Empirical Calculus

Calculus Level 3

A scientist has determined that an output quantity $z$ is related to input quantities $x$ and $y$ in the following way:

$\large{z = A \, x^B \, y^C}$

Initially, the scientist's experimental setup has $(x_0,y_0,z_0) = (2,4,57344)$ . These are the experimental baseline numbers. The scientist then perturbs the inputs and measures the changes in the output in order to determine constants $A,B,C$ .

The results of the first trial are $(x = x_0 + 0.0001, y = y_0, z = z_0 + 8.6020301)$ .
The results of the second trial are $(x = x_0, y = y_0 + 0.0001 , z = z_0 + 7.1683584)$ .

Based on the scientist's empirical results, calculate $A,B,C$ and round each one to the nearest integer. Enter your answer as the sum of the three rounded numbers.

The answer is 15.

1 solution

Steven Chase
Mar 18, 2018

By altering one variable at a time and comparing the change in the output to the change in the input, the scientist is effectively taking partial derivatives.

$\large{z = A \, x^B \, y^C \\ \frac{\partial{z}}{\partial{x}} = B A \, x^{B-1} \, y^C = B A \, x^B \, x^{-1} \, y^C = B \, z \, x^{-1} \\ B = \frac{\partial{z}}{\partial{x}} \, \frac{x}{z} \\ \frac{\partial{z}}{\partial{y}} = C A \, x^B \, y^{C-1} = C A \, x^B \, y^C \, y^{-1} = C \, z \, y^{-1} \\ C = \frac{\partial{z}}{\partial{y}} \, \frac{y}{z}}$

Evaluate the expressions for $B$ and $C$ given measurement data. Then calculate $A$ using the baseline numbers. The results come out to:

$\large{A = 7 \\ B = 3 \\ C = 5}$

@Steven Chase Hello somehow I have predicted your home location.
I just want know that is it correct or not??

Talulah Riley - 9 months, 2 weeks ago

Empirical Calculus

The answer is 15.

1 solution

0 pending reports