A cylinder has a radius of 4x + 1 and a height of 3x + 4. Write the
polynomial in standard form for the volume of the cylinder. Use the
formula: V = πr^2h. Leave answer in terms of π.
We can begin by substituting the given values into the formula:
V = π(4x+1)^2(3x+4)
To simplify this expression, we need to expand the squared term:
V = π(16x^2 + 8x + 1)(3x+4)
Next, we can use the distributive property to multiply the two binomials:
V = π(48x^3 + 64x^2 + 24x + 12x^2 + 16x + 4)
Simplifying the terms inside the parentheses, we get:
V = π(48x^3 + 76x^2 + 40x + 4)
Finally, we can multiply the entire expression by π and write it in standard form:
V = 48πx^3 + 76πx^2 + 40πx + 4π
Therefore, the polynomial in standard form for the volume of the cylinder is 48πx^3 + 76πx^2 + 40πx + 4π.
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To find the volume of a cylinder, we can use the formula V = πr^2h, where V represents the volume, r represents the radius, and h represents the height.
In this case, the radius of the cylinder is given as 4x + 1, and the height is given as 3x + 4. We can substitute these values into the formula to get:
V = π(4x + 1)^2(3x + 4)
To simplify this expression, we need to expand and multiply the terms inside the parentheses. Using the binomial expansion formula (a + b)^2 = a^2 + 2ab + b^2, we can expand (4x + 1)^2 as:
(4x + 1)^2 = (4x)^2 + 2(4x)(1) + (1)^2
= 16x^2 + 8x + 1
Substituting this back into the original expression, we have:
V = π(16x^2 + 8x + 1)(3x + 4)
Now, we can multiply the terms using the distributive property:
V = π * 16x^2 * 3x + π * 16x^2 * 4 + π * 8x * 3x + π * 8x * 4 + π * 1 * 3x + π * 1 * 4
Simplifying further, we get:
V = 48πx^3 + 64πx^2 + 24πx + 32π + 3πx + 4π
Combining like terms, we have:
V = 48πx^3 + 64πx^2 + 27πx + 36π
Therefore, the polynomial in standard form for the volume of the cylinder is 48πx^3 + 64πx^2 + 27πx + 36π.