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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 2020 centimeters. The new volume of the cylinder is given by the equation \newlineV(x)=20π(5+x)2V(x)=20\pi(5+x)^{2}.\newlinewhere xx 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) 20πx2+200πx+500π20\pi x^{2}+200\pi x+500\pi\newline(B) 20π(x2+10x+25)20\pi(x^{2}+10x+25)\newline(C) 20π(x2+10x+25)20\pi(x^{2}+10x+25)\newline(D) 20πx2+200πx+500π20\pi x^{2}+200\pi x+500\pi

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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 \newlineV(x)=20π(5+x)2V(x)=20\pi(5+x)^{2}.\newlinewhere xx 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) 20πx2+200πx+500π20\pi x^{2}+200\pi x+500\pi\newline(B) 20π(x2+10x+25)20\pi(x^{2}+10x+25)\newline(C) 20π(x2+10x+25)20\pi(x^{2}+10x+25)\newline(D) 20πx2+200πx+500π20\pi x^{2}+200\pi x+500\pi
  1. Understand Equation: Understand the given equation for the new volume of the cylinder.\newlineThe equation provided is V(x)=20π(5+x)2V(x) = 20\pi(5+x)^2, which represents the volume of the cylinder after increasing the radius by xx centimeters. The current volume of the cylinder can be found by setting xx to 00, since xx is the additional length added to the original radius.
  2. Calculate Current Volume: Calculate the current volume of the cylinder by setting xx to 00 in the given equation.V(0)=20π(5+0)2V(0) = 20\pi(5+0)^2V(0)=20π(5)2V(0) = 20\pi(5)^2V(0)=20π(25)V(0) = 20\pi(25)V(0)=500πV(0) = 500\piThis calculation gives us the current volume of the cylinder as a constant, which is 500π500\pi cubic centimeters.
  3. Compare with Answer Choices: Compare the result from Step 22 with the given answer choices to find the equivalent expression.\newlineWe are looking for an expression that has the constant 500π500\pi as part of it when xx is 00. Let's check each option:\newline(A) π(20x2+200x+500)\pi(20x^2 + 200x + 500) - When x=0x=0, this simplifies to π(500)\pi(500), which matches our calculation.\newline(B) 20π(x2+10x+10)+320\pi(x^2 + 10x + 10) + 3 - This does not simplify to 500π500\pi when x=0x=0.\newline(C) 20π(x2+10x+25)20\pi(x^2 + 10x + 25) - This does not simplify to 500π500\pi when x=0x=0.\newline(D) xx22 - This simplifies to 500π500\pi when x=0x=0, but it is not written as a single expression with a constant or coefficient.\newlineThe correct answer is the one that simplifies to 500π500\pi when x=0x=0 and is written as a single expression with a constant or coefficient. Option (A) meets these criteria.

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