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:
Answers
The function is f(x) = 100(x + 3.5)/(x + 7)
(a) The domain of f is (-3.5, ∞]
The range of f is (0%, 100%)
(b) The domain tells us about the mass of copper that must be added to vary its quantity in the bronze
The range tells us about the percentage of copper in the bronze that we will obtain by varying the mass of bronze
(c) If the function is abstract mathematical function then
Domain of f(x) is x ∈ R - {-7}
Range of f(x) is (-∞, ∞) - {100}
Step-by-step explanation:
Initially in 7 kg of Bronze, there is 3.5 kg of Copper and then the mass of copper is varied
If the function f(x) is the percentage of Copper in the mix then
If Copper is added then x > 0
If it is removed then x < 0
Copper that can be removed = -3.5 kg (slightly less than this)
Copper that can be added = unlimited
The domain of f(x) is (-3.5, ∞)
Minimum value of f(x) in the interval (-3.5, ∞) is just greater than 0 at x = -3.5 and maximum value is just less than 100
Therefore, the range of f(x) is (0, 100)
The domain tells us about the amount of copper that must be added to vary its quantity in the bronze, we cannot remove all the copper because then it will only be tin and not bronze
The range tells us about the percentage of copper in the bronze that we will obtain by varying the mass of bronze with varies from 0% to 100% both not included
We cannot have whole 100% of copper because then it will be only copper metal and not bronze.
If the function is abstract mathematical function then
The domain of f(x) will be (-∞, -7) ∪ (-7, ∞) or in other words x ∈ R - {-7}
The range will be (-∞, ∞) - {100}
Hope this answer is helpful.
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