A bowl is created with circular cross sections so that its radius (in cm) is a function of height (in cm) according to the equation r=4sqrt(h+1). Find the volume of soup that fills the bowl if the bow is 2 cm high.
Consider the interior of the bowl as a stack of thin circles. At height h, the circle has area
πr^2 = π(16(h+1))
So integrate that from 0 to 2
To find the volume of the soup that fills the bowl, we'll first need to calculate the volume of the bowl itself.
The bowl is created with circular cross-sections, which means that each cross-section is a circle. The radius of each circle is given by the equation r = 4√(h + 1), where h is the height of the bowl.
Since we're given that the height of the bowl is 2 cm, we can substitute h = 2 into the equation to find the radius of the cross-section at that height:
r = 4√(2 + 1)
= 4√3 cm
Now, let's calculate the volume of each cross-section. The volume of a circular cross-section is given by the formula V = πr^2, where r is the radius.
V = π(4√3)^2
= π(16)(3)
= 48π cm^2
Since each cross-section has the same area, the volume of the entire bowl is given by the integral of the cross-section area over the entire range of heights. In this case, the range of heights is from 0 to 2 cm.
Therefore, the volume of the bowl is:
V_bowl = ∫[0,2] 48π dh
To find the value of this integral, we integrate the function 48π with respect to h over the interval [0,2]:
V_bowl = [48πh]_[0,2]
= 48π(2) - 48π(0)
= 96π cm^3
So, the volume of the soup that fills the bowl is 96π cubic centimeters.