![\bold{\huge{\underline{ Solution }}}](https://img.qammunity.org/2023/formulas/mathematics/high-school/jdi2w7914cic76zpb2xuxp7e51pz44d9g8.png)
Given :-
- We have given one function f(x) = -45x + 270
- The given function represents the total distance in miles.
- The initial interval was x = 1 and final interval was x = 3
To Find :-
- We have to find the rate of change over the given interval
Let's Begin :-
Here, We have
- Function = f(x) = -45x + 270
The given function represents the total distance in miles covered by the traveller.
Therefore,
For initial interval that is x = 1 , Distance covered by the traveller
![\sf{ f(x) = -45x + 270 }](https://img.qammunity.org/2023/formulas/mathematics/high-school/lll9c498sllvrt1ys0yiolxqxidi7xs58b.png)
![\sf{ f(1) = - 45(1) + 270 }](https://img.qammunity.org/2023/formulas/mathematics/high-school/guefvwvzrc6em0ck1av9p0tq3efkxpkqe2.png)
![\sf{ f(1) = - 45 + 270 }](https://img.qammunity.org/2023/formulas/mathematics/high-school/5c80j2apqjt1rr83ivc3310y002snvoysz.png)
![\sf{ f(1) = 225 }](https://img.qammunity.org/2023/formulas/mathematics/high-school/34zkf7wuu9auqewzerttvhunxwf6t48m97.png)
For final interval that is x = 3, Distance covered by the traveller
![\sf{ f(x) = - 45x + 270 }](https://img.qammunity.org/2023/formulas/mathematics/high-school/p98r3jee18z916ewloj8j4s4mgmbxpihqr.png)
![\sf{ f(1) = - 45(3) + 270 }](https://img.qammunity.org/2023/formulas/mathematics/high-school/eavnspxomrf0x00yp6pod4146q5zpzesmg.png)
![\sf{ f(1) = - 135 + 270 }](https://img.qammunity.org/2023/formulas/mathematics/high-school/3upuxlmmdbjk52l4z8z35wovlaoqgx7smh.png)
Now,
We have to find the average rate of change over the given time interval
Therefore,
Average rate of change in the given time interval
![\sf{ = }{\sf{( f(1) + f(3) )/(3 - 1)}}](https://img.qammunity.org/2023/formulas/mathematics/high-school/h3sq5nrlvpw94r36ewf3ujssax1n5kinw9.png)
![\sf{ = }{\sf{( 225 + ( -135) )/( 2)}}](https://img.qammunity.org/2023/formulas/mathematics/high-school/6ao52ihnot1ntnxxbbhurpdrmid1x60sox.png)
![\sf{ = }{\sf{( 225 - 135 )/( 2)}}](https://img.qammunity.org/2023/formulas/mathematics/high-school/ykfq3dvzr20od72tpbalvj7yi47herot1w.png)
![\sf{ = }{\sf{( 90 )/( 2)}}](https://img.qammunity.org/2023/formulas/mathematics/high-school/o58zdyn6ir47q6woo6mozdxlssoqli3aaq.png)
Hence, The average rate of change over the given interval is 45 miles per hour.