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used for one of its products by enlarging the radius of the cylinder. The height must be 20 centimeters. The new volume of the cylinder is given by the equation 
V(x)=20 pi(5+x)^(2). where 
x is the additional length of the radius in centimeters. Which of the following equivalent expressions displays the current volume of the cylinder as a constant or coefficient?
Choose 1 answer:
(A) 
pi(20x^(2)+200 x+500)
(B) 
20 pi(x^(2)+10 x+10)+300 pi
(C) 
20 pi(x^(2)+10 x+25)
(D) 
20 pix^(2)+200 pi x+500 pi

used for one of its products by enlarging the radius of the cylinder. The height must be 2020 centimeters. The new volume of the cylinder is given by the equation V(x)=20π(5+x)2 V(x)=20 \pi(5+x)^{2} . where x x is the additional length of the radius in centimeters. Which of the following equivalent expressions displays the current volume of the cylinder as a constant or coefficient?\newline(A) π(20x2+200x+500) \pi\left(20 x^{2}+200 x+500\right) \newline(B) 20π(x2+10x+10)+300π 20 \pi\left(x^{2}+10 x+10\right)+300 \pi \newline(C) 20π(x2+10x+25) 20 \pi\left(x^{2}+10 x+25\right) \newline(D) 20πx2+200πx+500π 20 \pi x^{2}+200 \pi x+500 \pi

Full solution

Q. used for one of its products by enlarging the radius of the cylinder. The height must be 2020 centimeters. The new volume of the cylinder is given by the equation V(x)=20π(5+x)2 V(x)=20 \pi(5+x)^{2} . where x x is the additional length of the radius in centimeters. Which of the following equivalent expressions displays the current volume of the cylinder as a constant or coefficient?\newline(A) π(20x2+200x+500) \pi\left(20 x^{2}+200 x+500\right) \newline(B) 20π(x2+10x+10)+300π 20 \pi\left(x^{2}+10 x+10\right)+300 \pi \newline(C) 20π(x2+10x+25) 20 \pi\left(x^{2}+10 x+25\right) \newline(D) 20πx2+200πx+500π 20 \pi x^{2}+200 \pi x+500 \pi
  1. Expand Expression: Step 11: Given V(x)=20π(5+x)2V(x) = 20\pi(5 + x)^2. Expand the expression.
  2. Expand (5+x)2(5 + x)^2: \newlineStep 22: Expand (5+x)2(5 + x)^2.\newline(5+x)2=25+10x+x2(5 + x)^2 = 25 + 10x + x^2
  3. Multiply by 20π20\pi: Step 33: Multiply the expanded form by 20π20\pi. V(x)=20π(25+10x+x2)V(x) = 20\pi(25 + 10x + x^2)
  4. Distribute 20extextpi20 ext{ extpi}: Step 44: Distribute 20extextpi20 ext{ extpi} to each term inside the parentheses. V(x)=20extextpiimes25+20extextpiimes10x+20extextpiimesx2V(x) = 20 ext{ extpi} imes 25 + 20 ext{ extpi} imes 10x + 20 ext{ extpi} imes x^2 V(x)=500extextpi+200extextpix+20extextpix2V(x) = 500 ext{ extpi} + 200 ext{ extpi}x + 20 ext{ extpi}x^2
  5. Compare with Choices: Step 55: Compare the expanded form with the given choices. (A) π(20x2+200x+500)\pi(20x^2 + 200x + 500) (B) 20π(x2+10x+10)+300π20\pi(x^2 + 10x + 10) + 300\pi (C) 20π(x2+10x+25)20\pi(x^2 + 10x + 25) (D) 20πx2+200πx+500π20\pi x^2 + 200\pi x + 500\pi
  6. Identify Correct Choice: Step 66: Identify the correct choice that matches 500extπ+200extπx+20extπx2500 ext{π} + 200 ext{π}x + 20 ext{π}x^2. The correct choice is (D) 20extπx2+200extπx+500extπ20 ext{π}x^2 + 200 ext{π}x + 500 ext{π}.

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