Answer :
To solve this problem, we need to find out in how many years, [tex]\( x \)[/tex], the dairy company will be able to process 630 gallons of milk per day using the equation:
[tex]\[ y = x^6 - 9x^4 + 70x^2 \][/tex]
We are given that the company wants [tex]\( y \)[/tex] to be 630 gallons. So, we set up the equation:
[tex]\[ x^6 - 9x^4 + 70x^2 = 630 \][/tex]
We need to find the value of [tex]\( x \)[/tex] that satisfies this equation.
1. Set up the equation:
Replace [tex]\( y \)[/tex] with 630 in the equation:
[tex]\[
x^6 - 9x^4 + 70x^2 = 630
\][/tex]
2. Simplify the equation:
Rearrange it to set it to zero:
[tex]\[
x^6 - 9x^4 + 70x^2 - 630 = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
We need to find the roots (solutions) of this polynomial equation. The solutions are the values of [tex]\( x \)[/tex] that make the equation true.
4. Consider the valid solutions:
We only consider real and non-negative integer solutions since [tex]\( x \)[/tex] represents the number of years and it must be a whole number and positive.
5. Check the solutions within the range:
Since we are interested in the next five years and [tex]\( x \)[/tex] must be less than or equal to 5, check the integer solutions in this range.
After solving, the valid solution for [tex]\( x \)[/tex] that fits the criteria is:
- [tex]\( x = 3 \)[/tex]
Thus, it will take 3 years until the company can process 630 gallons of milk in a day. Therefore, the correct answer is:
B. 3 years
[tex]\[ y = x^6 - 9x^4 + 70x^2 \][/tex]
We are given that the company wants [tex]\( y \)[/tex] to be 630 gallons. So, we set up the equation:
[tex]\[ x^6 - 9x^4 + 70x^2 = 630 \][/tex]
We need to find the value of [tex]\( x \)[/tex] that satisfies this equation.
1. Set up the equation:
Replace [tex]\( y \)[/tex] with 630 in the equation:
[tex]\[
x^6 - 9x^4 + 70x^2 = 630
\][/tex]
2. Simplify the equation:
Rearrange it to set it to zero:
[tex]\[
x^6 - 9x^4 + 70x^2 - 630 = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
We need to find the roots (solutions) of this polynomial equation. The solutions are the values of [tex]\( x \)[/tex] that make the equation true.
4. Consider the valid solutions:
We only consider real and non-negative integer solutions since [tex]\( x \)[/tex] represents the number of years and it must be a whole number and positive.
5. Check the solutions within the range:
Since we are interested in the next five years and [tex]\( x \)[/tex] must be less than or equal to 5, check the integer solutions in this range.
After solving, the valid solution for [tex]\( x \)[/tex] that fits the criteria is:
- [tex]\( x = 3 \)[/tex]
Thus, it will take 3 years until the company can process 630 gallons of milk in a day. Therefore, the correct answer is:
B. 3 years
