You need 50 pounds of two commodities costing $1.25 and$1.60 per pound.
a) Verify that the total cost is y=1.25x + 1.60(50 - x), where x is the number of pounds of the less expensive commodity.
b) Find the inverse function of the cost function.
a) To verify that the total cost is given by the equation y = 1.25x + 1.60(50 - x), we need to understand how this equation is derived.
The cost of the first commodity is $1.25 per pound, and you need x pounds of it. So, the cost of the first commodity is 1.25x.
The cost of the second commodity is $1.60 per pound, and you need 50 - x pounds of it. The 50 - x represents the remaining pounds after subtracting the amount of the first commodity from the total required amount. So, the cost of the second commodity is 1.60(50 - x).
Adding these costs together, we get the total cost: 1.25x + 1.60(50 - x).
b) To find the inverse function of the cost function, we need to solve the equation for x in terms of y.
First, let's rewrite the equation y = 1.25x + 1.60(50 - x):
y = 1.25x + 80 - 1.60x
Next, let's combine like terms:
y - 80 = -0.35x
Dividing both sides by -0.35, we get:
x = (y - 80) / -0.35
So, the inverse function of the cost function is:
x = (y - 80) / -0.35
a) The y(x) expression is obviously correct. The first term is the amount spent on the least expensive commodity and the second is the amount spend on the more expensive commodity.
b) Solve for x in terms of y.