Final Answer:
The estimated volume of the fiberglass shell covering the cube with sides 14 inches long is approximately 739.2 cubic inches.
Step-by-step explanation:
To estimate the volume of the fiberglass shell, we can use differentials. Let V represent the volume of the cube, and let h represent the thickness of the fiberglass coating. The volume of the cube with the coating is given by V = (14 + 2h)^3, where the cube's original side length is 14 inches. Taking differentials, we get dV = 3(14 + 2h)^2 * 2dh. To find the estimated change in volume, we substitute h = 0.7 inches into the equation and find dh. Then, we can plug the values into the differential equation.
Starting with the given differential equation:
dV = 3(14 + 2h)^2 . 2dh.
Substituting h = 0.7 inches:
dV = 3(14 + 2(0.7))^2 . 2 . 0.7.
Simplifying the expression:
dV = 3(15.4)^2 . 1.4.
Calculating the result:
dV = 3(237.16) . 1.4 = 711.48.
This represents the change in volume. Adding this to the original volume of the cube gives the estimated volume of the fiberglass shell:
V_shell ≈ V + dV = (14 + 2(0.7))^3 + 711.48.
Finally, evaluating this expression gives the final answer:
V_shell ≈ 739.2 cubic inches
Therefore, the estimated volume of the fiberglass shell is approximately 739.2 cubic inches.