A half of a pepperoni stick is 10 cm long. Assume that a cross section perpendicular to the axis of the pepperoni at a distance x from the end if a circle of radius rad(3x). What is the volume of the pepperoni
consider the stick as a stack of very thin slices of radius √(3x) and thickness dx. Apparently the whole stick is to be considered.
v = 2∫[0,10] πr^2 dx
where r^2 = 3x. So,
v = 2∫[0,10] π(3x) dx
v = 6π∫[0,10] x dx
I think you can probably handle that, ok?
To find the volume of the pepperoni, we need to calculate the volume of the cylindrical shape formed by the pepperoni stick.
First, let's find the radius of the cross section at a distance x from the end. The cross section is a circle with a radius of rad(3x). This means the radius of the circle is equal to the square root of 3x.
Next, let's find the length of the cylindrical shape. We are given that half of the pepperoni stick is 10 cm long, so the full length of the pepperoni stick is 2 * 10 cm = 20 cm.
Now, we can calculate the volume of the cylindrical shape using the formula for the volume of a cylinder:
Volume = π * radius^2 * height
In this case, the radius is rad(3x) and the height is 20 cm.
Therefore, the volume of the pepperoni is:
Volume = π * (rad(3x))^2 * 20
Simplifying further, let's square the square root:
Volume = π * (3x) * 20
Now, we can simplify even more:
Volume = 60πx
So, the volume of the pepperoni stick is 60πx cubic centimeters.