the surface area of a cube is increased from 24 square centimeters to 26.016 square centimeters . use differentials to determine:
(1) by approximately how many centimeters does the side length of the cube increase?
(2) by approximately how many cubic centimeters does the volume of the cube increase?
(3) by approximately what percentage does the volume of the cube increased?
surface area = a = 6 x^2
da/dx = 12 x
so if a = 24, x = 2
da = 12 x dx = 12*2 * dx = 24 dx
if da = 2.016
then dx = da/24 = .084 (answer part 1)
volume = v = x^3 = 8 when x = 2
so
dv/dx = 3 x^2
dv = 3 (2^2)(.084) (part b)
100 (dv / v)
To find the approximate increase in side length, volume, and percentage increase, we can use differentials.
Let's start with finding the increase in side length:
1) The surface area of a cube is given by 6s^2, where s is the side length.
So, the differential of the surface area dA with respect to the side length ds is given by: dA = 12s ds.
We know that dA is the difference in surface area, which is 26.016 - 24 = 2.016 square centimeters.
Plugging in the values, we have: 2.016 = 12s ds.
Now, we can solve for ds:
ds = 2.016 / (12s)
To approximate the increase in side length, we can assume that ds is very small and neglect the term (12s) in the denominator.
Therefore, approximate increase in side length = ds = 2.016 / 12 = 0.168 centimeters.
So, the side length of the cube increases by approximately 0.168 centimeters.
Next, let's find the increase in volume:
2) The volume of a cube is given by V = s^3.
The differential of the volume dV with respect to the side length ds is given by: dV = 3s^2 ds.
We know that the initial volume of the cube is s^3 = (24/6)^3 = 2^3 = 8 cubic centimeters.
The final volume is (s+ds)^3.
To find the increase in volume, we subtract the initial volume from the final volume:
dV = (s+ds)^3 - s^3.
Expanding the equation:
dV = s^3 + 3s^2 ds + 3s ds^2 + ds^3 - s^3.
Since ds is very small, we can neglect the terms involving ds^2 and ds^3.
Therefore, approximate increase in volume = dV = 3s^2 ds = 3(8) ds = 24 ds cubic centimeters.
So, the volume of the cube increases by approximately 24 ds cubic centimeters.
Finally, let's find the percentage increase in volume:
3) The initial volume of the cube is 8 cubic centimeters.
The final volume is V + dV = 8 + 24 ds.
To find the percentage increase in volume, we can use the formula:
percentage increase = (dV / V) * 100.
Plugging in the values, we have:
percentage increase = (24 ds / 8) * 100 = 3ds * 100.
Therefore, the volume of the cube increases by approximately 300 ds percent.
Note: The values of ds we obtained in the previous calculations are approximations, so the answers given for the increase in side length, volume, and percentage increase are also approximate.