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A cereal company wants to enlarge the volume of the cylindrical container 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)+3 (c) 20 pi(x^(2)+10 x+25) (D) 20 pix^(2)+200 pi x+500 pi

A cereal company wants to enlarge the volume of the cylindrical container 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?\newlineChoose 11 answer:\newline(A) π(20x2+200x+500) \pi\left(20 x^{2}+200 x+500\right) \newline(B) 20π(x2+10x+10)+3 20 \pi\left(x^{2}+10 x+10\right)+3 \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

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Full solution

Q. A cereal company wants to enlarge the volume of the cylindrical container 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?\newlineChoose 11 answer:\newline(A) π(20x2+200x+500) \pi\left(20 x^{2}+200 x+500\right) \newline(B) 20π(x2+10x+10)+3 20 \pi\left(x^{2}+10 x+10\right)+3 \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. Volume Formula: The volume of a cylinder is given by the formula V=πr2hV = \pi r^2 h, where rr is the radius and hh is the height. The problem states that the height of the cylinder is 2020 centimeters and the volume of the cylinder after increasing the radius by xx centimeters is given by V(x)=20π(5+x)2V(x) = 20\pi(5+x)^2. To find the current volume of the cylinder, we need to substitute x=0x = 0 into the equation V(x)V(x).
  2. Substituting x=0x = 0: Substituting x=0x = 0 into the equation V(x)=20π(5+x)2V(x) = 20\pi(5+x)^2 gives us V(0)=20π(5+0)2=20π(5)2=20π(25)=500πV(0) = 20\pi(5+0)^2 = 20\pi(5)^2 = 20\pi(25) = 500\pi. This is the current volume of the cylinder before the radius is increased.
  3. Comparing Given Options: Now we need to compare the given options to find which one displays the current volume of the cylinder as a constant or coefficient. The current volume is 500π500\pi, so we are looking for an expression that has this term as a constant or coefficient.
  4. Option (A): Option (A) is π(20x2+200x+500)\pi(20x^2 + 200x + 500). This option has 500500 as a coefficient of π\pi, which matches the current volume of 500π500\pi.
  5. Option (B): Option (B) is 20π(x2+10x+10)+320\pi(x^2 + 10x + 10) + 3. This option does not have 500π500\pi as a constant or coefficient, so it cannot be the correct answer.
  6. Option (C): Option (C) is 20π(x2+10x+25)20\pi(x^2 + 10x + 25). This option does not have 500π500\pi as a constant or coefficient, so it cannot be the correct answer.
  7. Option (D): Option (D) is 20πx2+200πx+500π20\pi x^2 + 200\pi x + 500\pi. This option has 500π500\pi as a constant, which matches the current volume of 500π500\pi.
  8. Final Comparison and Conclusion: Comparing the options, we see that both (A) and (D) have the term 500π500\pi. However, we need to find the expression that displays the current volume as a constant or coefficient in the same form as the given volume equation V(x)V(x). Option (D) is in the form of 20πx2+200πx+500π20\pi x^2 + 200\pi x + 500\pi, which is similar to the given equation V(x)=20π(5+x)2V(x) = 20\pi(5+x)^2 when expanded. Therefore, option (D) is the correct answer.

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