Question

The function f(x)= -45x + 270 represents the total distance, in miles, a traveler is from home x hours after beginning the trip home. What is the average rate of change over the interval from x = 1 to x = 3, in miles per hour?

Answer 1

$\bold{\huge{\underline{ Solution }}}$

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

$\sf{ f(x) = -45x + 270 }$

$\sf{ f(1) = - 45(1) + 270 }$

$\sf{ f(1) = - 45 + 270 }$

$\sf{ f(1) = 225 }$

For final interval that is x = 3, Distance covered by the traveller

$\sf{ f(x) = - 45x + 270 }$

$\sf{ f(1) = - 45(3) + 270 }$

$\sf{ f(1) = - 135 + 270 }$

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

$\sf{ = }{\sf{\dfrac{ f(1) + f(3) }{3 - 1}}}$

$\sf{ = }{\sf{\dfrac{ 225 + ( -135) }{ 2}}}$

$\sf{ = }{\sf{\dfrac{ 225 - 135 }{ 2}}}$

$\sf{ = }{\sf{\dfrac{ 90 }{ 2}}}$

$\sf{ = 45 }$

Hence, The average rate of change over the given interval is 45 miles per hour.