Final answer:
The maximum possible error in the volume of a cube with an edge length of 15 cm and a measurement error of 0.2 cm is 135 cm³. The relative error is 0.04, which translates into a percentage error of 4% for the volume of the cube.
Step-by-step explanation:
You've been tasked with calculating the maximum possible error, relative error, and percentage error in the volume of a cube with an edge length of 15 cm and a possible measurement error of 0.2 cm. To do this, you will use differentials to estimate the errors.
The volume, V, of a cube with edge length x is given by V = x^3. The differential dV, which represents the error in the volume, can be found by differentiating the volume equation with respect to x, which gives dV = 3x^2 dx. Here, dx is the error in the measurement of the edge length. Substituting x = 15 cm and dx = 0.2 cm into the differential equation yields dV = 3(15)^2(0.2) = 135 cm^3. This is the maximum possible error in the volume.
The relative error is the maximum possible error divided by the actual measured volume, and the percentage error is the relative error expressed as a percentage. The volume of the cube is 15^3 = 3375 cm^3. Therefore, the relative error is 135 / 3375 = 0.04, or 4%. Hence, the percentage error in the volume is 4%.