A cube made out of lead has mass 12 grams. A side of the cube has length 1 cm. What would be the mass of a lead cube whose sides
2 answers:
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See the attached picture.
- Create an equation for the volume of the box, find the zeroes, and sketch the graph of the function.
The resulting box has a volume
because the volume of a box is the product of its width , length
, and height
.
- find the zeroes
You know right away from the factored form of that the zeroes are
. (zero product property)
- sketch the graph of the function
Easy to plot by hand. You know the zeroes, and you can check the sign of for any values of
between these zeros to get an idea of what the graph of
looks like. See the second attached picture.
Here's what I mean by "check the sign" in case you don't follow. We know when
and
. So we pick some value of
between them, say
, and find that
which is positive, so will be positive for any other
between 0 and 8. Similarly we would find that
for
between 8 and 12, and so on.
- What is the size of the cutout he needs to make so that he can fit the most marbles in the box?
It's impossible to answer this without knowing the volume of each marble...
- If Thomas wants a volume of 12 cubic inches, what size does the cutout need to be?
Thomas wants , so you solve
While this is possible to do by hand, the procedure is tedious (look up "solving the cubic equation"). With a calculator, you'd find three approximate solutions
but you throw out the third solution because, realistically, the cutout length can't be greater than either of the sheet's dimensions.
- What would be the dimensions of this box?
The box's dimensions are ( in) x (
in) x (
in).
If , then
and
.
If , then
and
.
The first thing we notice is that the function is reflected. So we can start by reflecting it with respect to the x-axis.
We do this by adding a negative sign in the function:
Now, we have the function reflected, but in the wrong position. We can track its position by the vertex. It was originally at (0,0) and remains at (0,0) after the reflection.
But the final function have its vertex at (-5,0), so we have to translate the function 5 units to the left. we do this by adding 5 to the x in the function:
Now, to check if there isn't any dilatation, we can check on other point in the graph to see if it checks out.
In the blue graph, we see the point (-3,-4), so let's input x = -3 and see if it checks out:
We got y = -4, so it checks out.
Thus, the answer is: