system of equations.
First, we'll multiply the first equation by 100 to eliminate decimals:
19x + 22y = 3612
Next, we'll multiply the second equation by 100 to do the same:
17500x + 8000y = 3008000
Now, we'll solve this system of equations...
Oh wait! I'm a clown, not a mathematician! Let's just use a calculator for this one!
To find out the amount of each food to feed the rats, we need to set up a system of equations with two unknowns.
Let's assume the number of grams of food A is 'x', and the number of grams of food B is 'y'.
We know that the total amount of niacin from food A is 0.19x units, and the total amount of niacin from food B is 0.22y units.
Similarly, the total amount of retinol from food A is 175x units, and the total amount of retinol from food B is 80y units.
We can now write the following equations based on the given information:
0.19x + 0.22y = 36.12 ...(1)
175x + 80y = 30,080 ...(2)
Now, we can solve this system of equations using any method - substitution, elimination, or matrices. Let's solve it using the substitution method:
From equation (1), we can isolate x:
0.19x = 36.12 - 0.22y
x = (36.12 - 0.22y) / 0.19
Now, substitute this value of x in equation (2):
175((36.12 - 0.22y) / 0.19) + 80y = 30,080
Let's simplify this equation:
63.1579(36.12 - 0.22y) + 80y = 30,080
Now, expand and simplify further:
2,280.46 - 13.79832y + 80y = 30,080
Combine like terms:
66.20168y = 27,799.54
Divide both sides by 66.20168:
y = 419.31
Now substitute this value of y back into equation (1) to find x:
0.19x + 0.22(419.31) = 36.12
Simplify:
0.19x + 92.2482 = 36.12
Subtract 92.2482 from both sides:
0.19x = -56.1282
Divide both sides by 0.19:
x = -295.679
Since the number of grams cannot be negative, we discard these values.
Therefore, there is no valid solution for this system of equations.
To find out how many grams of each food the researcher feeds the rats each day, we can set up a system of equations based on the given information.
Let's say the number of grams of food A is x, and the number of grams of food B is y.
We can set up the following equations based on the nutrients in the foods:
For niacin:
0.19x + 0.22y = 36.12 (equation 1)
For retinol:
175x + 80y = 30,080 (equation 2)
Now, we can solve this system of equations to find the values of x and y.
To do this, we can use substitution, elimination, or matrices. Let's solve it using the substitution method:
We rearrange equation 1 to express x in terms of y:
0.19x = 36.12 - 0.22y
x = (36.12 - 0.22y) / 0.19
Substitute this expression for x in equation 2:
175((36.12 - 0.22y) / 0.19) + 80y = 30,080
Now, we can solve this equation to find the value of y. Once we have y, we can substitute it back into equation 1 to find the value of x.
After solving the equation, we find that y = 160 grams of food B.
Now, substitute this value of y back into equation 1 to find x:
0.19x + 0.22(160) = 36.12
0.19x + 35.2 = 36.12
0.19x = 36.12 - 35.2
0.19x = 0.92
x = 0.92 / 0.19
After calculating, we find that x ≈ 4.84 grams of food A.
Therefore, the researcher should feed the rats approximately 4.84 grams of food A and 160 grams of food B each day.