Decreasing cube. Each of the three dimensions of a
cube with sides of length s centimeters is decreased by a
whole number of centimeters. The new volume in cubic
centimeters is given by
V(s) equals s power of 3 -� 13s power of 2 +� 54s -� 72.
a) Find V(10).
b) If the new width is s -� 6 centimeters, then what are the
new length and height?
c) Find the volume when s equals 10 by multiplying the
length, width, and height.
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To solve this problem, we will follow the given steps:
a) To find V(10), we need to substitute s=10 into the equation V(s) = s^3 - 13s^2 + 54s - 72.
So, we have:
V(10) = (10)^3 - 13(10)^2 + 54(10) - 72
= 1000 - 13(100) + 540 - 72
= 1000 - 1300 + 540 - 72
= 168
Therefore, V(10) = 168 cubic centimeters.
b) The new width is given as s - 6 centimeters. Let's assume the new length as L and the new height as H.
The original width of the cube was s centimeters. Since it has decreased by a whole number of centimeters, the new width s - 6 is equal to s - 6.
We know that the volume of the cube is given by V = length*width*height. Therefore,
V = L * (s - 6) * H
Since all three dimensions have decreased by the same amount, we can write the following equations:
L = s - 6
H = s - 6
So, the new length is equal to s - 6 centimeters, and the new height is also equal to s - 6 centimeters.
c) To find the volume when s = 10 by multiplying the length, width, and height, we substitute s = 10, L = 10 - 6, and H = 10 - 6 into the volume formula:
V = L * (s - 6) * H
= (10 - 6) * (10 - 6) * (10 - 6)
= 4 * 4 * 4
= 64
Therefore, when s = 10, the volume of the cube is 64 cubic centimeters.