b The radius is now 2 meters. Repeat the calculations from Part A.
C
c The length of the tank is unchanged which means the formula for both the cylinder and the hemispheres will depend on the radius.
A
a V≈ 16.76 m^3
B
b No, it does not
V≈ 83.78 m^3
C
c V(r)=4/3π r^3+4π r^2
Practice makes perfect
a We know that the tank's cylindrical portion is 4m long and its radius is 1m. If we detach the hemispheres and put them together, we can think of this as two separate solids — an open cylinder with a radius of 1m and a length of 4m, and a sphere with a radius of 1m.
To calculate the total volume of the gas tank, we have to add the volume of the sphere and the volume of the cylinder.
|c|c|c|
[-1em]
Solid & Calculation & Volume [0.2em]
[-0.8em]
Sphere & 4/3π( 1)^3 &4π/3 [0.8em]
[-0.4em]
Cylinder & π( 1)^2( 4) &4π [0.8em]
Finally, we can add the volumes of the sphere and the cylinder to get the volume of the tank.
4π/3+4π ≈ 16.76 m^3
b If we double the radius, the sphere and cylinder will have a new radius of 2 m. Let's calculate the new volume of the two parts using this radius.
|c|c|c|
[-1em]
Part & Calculation & = [0.2em]
[-0.8em]
Sphere & 4/3π(2)^3 &32π/3 [0.8em]
[-0.4em]
Cylinder & π(2)^2(4) &16π [0.8em]By adding the volume of the parts, we get the volume of the tank
32π/3+16π ≈ 83.78m^2
Finally, we can divide the larger volume by the smaller volume to calculate how much more the second tank holds compared to the first. We will use the exact volumes for this.
c Notice that the volume only depends on the radius since the length of the cylinder is set. With this information, we can write expressions for each body's volume.
Volume of Sphere:& V=4/3π r^3
Volume of Cylinder:& V=4π r^2
The function, V(r), that will give you the total volume of the tank is the sum of the right-hand sides in the above equations.
V(r)=4/3π r^3+4π r^2
