Jan built a cube from unit cubes (each 1 by 1 by 1 unit). Ken took this cube apart,and used the unit cubes to build a rectangular solid that was the same height asJan's cube but 2 units greater in width and 2 units less in length. Ken had 24 unit cubes left over. How many unit cubes did Ken use to build his figure?
solve
x^3 - x(x+2)(x-2) = 24
x^3 - x(x^2-4) = 24
x^3 - x^3 + 4x = 24
4x=24
x=8
So Jan used 8^3 or 515 cubes
Ken used 24 less or 488 cubes
check:
original:
8^3 = 512
Ken's cube = 8 x 6 x 10 = 480
512-480=24
Thanks much.
The answer is X=6, not X=8. Ken's cube was 6 x 4 x 8 = 192, Jan's was 6 x 6 x 6 = 216. 216 - 192 = 24.
To find the number of cubes Ken used to build his figure, we need to calculate the volume of Jan's cube and the volume of Ken's rectangular solid.
Let's start by finding the volume of Jan's cube. A cube is a three-dimensional shape with equal lengths, widths, and heights. Since Jan's cube is built from unit cubes, each side of the cube has a length of 1 unit. Therefore, Jan's cube has a volume of 1^3 = 1 unit^3.
Now let's find the volume of Ken's rectangular solid. We know that Ken's figure has the same height as Jan's cube but is 2 units wider and 2 units shorter in length.
Let's say the width of Ken's solid is w units and the length is l units. Since the height is the same as Jan's cube, it is also 1 unit.
Given that Ken had 24 unit cubes left over, we can set up the following equation:
Volume of Jan's cube - Volume of Ken's solid = Cubes left over
1 unit^3 - (w + 2)(l - 2)(1) = 24
Now we need to solve this equation to find the values of w and l.
Expanding the expression on the left side, we get:
1 - (w + 2)(l - 2) = 24
Simplifying further:
1 - wl + 2w - 2l - 4 = 24
Combine like terms:
- wl + 2w - 2l - 3 = 24
Rearranging the terms:
- wl + 2w - 2l = 24 + 3
- wl + 2w - 2l = 27
To solve this equation, we need more information.