Problem 1
1a) Volume measures the amount of space an object takes up while the surface area is the amount of outer area that we see of the object. Basically volume is the amount of stuff inside and the surface area is the amount of stuff outside.
1b) Square units are used for surface area. Cubic units are used for volume. For example: square inches are used for surface area and cubic inches are used for volume. The square units are two dimensional (think flat plane) and the cubic units are 3D space.
1c) You use volume if you want to know how much space an object takes up. You can also use volume to find the mass assuming you know the density. From there you can find things like center of mass. Surface area will tell you how much paint you'll need to cover the outer faces of the object, which is handy if you want to know how to wrap it efficiently.
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Problem 2
To graph (3,0,0), you would start at the origin, move 3 units along the x axis (axis coming out of the board) and then plot a point. We don't move along the y or z axis at all.
Plot (0,8,0) by plotting a dot at 8 on the y axis. Similarly, plot (0,0,6) by placing a dot at 6 on the z axis.
Draw a triangle with these three points. This will give you a rough idea of what the plane looks like. The plane extends infinitely in all directions but truncated as a triangle makes it easier to see I think.
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Problem 3
Focus on triangle AED. This is one of the base faces of the triangular prism. The area of this triangle is 6 square units (base = AE = 3, height = ED = 4; base*height/2 = 3*4/2 = 12/2 = 6)
Multiply the area of the prism face by the height or depth of the prism
Volume = (area of face)*(depth)
Volume = (6)*(AB)
Volume = (6)*(4)
Volume = 24
The volume of this 3D solid is 24 cubic units
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Problem 4
The value t can be anything you want. Let's say t = 0 as it's easiest to work with
If t = 0, then...
x = t
x = 0
y = -3*t+6
y = -3*0+6
y = 6
z = t-3
z = 0-3
z = -3
So t = 0 leads to the ordered triple (x,y,z) = (0,6,-3)
If t = 1, then...
x = t
x = 1
y = -3*t+6
y = -3*1+6
y = 3
z = t-3
z = 1-3
z = -2
So t = 1 leads to the ordered triple (x,y,z) = (1,3,-2)
If t = 2, then...
x = t
x = 2
y = -3*t+6
y = -3*2+6
y = 0
z = t-3
z = 2-3
z = -1
So t = 2 leads to the ordered triple (x,y,z) = (2,0,-1)
This process is repeated over and over to generate as many points as you need. The more points, the more accurate the graph. A straight line should form. This straight line isn't restricted to the 2D plane as it normally was in the past. Instead, it passes through 3D space. Though of course you'll have to try your best to plot a 3D figure on a 2D flat surface.
