For problem 15):
The object whose volume you are trying to calculate can be broken down into three different shapes: a half-sphere, a cylinder, and a cone.
This requires knowledge of three different volume formulas:
Formula for the volume of a cylinder:
(pi)*r^2*length.
Volume of a sphere:
(4/3)(pi)(r^3)
volume of a cone:
(pi)(r^2)(h/3)
You can then add the formulas up to get an expression for the volume of the whole object:
(1/2)(sphere volume) + (cone volume) + (cylinder volume)
For problem 16:
Here we have a square pyramid. To find the volume of a pyramid with a square base, we use the formula:
(b^2)(h/3)
You know that the base is 6 units, and the height is 4 units, so for part a) just plug in those values and solve.
Parts b) and c) require you to compare the volume formula to two different values of the volume.
I'll show you part b, and let you work out c) on your own:
If volume = (b^2)(h/3)
and volume also equals (1/3)(8), we can set these volumes to be equal to each other and solve for b and h.
(b^2)(h/3) = (8)(1/3)
h/3 corresponds to 1/3, so h must equal 1.
(b^2) corresponds to 8, so b = sqrt(8)
You can apply this method to part c as well.
For part d, it's asking for the height of a pyramid with a volume of 9x^3 cubic units.
Again, volume = (b^2)(h/3), or (b^2)(1/3)(h)
The volume of the pyramid is 9x^3.
We want to rearrange 9x^3 into an expression that looks like (b^2)(1/3)(h).
First pull out the 1/3:
(1/3) (3x^3)
b^2 * h = 3x^3
What I'm not sure about here, is whether or not they're telling you that x is the length of the base. If that's the case, then we can solve this pretty easily:
x^2 * h = 3x^3
h = 3x
If x is a new variable, then you're going to have to solve for h in terms of x and b:
b^2 * h = 3x^3
h = (3x^3)/(b^2)
Hope this helps!
