Find the volume of the solid that lies above the cone 3 and below the sphere 4 cos
To find the volume of the solid that lies between the cone and the sphere, we will need to use the formula for the volume of a cone and the formula for the volume of a sphere. The volume of a cone can be found using the following formula:
V = 1/3 * pi * r^2 * h
where V is the volume, r is the radius of the base, h is the height, and pi is 3.14. The volume of a sphere can be found using this formula:
V = 4/3 * pi * r^3
where V is the volume, r is the radius, and pi is 3.14. In order to find the desired volume, we will need to subtract the volume of the cone from the volume of the sphere. We will also need to account for the fact that only part of each object is being considered by multiplying each by a certain factor.
For instance, if we want to find how much water will fit in a container that has a cone-shaped bottom and is half full, we would use these calculations:
ContainerVolume = 1/2 * SphereVolume – 1/2 * ConeVolume
Now let's plug in what we know. We are looking for how much water will fit in a container with a radius of 4 cm and a height of 3 cm that has a cone-shaped bottom and is half full. This means our calculation would look like this:
ContainerVolume = 1/2 * (4/3 * 3.14 * 4^3) – 1/2 * (1/3 * 3.14 * 4^2* 3)
Which can be simplified to:
ContainerVolume = 1/6 * 1256.64 – 1/6* 201.06
Which equals approximately 843 mL</p><br /><h2>Use the method of disks or washers to find the volume </h2><br /><p>
To find the volume of the solid that lies above the cone z=3 and below the sphere z=4 cos , we can use the method of disks or washers.
We slice the solid into thin disks or washers, with each one perpendicular to the z-axis. For each disk or washer, we need to find its radius and height.
The radius of each disk or washer will be equal to the radius of the sphere at that height – that is, r = 4 cos h.
The height of each disk or washer will be equal to the difference in height between the top and bottom of that slice – that is, h = 4 cos – 3.
Therefore, the volume of each disk or washer will be V = πr2h.
Now we need to sum up the volumes of all of these disks or washers. We can do this by integrating V(h) from h = 0 to h = π/2:
V(h) = πr2h ∫V(h)dh = ∫πr2dh ∫V(h)dh = [πr2dh]0π/2 ∫V(h)dh = [π(4 cos h)2(4 cos h - 3)]0π/2 Vtotal = [π(4 cos h)3 - 3π(4 cos h)4]0π/2</p><br /><h2>Express the volume in terms of π</h2><br /><p>
The volume of the solid that lies above the cone and below the sphere is given by:
volume = π * 3 * (4 – 3 cos θ)