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) _ s3 _ 13s2 _ 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 _ 10 by multiplying the
length, width, and height
a.) V(s)= S^3 -13s^2 +54s -72
V(10)= 10^3-13(10)^2+54(10)-72
V(10)= 1000-1300+540-72
V(10)= 168
To find the solutions to these questions, we need to substitute the given values of s into the equation V(s) = s^3 - 13s^2 - 54s - 72.
a) To find V(10), substitute s = 10 into the equation:
V(10) = (10)^3 - 13(10)^2 - 54(10) - 72
= 1000 - 13(100) - 540 - 72
= 1000 - 1300 - 540 - 72
= -912.
Therefore, V(10) = -912 cubic centimeters.
b) Given the new width s = 6 centimeters, we can find the new length and height by substituting s = 6 into the equation V(s) = s^3 - 13s^2 - 54s - 72, and then solving for the remaining unknowns.
V(s) = s^3 - 13s^2 - 54s - 72
V(6) = (6)^3 - 13(6)^2 - 54(6) - 72
V(6) = 216 - 13(36) - 54(6) - 72
V(6) = 216 - 468 - 324 - 72
V(6) = -648.
Therefore, the new volume when the width is 6 centimeters is -648 cubic centimeters.
c) To find the volume when s = 10 by multiplying the length, width, and height, we need to find the values for the length and height. Since the cube has equal dimensions, the length, width, and height are all equal to s.
Therefore, when s = 10, the length, width, and height are all 10 centimeters.
The volume is then calculated by:
Volume = Length * Width * Height
Volume = 10 * 10 * 10
Volume = 1000 cubic centimeters.
Therefore, when s = 10, the volume of the cube is 1000 cubic centimeters.