a cub measures n inches on each edge. If a slice 1 inch thick is cut from one face of the cube and then a slice 3 inches thick is cut from another face of the cube, the resulting solid has a volume of 1560 cubic inches. Find the dimensions of the original cube.
original side = n
slice = (1)n^2
is the 2nd slice from the other square face, or one of the four nx(n-1) faces?
Assuming the latter, lest the problem be too, too trivial, then the 2nd
slice = 3(n)(n-1)
n^3 - n^2 - 3n(n-1) = 1560
n^3 - 4n^2 + 3n - 1560 = 0
Hmmm. 1560 = 13*12*10
(n-13)(n^2 + 9n + 120)
so, n=13
original volume: 2197
1st slice: 13x13 = 169
2nd slice: 3x13x12 = 468
remaining: 1560
To solve this problem, let's break it down step by step:
1. Start with a cube measuring n inches on each edge.
2. If a 1-inch thick slice is cut from one face, the height of the resulting solid will be reduced by 1 inch. So the height of the solid is now (n - 1) inches.
3. Similarly, if a 3-inch thick slice is cut from another face, the width of the resulting solid will be reduced by 3 inches. So the width of the solid is now (n - 3) inches.
4. Since the original cube was perfect, both in height and width, the length of the solid remains unchanged at n inches.
Now, let's calculate the volume of the resulting solid using the formula:
Volume = Length × Width × Height
Given that the resulting solid has a volume of 1560 cubic inches, we can plug in the values we determined earlier:
1560 = n × (n - 3) × (n - 1)
Simplify the equation:
1560 = (n^2 - 3n) × (n - 1)
Expand:
1560 = n^3 - 4n^2 + 3n
Rearrange the equation:
n^3 - 4n^2 + 3n - 1560 = 0
At this point, we can use numerical methods or synthetic division to find the value of n. Let's use synthetic division to find a root.
By trying different values, we find that n = 12 satisfies the equation. Therefore, the dimensions of the original cube are 12 inches on each edge.