# 9.7 Solve a formula for a specific variable

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By the end of this section, you will be able to:
• Use the distance, rate, and time formula
• Solve a formula for a specific variable

Before you get started, take this readiness quiz.

1. Write $35$ miles per gallon as a unit rate.
If you missed this problem, review Ratios and Rates .
2. Solve $6x+24=96.$
If you missed this problem, review Solve Equations with Variables and Constants on Both Sides .
3. Find the simple interest earned after $5$ years on $\text{1,000}$ at an interest rate of $\text{4%}.$
If you missed this problem, review Solve Simple Interest Applications .

## Use the distance, rate, and time formula

One formula you’ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you travel if you drove at a steady rate of $60$ miles per hour for $2$ hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said $120$ miles, you already know how to use this formula!

The math to calculate the distance might look like this:

$\begin{array}{}\\ \text{distance}=\left(\frac{60\phantom{\rule{0.2em}{0ex}}\text{miles}}{1\phantom{\rule{0.2em}{0ex}}\text{hour}}\right)\left(2\phantom{\rule{0.2em}{0ex}}\text{hours}\right)\hfill \\ \text{distance}=120\phantom{\rule{0.2em}{0ex}}\text{miles}\hfill \end{array}$

In general, the formula relating distance, rate, and time is

$\text{distance}\phantom{\rule{0.2em}{0ex}}\text{=}\phantom{\rule{0.2em}{0ex}}\text{rate}·\text{time}$

## Distance, rate and time

For an object moving in at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula

$d=rt$

where $d=$ distance, $r=$ rate, and $t=$ time.

Notice that the units we used above for the rate were miles per hour, which we can write as a ratio $\frac{miles}{hour}.$ Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles.

Jamal rides his bike at a uniform rate of $12$ miles per hour for $3\frac{1}{2}$ hours. How much distance has he traveled?

## Solution

 Step 1. Read the problem. You may want to create a mini-chart to summarize the information in the problem. $d=?$ $r=12\phantom{\rule{0.2em}{0ex}}\text{mph}$ $t=3\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\text{hours}$ Step 2. Identify what you are looking for. distance traveled Step 3. Name. Choose a variable to represent it. let d = distance Step 4. Translate. Write the appropriate formula for the situation. Substitute in the given information. $d=rt$ $d=12\cdot 3\frac{1}{2}$ Step 5. Solve the equation. $d=42\phantom{\rule{0.2em}{0ex}}\text{miles}$ Step 6. Check: Does 42 miles make sense? Step 7. Answer the question with a complete sentence. Jamal rode 42 miles.

Lindsay drove for $5\frac{1}{2}$ hours at $60$ miles per hour. How much distance did she travel?

330 mi

Trinh walked for $2\frac{1}{3}$ hours at $3$ miles per hour. How far did she walk?

7 mi

Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of $520$ miles. If he can drive at a steady rate of $65$ miles per hour, how many hours will the trip take?

## Solution

 Step 1. Read the problem. Summarize the information in the problem. $d=520\phantom{\rule{0.2em}{0ex}}\text{miles}$ $r=65\phantom{\rule{0.2em}{0ex}}\text{mph}$ $t=?$ Step 2. Identify what you are looking for. how many hours (time) Step 3. Name: Choose a variable to represent it. let t = time Step 4. Translate. Write the appropriate formula. Substitute in the given information. $d=rt$ $520=65t$ Step 5. Solve the equation. $t=8$ Step 6. Check: Substitute the numbers into the formula and make sure the result is a true statement. $d=rt$ $520\stackrel{?}{=}65\cdot 8$ $520=520>✓$ Step 7. Answer the question with a complete sentence. We know the units of time will be hours because we divided miles by miles per hour. Ray's trip will take 8 hours.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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