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The number of people waiting in line to buy a new piece of technology is measured by the differentiable function
f, where
f(t) is measured in people and
t is measured in minutes after the store opened. What are the units of
(1)/(4)int_(1)^(5)f(t)dt ?
minutes
people
minutes / person
people / minute
minutes / person
^(2)
people / minute
^(2)

The number of people waiting in line to buy a new piece of technology is measured by the differentiable function $f$ , where $f(t)$ is measured in people and $t$ is measured in minutes after the store opened. What are the units of $\frac{1}{4} \int_{1}^{5} f(t) d t$ ? $\newline$ minutes $\newline$ people $\newline$ minutes / person $\newline$ people / minute $\newline$ minutes / person ${ }^{2}$ $\newline$ people / minute ${ }^{2}$

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Identify discrete and continuous random variables

Full solution

Q. The number of people waiting in line to buy a new piece of technology is measured by the differentiable function

f

, where

f(t)

is measured in people and

t

is measured in minutes after the store opened. What are the units of

\frac{1}{4} \int_{1}^{5} f(t) d t

?

\newline

minutes

\newline

people

\newline

minutes / person

\newline

people / minute

\newline

minutes / person

{ }^{2}

\newline

people / minute

{ }^{2}

Understand integral and units: Understand the integral and its units. $\newline$ The integral $\int_{1}^{5}f(t)\,dt$ represents the accumulation of the quantity $f(t)$ over the interval from $t=1$ to $t=5$ . Since $f(t)$ is measured in people and $t$ is measured in minutes, the integral itself has units of people multiplied by minutes, because we are summing up ＂people＂ over an interval of ＂minutes＂.
Consider constant effect: Consider the constant $(1)/(4)$ and its effect on units. Multiplying the integral by $(1)/(4)$ does not change the units. It simply scales the quantity by a factor of $1/4$ . Therefore, the units remain the same as the integral's units.
Determine final units: Determine the final units. $\newline$ Since the integral $\int_{1}^{5}f(t)\,dt$ has units of people*minutes, and multiplying by $(1)/(4)$ does not change the units, the units of $(1)/(4)\int_{1}^{5}f(t)\,dt$ are also people*minutes.

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