A cylinder has a radius of 6x + 4 and a height of 8x + 7. Which polynomial in standard form best describes the total volume of the cylinder? Use the formula 𝑉 = 𝜋𝑟^2h for the volume of a cylinder.
a. 288𝜋𝑥^3 + 636𝜋𝑥^2 + 464𝜋𝑥 + 112𝜋
b. 288𝜋𝑥^3 − 636𝜋𝑥^2 + 464𝜋𝑥 + 112𝜋
c. 288𝜋𝑥^3 + 636𝜋𝑥^2 + 464𝜋𝑥 + 112
d. 288𝜋𝑥^3 + 636𝜋𝑥^2 − 464𝜋𝑥 + 112𝜋
its b
Well, I hope this answer doesn't make you cylinder with laughter, but let's calculate the volume of this cylinder!
The formula for the volume of a cylinder is 𝑉 = 𝜋𝑟^2h, where r is the radius and h is the height.
Substituting the given expressions for the radius and height, we get:
𝑉 = 𝜋(6x + 4)^2(8x + 7)
Simplifying the expression inside the brackets, we have:
𝑉 = 𝜋(36x^2 + 48x + 16)(8x + 7)
Now, let's distribute the 𝜋 to each term:
𝑉 = 𝜋(288x^3 + 384x^2 + 128x + 224x^2 + 384x + 128)
Simplifying further, we combine like terms:
𝑉 = 𝜋(288x^3 + 608x^2 + 512x + 128)
Now, we have the volume of the cylinder. Let me tell you, this calculation was only mildly cylindrical, but let's see which polynomial is the most accurate representation!
Comparing the expression we derived with the given options, we can see that option (c) is the closest match: 288𝜋𝑥^3 + 636𝜋𝑥^2 + 464𝜋𝑥 + 112.
So, the correct answer is option C. It looks like we've got a winner!
I hope this answer didn't twist you out of shape, and that it was just the right amount of silly to make you smile!
To find the total volume of the cylinder, we use the formula V = πr^2h, where r is the radius and h is the height.
Given that the radius of the cylinder is 6x + 4 and the height is 8x + 7, we substitute these values into the formula:
V = π(6x + 4)^2(8x + 7)
Expanding the square of the radius:
V = π(36x^2 + 48x + 16)(8x + 7)
Simplifying further by multiplying each term:
V = π(288x^3 + 384x^2 + 128x + 112x^2 + 144x + 48)
Combining like terms:
V = π(288x^3 + 496x^2 + 272x + 48)
So, the polynomial in standard form that best describes the total volume of the cylinder is:
288πx^3 + 496πx^2 + 272πx + 48π.
Therefore, the correct answer is option c. 288πx^3 + 636πx^2 + 464πx + 112.
To find the total volume of a cylinder, we can use the formula V = πr^2h, where V represents the volume, π is a mathematical constant (pi), r is the radius, and h is the height.
In this case, the given cylinder has a radius of 6x + 4 and a height of 8x + 7. Substituting these values into the formula, we get:
V = π(6x + 4)^2(8x + 7)
Now, let's simplify this expression to determine the polynomial that best represents the total volume of the cylinder.
First, let's expand the squared term (6x + 4)^2 using the exponent rule (a + b)^2 = a^2 + 2ab + b^2:
V = π(36x^2 + 48x + 16)(8x + 7)
Then, we can distribute π to each term inside the parentheses:
V = 288πx^3 + 384πx^2 + 128πx + 224πx + 252π
Combining like terms, we have:
V = 288πx^3 + 384πx^2 + 352πx + 252π
Now, we can rewrite the expression in standard polynomial form by arranging the terms in descending order of the exponent:
V = 288πx^3 + 384πx^2 + 352πx + 252π
Therefore, the correct polynomial in standard form that best describes the total volume of the cylinder is:
c. 288𝜋𝑥^3 + 384𝜋𝑥^2 + 352𝜋𝑥 + 252𝜋
given that V =pi r^2 h
r = 2x + 3
h = 6x + 1
v = pi ( 2x + 3 )^2 ( 6x + 1 )
V = pi ( 4x^2 + 12x + 9 ) ( 6x + 1 )
V = pi ( 24x^3 + 4x^2 + 72x^2 + 12x + 54x + 9 )
V = pi ( 24x^3 + 76x^2 + 66x + 9 )
V = 24pix^3 + 76pix^2 + 66pix + 9pi