Algebra 2 problem??? Please help?
Bob is concerned about the diet he is feeding his dog. A veterinary nutritionist has recommended that the dog's diet include at least 32 g of protein and at least 18 g of fat per day.
Bob has two types of food available – Purina and science diet. Each ounce of Purina supplies 2 g of protein and 4 g of fat. Each ounce of science diet supplies 6 g of protein and 2 g of fat. The dog must not eat a total of more than 12 ounces of food per day.
Bob would like to vary the diet for the dog within these requirements, and so he needs to know what his options are.
Your task:
1.) Find some combination of food (some ounces of pure Warrena and some ounces of science diet) that works with all of the constraints and prove that it works.
2.) Clearly define your variables
3.) Use your variables to write inequalities to describe the constraints of the problem.
4.) Given the following additional information; Purina costs $0.38 cents per ounce and science diet $0.51 cents per ounce; since the dog is a show dog his food costs are tax-deductible - therefore how much of each food should Bob feed the dog in order to maximize his costs?
5.) describe your work in a paragraph or two summary.
Please help I'm preparing for a major test and I can't get this??? I beg you please help me thank you soooooooooo much.
Comments
1.) Find some combination of food (some ounces of pure Warrena and some ounces of science diet) that works with all of the constraints and prove that it works.
I assume "pure Warrena" was some kind of typo for "Purina."
Anyway, 4 ounces of each food should do the trick:
4 oz of Purina supplies 8 g protein and 16 g fat.
4 oz science diet supplies 24 g protein and 8 g fat.
The total nutrients is 32 g protein and 24 g fat, in only 8 oz of food, which meets all requirements.
2.) Clearly define your variables
Let x = the number of oz of Purina fed the dog in a day,
and y = the number of oz of science diet fed the dog in a day.
3.) Use your variables to write inequalities to describe the constraints of the problem.
Notation: I'll use the computer-language symbols
<= for "less than or equal to" and
>= for "greater than or equal to."
2x + 6y >= 32 [grams of protein in the diet]
4x + 2y >= 18 [grams of fat in the diet]
x + y <= 12 [limit on quantity of food per day]
Of course, x >= 0 and y >= 0, because we can't feed the dog a negative quantity.
4.) Given the following additional information; Purina costs $0.38 cents per ounce and science diet $0.51 cents per ounce; since the dog is a show dog his food costs are tax-deductible - therefore how much of each food should Bob feed the dog in order to maximize his costs?
To maximize (or minimize) food costs, we need to find the "feasible region," the set of combinations of x and y that satisfy all the constraints. This region is defined by the pairwise solutions of the equations for the borderlines of each of our constraint inequalities:
{1} 2x + 6y = 32
{2} 4x + 2y = 18
{3} x + y = 12
{4} x = 0
{5} y = 0
The common solution of {1} and {2} is (2.2, 4.6) [which satisfies x+y<=12]
The common solution of {1} and {3} is (10, 2) [which satisfies 4x+2y>=18]
The common solution of {2} and {3} is (-3, 15), which violates x >= 0,
so we'll have to examine where {2} and {3} intersect the y-axis, at (0,9) and (0,12).
We don't have to worry about y=0 because if y=0, then x must be at least 16, exceeding the constraint on x + y. This means we can't feed the dog only Purina, because meeting the requirement for minimum protein would require exceeding the total-food limit.
Our feasible region is a quadrilateral with vertices (0.9), (0.12), (2.2, 4.6), and (10,2).
The cost function is
c(x,y) = $0.38x + $0.51y
Cost of the diet represented by each vertex:
at (0.9): c(x,y) = $4.59
at (0.12): c(x,y) = $6.12
at (2.2, 4.6): c(x,y) = $0.836 + $2.346 = $3.182
[where the fraction of a cent is relevant over multiple days]
at (10,2): c(x,y) = $3.80 + $1.02 = $4.82
So if Bob wanted to MINIMIZE his food costs, a daily diet of
2.2 oz Purina and 4.6 oz science diet would be suitable.
But since he wants to maximize costs, he can feed the dog 12 oz a day of science diet.
the gap between 2 factors is predicated on the pythagorean theorem. You calculate the entire upward push (distinction between the y coordinates) and the run (distinction between the x coordinates), and then build a proper triangle such that the hypotenuse is the line between the two factors. -14 - (-2) = -12 (your "upward push") 12 - 9 = 3 (your "run") by skill of pyth thm, (-12)^2 + 3^2 = d^2, the place d is the gap between the two factors. (-12)^2 + 3^2 = d^2 a hundred and forty four + 9 = d^2 153 = d^2 d = sqrt(153), or the sq. root of 153 = approximately 12.369. wish it facilitates.