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Answer :
To find out how many minutes are needed for there to be 3000 bacteria left in the swimming pool after disinfection, we start with the given equation:
[tex]\[ y = 5500 \sqrt{0.025x + 0.1} \][/tex]
We want to find the value of [tex]\( x \)[/tex] when [tex]\( y = 3000 \)[/tex].
### Step-by-Step Solution:
1. Set the equation for the number of bacteria to 3000:
[tex]\[
3000 = 5500 \sqrt{0.025x + 0.1}
\][/tex]
2. Divide both sides of the equation by 5500 to isolate the square root term:
[tex]\[
\frac{3000}{5500} = \sqrt{0.025x + 0.1}
\][/tex]
3. Calculate the left side of the equation:
[tex]\[
0.545 \approx 0.5454545454545454
\][/tex]
4. Square both sides to eliminate the square root:
[tex]\[
(0.545)^2 \approx (0.5454545454545454)^2 \approx 0.29752066115702475
\][/tex]
5. Equate to the expression under the square root:
[tex]\[
0.29752066115702475 = 0.025x + 0.1
\][/tex]
6. Subtract 0.1 from both sides to solve for the term with [tex]\( x \)[/tex]:
[tex]\[
0.19752066115702475 = 0.025x
\][/tex]
7. Divide both sides by 0.025 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{0.19752066115702475}{0.025} \approx 7.900826446280989
\][/tex]
8. Round the result to the nearest whole number:
[tex]\[
x \approx 8
\][/tex]
Therefore, it will take approximately 8 minutes for the number of bacteria in the swimming pool to be reduced to 3000 after disinfection.
[tex]\[ y = 5500 \sqrt{0.025x + 0.1} \][/tex]
We want to find the value of [tex]\( x \)[/tex] when [tex]\( y = 3000 \)[/tex].
### Step-by-Step Solution:
1. Set the equation for the number of bacteria to 3000:
[tex]\[
3000 = 5500 \sqrt{0.025x + 0.1}
\][/tex]
2. Divide both sides of the equation by 5500 to isolate the square root term:
[tex]\[
\frac{3000}{5500} = \sqrt{0.025x + 0.1}
\][/tex]
3. Calculate the left side of the equation:
[tex]\[
0.545 \approx 0.5454545454545454
\][/tex]
4. Square both sides to eliminate the square root:
[tex]\[
(0.545)^2 \approx (0.5454545454545454)^2 \approx 0.29752066115702475
\][/tex]
5. Equate to the expression under the square root:
[tex]\[
0.29752066115702475 = 0.025x + 0.1
\][/tex]
6. Subtract 0.1 from both sides to solve for the term with [tex]\( x \)[/tex]:
[tex]\[
0.19752066115702475 = 0.025x
\][/tex]
7. Divide both sides by 0.025 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{0.19752066115702475}{0.025} \approx 7.900826446280989
\][/tex]
8. Round the result to the nearest whole number:
[tex]\[
x \approx 8
\][/tex]
Therefore, it will take approximately 8 minutes for the number of bacteria in the swimming pool to be reduced to 3000 after disinfection.
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