2.3: Evaluating Expressions

Last updated
Save as PDF

Page ID: 20034

Kelly Brooks
Eastern Gateway Community College

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

In the Water Industry, we use many formulas to make calculations. As a result, we need to know how to evaluate expressions, that is, substitute known values for variables, and simplify the expression using the order of operations.

Example \(\PageIndex{1}\)

If \(x=4\) and \(y=3\) evaluate \(3xy^2+2\sqrt{x}\).

Solution

First, substitute the given values for \(x\) and \(y\):

\[3(4)(3)^2 + 2 \sqrt{4} \nonumber\]

Next, perform order of operations to simplify:

\[3\left(4\right){\left(3\right)}^2+2\sqrt{4}=3\left(4\right)\left(9\right)+2\left(2\right)=108+4=112 \nonumber\]

Example \(\PageIndex{2}\)

If \(a=9\ and\ b=7\), evaluate \(\frac{2\sqrt{a}+3}{b-5}\).

Solution

First, substitute the given values for a and b:

\( \frac{2\sqrt{9}+3}{7-5}\)

Next, simplify using order of operations:

\(\frac{2\cdot 3+3}{7-5}=\frac{6+3}{7-5}=\frac{9}{2}\)

Recall from Unit 1, we could convert this fraction to a decimal to obtain 4.5, however, the general rule of thumb is to use decimals only when you start with decimals, or the directions states to use decimals.

Example \(\PageIndex{3}\)

A water plant has a rectangular basin with a base that is in the shape of a square with sides of length x and the height of the basin is y. As a result, the volume is calculated using the following expression: \(x^2y\). If the length of the base of the basin is 50 feet and the height of the basin is 20 feet, calculate the volume of the basin.

Solution

First, we substitute the values for x and y: \(x^2y={\left(50\ feet\right)}^2\cdot \left(20\ feet\right)\)

Next, we simplify the expression using order of operations to obtain:

\[\left(2500\ ft^2\right)\left(20\ ft\right)=50,000\ ft^3\]

Example \(\PageIndex{4}\)

To estimate the extra water a town needs to store for emergencies, we must multiply the population by 45 gallons, then add 42,000 gallons to account for what the fire fighters would need in case of a fire, and then multiply the result by a safety factor of 1.8.

a. Write an expression for calculating the amount of extra water a town needs to store in case of an emergency.

b. Use the expression to calculate the amount of extra water a town needs to store in case of an emergency if the population of the town is 48,000.

Solution

a. We begin by converting the English to math:

Multiply the population, let’s call the population P, by 45 gallons: \(45P\)

Then add 42,000 gallons: \(45P+42,000\)

Then multiply by 1.8: \(1.8(45P+42,000)\)

Hence, the amount of extra water needed is \(1.8(45P+42,000)\)

b. If the population is 48,000, we will replace P in the above expression with 48,000 to obtain: \(1.8\left(45\cdot 48,000+42,000\right)=1.8\left(2,160,000+42,000\right)=1.8\left(2,202,000\right)=3,963,600\)

Hence, the town needs to reserve 3,963,600 gallons of water in case of an emergency.