Olive Oyl sent Popeye and Wimpy out to get spinach and burgers. They bought the following combination: If the square of the number of burgers is doubled and added to the square of the number of cans of spinach, the total equals 59. When the square of the number of cans of spinach is doubled and added to the square of the number of burgers, the total equals 43. Find the number of burgers and cans of spinach.
How many "mathematical" solutions does this problem have? What is the feasible number of burgers and cans of spinach purchased?
When the guys got home, Olive Oyl (a stingy harridan) nagged them about spending $2.98 per can of spinach and $4.58 per burger (without fries!). With 6% sales tax, what was the total spent?
2b^2+s^2 = 59
2s^2+b^2 = 43
b^2 = 43-2s^2, so
2(43-2s^2)+s^2 = 59
3s^2 = 27
s=±3
so, b=±5
Only positive values are "feasible", so
s=3,b=5
(2.98*3 + 4.58*5)*1.06 = 33.75
To solve this problem, we'll use algebraic equations to represent the given information. Let's assign variables to the number of burgers and cans of spinach purchased.
Let's assume the number of burgers is represented by 'b' and the number of cans of spinach is represented by 's'.
According to the problem:
1. The square of the number of burgers, doubled and added to the square of the number of cans of spinach, equals 59:
(2b^2) + s^2 = 59
2. The square of the number of cans of spinach, doubled and added to the square of the number of burgers, equals 43:
(2s^2) + b^2 = 43
To find the number of "mathematical" solutions, we need to solve these two equations simultaneously.
To solve these equations, we can use a variety of methods, such as substitution, elimination, or matrices. Here, we'll use the substitution method:
From equation 1, we can rearrange it to solve for s^2:
s^2 = 59 - (2b^2)
Substituting this expression for s^2 in equation 2:
(2(59 - (2b^2))) + b^2 = 43
Simplifying the equation:
118 - 4b^2 + b^2 = 43
-3b^2 = -75
b^2 = 25
b = ±5
So the feasible number of burgers is 5 or -5. However, the number of burgers cannot be negative, so we have one feasible solution: 5 burgers.
To find the value of s, we substitute the value of b into equation 1:
(2(5^2)) + s^2 = 59
50 + s^2 = 59
s^2 = 9
s = ±3
Again, since the number of cans of spinach cannot be negative, we have one feasible solution: 3 cans of spinach.
Therefore, the number of burgers purchased is 5, and the number of cans of spinach purchased is 3.
Now, let's calculate the total amount spent on spinach and burgers, including tax:
Cost per can of spinach = $2.98
Cost per burger = $4.58
Total cost of spinach = 3 * $2.98 = $8.94
Total cost of burgers = 5 * $4.58 = $22.90
Total cost before tax = $8.94 + $22.90 = $31.84
Sales tax rate = 6%
Sales tax amount = 6% of $31.84 = 0.06 * $31.84 = $1.91
Total amount spent = $31.84 + $1.91 = $33.75
Therefore, the total amount spent on spinach and burgers including tax is $33.75.