## Related questions with answers

Personal consumption expenditures (in billions of dollars) for several types of recreation from 2012 through 2017 are shown in the table, where $x$ is the expenditures on amusement parks and campgrounds, $y$ is the expenditures on live entertainment (excluding sports), and $z$ is the expenditures on spectator sports.

Year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
---|---|---|---|---|---|---|

x | 44.3 | 46.8 | 49.8 | 54.2 | 57.9 | 61.9 |

y | 24.9 | 24.8 | 26.5 | 28.9 | 30.7 | 32.6 |

z | 20.9 | 22.1 | 23.2 | 23.1 | 24.2 | 26.1 |

A model for the data is given by

z = 0.54x - 0.6y + 11.98

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

Solution

VerifiedThis is the first part of a two-part exercise, in which, based on collected data, we are given a model for $z$, expenditures on spectator sports, which is interpreted as a function of two variables:

- $x$, expenditures on amusement parks and campgrounds,
- $y$, expenditures on live entertainment excluding sports.

The model is

$z(x,y)=0.54x-0.6y+11.98$

Our job here is to find the two first partial derivatives, $\partial z/\partial x$ and $\partial z/\partial y$.

*How are partial derivatives calculated?*

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