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 = πr2h. Leave answer in terms of π.
Oh, I love a good cylinder question! Let's calculate the volume of this cylindrical wonder.
According to the formula V = πr^2h, we need to substitute the given expressions for the radius and height into the equation. So, we have:
V = π(4x + 1)^2(3x + 4)
To simplify this, we'll start by expanding the squared term:
V = π(16x^2 + 8x + 1)(3x + 4)
Now, let's distribute the π through the parentheses:
V = π(48x^3 + 64x^2 + 24x^2 + 32x + 3x + 4)
Combining like terms, we get:
V = π(48x^3 + 88x^2 + 35x + 4)
And voila! That's your polynomial in standard form for the volume of the cylinder. So, whenever you need to calculate the volume of this particular cylinder, just plug in the value of x and you're good to go!
To find the volume of a cylinder, we'll use the formula V = πr^2h, where r represents the radius of the cylinder and h represents the height. In this case, the radius is 4x + 1 and the height is 3x + 4.
We'll substitute these values into the formula to calculate the volume:
V = π(4x + 1)^2(3x + 4)
Now, we'll expand the square of the radius:
V = π(16x^2 + 8x + 1)(3x + 4)
To multiply two binomials, we'll use the distributive property:
V = π(48x^3 + 24x^2 + 3x + 64x^2 + 32x + 4)
Next, let's combine like terms:
V = π(48x^3 + 88x^2 + 35x + 4)
Finally, we can write the polynomial in standard form:
V = 48πx^3 + 88πx^2 + 35πx + 4π
Therefore, the polynomial in standard form for the volume of the cylinder is 48πx^3 + 88πx^2 + 35πx + 4π.
To find the volume of a cylinder, we can use the formula V = πr²h, 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 and simplify the expression to get the polynomial in standard form.
V = π(4x + 1)²(3x + 4)
First, let's square the quantity (4x + 1) using the distributive property:
V = π((4x + 1)(4x + 1))(3x + 4)
Expanding this expression, we get:
V = π(16x² + 4x + 4x + 1)(3x + 4)
Simplifying further:
V = π(16x² + 8x + 1)(3x + 4)
Now, we multiply each term in the first bracket by each term in the second bracket:
V = π(48x³ + 64x² + 24x² + 32x + 3x + 4)
Combining like terms:
V = π(48x³ + 88x² + 35x + 4)
Hence, the polynomial in standard form for the volume of the cylinder is 48x³ + 88x² + 35x + 4, expressed in terms of π.
v = πr^2 h = π(4x+1)^2 (3x+4)
now just expand that out.