The volume of a cube is increased from 729 cubic centimetres to 849.285 cubic centimetres
Use differentials to determine :
(a) by approximately how many centimetres does the side length of the cube increase?
(b)by approximately how many centimetres does the surface area of the cube increase?
(c) by approximately what percentage does the surface area of the cube increase?
Hint:
http://www.jiskha.com/display.cgi?id=1413053733
v = x^3
729 = 9^3 so x = 9
dv = 3 x^2 dx = 243 dx = 849.285 - 729
etc
Use differentials to determine by approximately what percentage does the perimeter of a square table increase if its diagonal is increased from 30 centimetres to 31.245 centimetres
To determine the approximate increase in the side length, surface area, and percentage increase in surface area of the cube, we can make use of differentials.
(a) To find the approximate increase in the side length of the cube, we can use the differential of the volume equation for a cube.
Let's denote the side length of the original cube as "x" and the increase in volume as "ΔV". We know that the volume of the original cube is 729 cubic centimeters, and the volume after the increase is 849.285 cubic centimeters.
The volume of a cube is given by V = x^3. So we have:
729 + ΔV = (x + Δx)^3
Expanding the right side using the binomial expansion formula (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 and neglecting higher powers of Δx:
729 + ΔV ≈ x^3 + 3x^2Δx
Subtracting x^3 from both sides and neglecting higher powers of Δx:
ΔV ≈ 3x^2Δx
Now, we can solve for Δx by rearranging the equation:
Δx ≈ ΔV / (3x^2)
Substituting the values, ΔV = 849.285 - 729 and x = cube root of 729, we can calculate Δx.
(b) To find the approximate increase in the surface area of the cube, we use the differential of the surface area equation for a cube.
The surface area of a cube is given by S = 6x^2. Taking the differential of this equation, we have:
dS = 12x dx
Here, dx represents the change in the side length of the cube.
(c) To find the approximate percentage increase in the surface area of the cube, we can use the formula:
Percentage increase = (Change in surface area / Original surface area) * 100
Let's denote the original surface area as "S1" and the surface area after the increase as "S2". The change in surface area is ΔS = S2 - S1.
Substituting the values into the percentage increase formula:
Percentage increase ≈ (ΔS / S1) * 100
Now, you can plug in the values and calculate the approximate increase in the side length, surface area, and the percentage increase in surface area using differentials.