Bronze is an alloy or mixture of the metals copper and tin. The properties of bronze depends on the percentage of copper in the mix. A chemist decides to study the properties of a given alloy of bronze as the proportion of copper is varied. She starts with 7 kg of bronze that contain 3.5 kg of copper and 3.5 kg of tin and either adds or removes copper. Let f(x) be the percentage of copper in the mix if x kg of copper are add (x>0) or removed (x<0).
(a) State the domain and range of f (assume that the chemist wants a non-negative amount of both copper and tin).
Domain:
Range:
What do each of the answers mean in the context of bronze?
The domain tells us
The range tells us
(b) Write a formula in terms of x for f(x).
(c) If the formula you found in part (b) was not intended to represent the percentage of copper in an alloy of bronze, but instead simply defined an abstract mathematical function, what would be the domain and range of this function?
Domain:
Range:
(a) In the context of bronze, the domain of f(x) represents the amount of copper that can be added or removed in kilograms to the initial 7 kg of bronze. Since we want a non-negative amount of both copper and tin, the domain would be x ≥ -3.5 and x ≤ 3.5.
The range of f(x) represents the percentage of copper in the mix. As copper is added or removed, the percentage of copper in the mix will change. The range would be the set of all non-negative percentages, 0% ≤ f(x) ≤ 100%.
(b) To write a formula for f(x), we need to determine how the percentage of copper in the mix changes with the amount of copper added or removed.
Since the initial amount of copper is 3.5 kg and the total weight of the alloy remains constant at 7 kg, we can write the formula as:
f(x) = (3.5 + x) / 7 * 100
(c) If the formula f(x) is not intended to represent the percentage of copper in an alloy of bronze, but instead defines an abstract mathematical function, the domain and range would not be limited by the physical constraints of bronze.
The domain of the abstract function would typically encompass all possible real numbers. Therefore, the domain would be -∞ < x < +∞.
Similarly, the range of the abstract function would depend on the specific behavior of the function. Without additional information, it could be any set of real numbers.
Domain: -∞ < x < +∞
Range: Any set of real numbers.