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Answer :
Sure, let's solve this problem step-by-step without using the Python code.
The equation given is:
[tex]\[ y = x^5 - 9x^4 + 70x^2 \][/tex]
We need to find for which value of [tex]\( x \)[/tex], the company can process 630 gallons of milk in a day, i.e., when [tex]\( y = 630 \)[/tex]:
[tex]\[ x^5 - 9x^4 + 70x^2 = 630 \][/tex]
Let's simplify and solve this equation:
[tex]\[ x^5 - 9x^4 + 70x^2 - 630 = 0 \][/tex]
Given that we're looking for integer solutions, we can test the values of [tex]\( x \)[/tex] from the provided options.
### Testing [tex]\( x = 2 \)[/tex]:
[tex]\[ 2^5 - 9 \cdot 2^4 + 70 \cdot 2^2 \][/tex]
[tex]\[ = 32 - 9 \cdot 16 + 70 \cdot 4 \][/tex]
[tex]\[ = 32 - 144 + 280 \][/tex]
[tex]\[ = 32 - 144 + 280 \][/tex]
[tex]\[ = 168 \][/tex]
This does not match 630 gallons.
### Testing [tex]\( x = 3 \)[/tex]:
[tex]\[ 3^5 - 9 \cdot 3^4 + 70 \cdot 3^2 \][/tex]
[tex]\[ = 243 - 9 \cdot 81 + 70 \cdot 9 \][/tex]
[tex]\[ = 243 - 729 + 630 \][/tex]
[tex]\[ = 243 - 729 + 630 \][/tex]
[tex]\[ = 144 \][/tex]
This does not match 630 gallons either.
### Testing [tex]\( x = 4 \)[/tex]:
[tex]\[ 4^5 - 9 \cdot 4^4 + 70 \cdot 4^2 \][/tex]
[tex]\[ = 1024 - 9 \cdot 256 + 70 \cdot 16 \][/tex]
[tex]\[ = 1024 - 2304 + 1120 \][/tex]
[tex]\[ = 1024 - 2304 + 1120 \][/tex]
[tex]\[ = -160 \][/tex]
This also does not match 630 gallons.
### Testing [tex]\( x = 5 \)[/tex]:
[tex]\[ 5^5 - 9 \cdot 5^4 + 70 \cdot 5^2 \][/tex]
[tex]\[ = 3125 - 9 \cdot 625 + 70 \cdot 25 \][/tex]
[tex]\[ = 3125 - 5625 + 1750 \][/tex]
[tex]\[ = 3125 - 5625 + 1750 \][/tex]
[tex]\[ = 1250 - 5625 + 1750 \][/tex]
[tex]\[ = 250 \][/tex]
This too does not match 630 gallons.
It appears that the given options (2, 3, 4, 5 years) do not yield the desired number of 630 gallons. Therefore, none of the provided options A, B, C, or D is correct.
However, the problem appears straightforward, and no apparent typographical mistakes are present. Consequently, it's critical to walk through the calculations or recheck the given equation for feasibility with the provided options.
The most likely conclusion based on a detailed check indicates typographical or problem configuration errors, or alternative years could resolve the obtained 630-gallon requirement.
The equation given is:
[tex]\[ y = x^5 - 9x^4 + 70x^2 \][/tex]
We need to find for which value of [tex]\( x \)[/tex], the company can process 630 gallons of milk in a day, i.e., when [tex]\( y = 630 \)[/tex]:
[tex]\[ x^5 - 9x^4 + 70x^2 = 630 \][/tex]
Let's simplify and solve this equation:
[tex]\[ x^5 - 9x^4 + 70x^2 - 630 = 0 \][/tex]
Given that we're looking for integer solutions, we can test the values of [tex]\( x \)[/tex] from the provided options.
### Testing [tex]\( x = 2 \)[/tex]:
[tex]\[ 2^5 - 9 \cdot 2^4 + 70 \cdot 2^2 \][/tex]
[tex]\[ = 32 - 9 \cdot 16 + 70 \cdot 4 \][/tex]
[tex]\[ = 32 - 144 + 280 \][/tex]
[tex]\[ = 32 - 144 + 280 \][/tex]
[tex]\[ = 168 \][/tex]
This does not match 630 gallons.
### Testing [tex]\( x = 3 \)[/tex]:
[tex]\[ 3^5 - 9 \cdot 3^4 + 70 \cdot 3^2 \][/tex]
[tex]\[ = 243 - 9 \cdot 81 + 70 \cdot 9 \][/tex]
[tex]\[ = 243 - 729 + 630 \][/tex]
[tex]\[ = 243 - 729 + 630 \][/tex]
[tex]\[ = 144 \][/tex]
This does not match 630 gallons either.
### Testing [tex]\( x = 4 \)[/tex]:
[tex]\[ 4^5 - 9 \cdot 4^4 + 70 \cdot 4^2 \][/tex]
[tex]\[ = 1024 - 9 \cdot 256 + 70 \cdot 16 \][/tex]
[tex]\[ = 1024 - 2304 + 1120 \][/tex]
[tex]\[ = 1024 - 2304 + 1120 \][/tex]
[tex]\[ = -160 \][/tex]
This also does not match 630 gallons.
### Testing [tex]\( x = 5 \)[/tex]:
[tex]\[ 5^5 - 9 \cdot 5^4 + 70 \cdot 5^2 \][/tex]
[tex]\[ = 3125 - 9 \cdot 625 + 70 \cdot 25 \][/tex]
[tex]\[ = 3125 - 5625 + 1750 \][/tex]
[tex]\[ = 3125 - 5625 + 1750 \][/tex]
[tex]\[ = 1250 - 5625 + 1750 \][/tex]
[tex]\[ = 250 \][/tex]
This too does not match 630 gallons.
It appears that the given options (2, 3, 4, 5 years) do not yield the desired number of 630 gallons. Therefore, none of the provided options A, B, C, or D is correct.
However, the problem appears straightforward, and no apparent typographical mistakes are present. Consequently, it's critical to walk through the calculations or recheck the given equation for feasibility with the provided options.
The most likely conclusion based on a detailed check indicates typographical or problem configuration errors, or alternative years could resolve the obtained 630-gallon requirement.
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