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### Course: Digital SAT Math > Unit 12

Lesson 10: Quadratic and exponential word problems: advanced- Quadratic and exponential word problems | Lesson
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems: advanced

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# Quadratic and exponential word problems | Lesson

A guide to quadratic and exponential word problems on the digital SAT

## What are quadratic and exponential word problems?

Both

**quadratic functions**and**exponential functions**can be used to model nonlinear relationships in everyday life, such as the height of a falling object or the population change of a city.In this lesson, we'll learn to:

- Interpret quadratic and exponential functions
- Solve quadratic and exponential word problems
- Write quadratic and exponential functions, including rewriting exponential functions using different time units

**Note:**An understanding of percentages is useful for interpreting exponential functions.

**You can learn anything. Let's do this!**

## What are some common SAT scenarios modeled by quadratic functions?

### Interpret quadratic models: Factored form

### Area of a rectangle

The formula for $A$ , the area of a rectangle with length $\ell $ and width $w$ is:

In a quadratic function dealing with area, the area is the output, one of the linear dimensions is the input, and the other linear dimension is described in terms of the input. The quadratic expression is usually written in

**factored form**, with the length and width represented by a factors.For example, if the length of a rectangular piece of paper is $x$ inches and the width is $2$ inches shorter than the length, then the area of the rectangle is equal to:

### Height versus time

The quadratic function $h$ for the height of an object at time $t$ looks like a quadratic function in

**standard form**:Where:

is the input variable. It usually represents time in seconds.$t$ describes the$c$ *initial height*of the object, or the object's height when .$t=0$

For example, if the height of a projectile in feet is modeled by the function $h(t)=-16{t}^{2}+144t+32$ , where $t$ is time in seconds:

- The initial height of the projectile is
feet because the constant term is equal to$32$ .$32$

### Try it!

## What are some common SAT scenarios modeled by exponential functions?

### Interpreting exponential expression word problem

### Population growth and decline

The exponential function $P$ for population looks like the following:

Where:

is the input variable representing the number of time periods elapsed.$t$ is the${P}_{0}$ *initial population*, or the population when .$t=0$ describes how the population is changing.$r$

For example, if $P(t)=75(1.04{)}^{t}$ describes the population of a village $t$ years after $2010$ :

represents the number of years after$t$ . In the year$2010$ ,$2011$ ; in the year$t=1$ ,$2020$ .$t=10$ - The initial population of the village is
.$75$ tells us that for each year after$1.04$ , the population of the town is$2010$ times the population in the previous year.$1.04$

If we convert $1.04$ to $104\mathrm{\%}$ , we can also say that the population of the town grows by $104\mathrm{\%}-100\mathrm{\%}=4\mathrm{\%}$ each year.

If $r>1$ , then the population is growing. If $0<r<1$ , then the population is declining.

### Compounding interest

The exponential function $P$ for an amount of money accruing compounding interest looks like the following:

Where:

is the input variable representing the number of time periods elapsed.$t$ is the${P}_{0}$ *initial amount*of money, or the amount of money before any interest is accrued. is the interest rate applied for each time period expressed as a decimal.$r$

**Note:**there is a more complex version of the formula in which the interest can be applied multiple times within a single time period (for example, an annual interest rate with monthly interest calculations), but that version typically does not appear on the SAT.

For example, if $P(t)=500(1.01{)}^{t}$ models the amount of money, in dollars, in a savings account after $t$ years:

represents the number of years after the initial deposit.$t$ represents the initial amount put into the savings account:$500$ dollars.$500$ , which means$1.01=1+0.01$ and the (annual) interest rate is$r=0.01$ .$1\mathrm{\%}$

### Try it!

## How do I solve quadratic and exponential word problems?

### Quadratic word problem: ball

### Exponential expressions word problems (algebraic)

### Common word problem scenarios on the SAT

#### Area of a rectangle

Because the formula for the area of a rectangle, $A=\ell w$ , is commonly known, you're expected to be able to write quadratic equations modeling rectangular areas and solve for length or width when area is given.

#### Height versus time

You're not expected to know the physics of falling objects; functions that describe the relationship between height and time will be given to you. However, you might be asked to:

- Calculate the height of the object at a given time
- Calculate the time at which the object is at a given height

A common given height is "the ground", which means a height of $0$ units.

**Example:**

The function above models the height $h$ , in feet, of an object above ground $t$ seconds after being launched straight up in the air. At how many seconds after launch does the object fall back to the ground?

#### Population growth and decay / Compounding interest

For both of these topics, you'll either be asked to write a function based on a verbal description or to evaluate the function when one is given. The SAT generally doesn't ask you to do both in the same question.

Functions modeling these topics typically look like $f(t)=a(b{)}^{t}$ , where $a$ is the $b$ describes , and $t$ is the variable representing time.

**initial value**,**Example:**

The function above models $P$ , the population of Pallet Town, $t$ years after $1996$ . To the nearest whole number, what was the $1996$ to $2002$ ?

*net increase*of Pallet Town's population from### Try it!

## How do I rewrite exponential functions using different time units?

### Interpreting time in exponential models

### How do we change time units in an exponential expression?

On the SAT, we're sometimes given an exponential model in one time unit (for example:

*years*), and asked to re-write it using another time unit (for example:*months*), showing the same rate of growth or decay as the original model.For example, let's look at the equation $M=100(1.62{)}^{t}$ , where $M$ is the number of members a club has $t$ years after it opens. How do we write an equation for the number of club members $m$

__months__after opening?In this type of question, the base of the exponent is kept the same, and only the exponent changes. We know that the number of club members increases by a factor of $1.62$ after $t=1$ year, which means the number of club members also needs to increase by a factor of $1.62$ after $m=12$ months.

To make sure the $1.62$ is $1$ when $m=12$ and $2$ when $m=24$ (two years), we need to divide $m$ by $12$

*exponent*of*in the exponent*:When changing the time units in an exponential expression:

- Identify the unit equivalence used for the question, e.g.,
.$1{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{year}=12{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{months}$ **Tip:**If converting the variable from a larger time unit to a smaller time unit, we need to make the exponent*smaller*. For example, when converting from years to months, we*divide*the variable representing the number of months by .$12$ **Tip:**If converting the variable from a smaller time unit to a larger time unit, we need to make the exponent*larger*. For example, when converting from months to years, we*multiply*the variable representing the number of years by .$12$ - Verify that the two expressions give us the same values when evaluated using equivalent amounts of time, e.g.,
year and$1$ months,$12$ years and$2$ months, etc.$24$

### Try it!

## Your turn!

## Want to join the conversation?

- My brain is melting down(91 votes)
- what on the earth are these(53 votes)
- The more complex version of the compound interest formula doesn't "typically" appear on the SAT. Is that supposed to be a guarantee or it may appear depending on CollegeBoard's mood? Because I don't want to be surprised on the test .(25 votes)
- It's unlikely to appear, but it doesn't mean the chance is zero. So, indeed it could show up in the test depending on their mood. So be prepared for it.(17 votes)

- i dont think my brain works(30 votes)
- Isn't Pallet town the town that Ash Ketchum comes from?(10 votes)
- Yes, mate.. It is! (Happy and proud crying smile)(6 votes)

- Does the DSAT give all formulas, and if not, which ones please? Tanks in advance! Chaaarge!(6 votes)
- Hi Joshi Jake!

I know this is what they at least used to give for the paper SAT. Not sure if they've changed it though. I couldn't find any specific info regarding the DSAT:

https://satsuite.collegeboard.org/sat/whats-on-the-test/math/reference-information

Hope this helps!(8 votes)

- Thank you Khan Academy for giving my energy back with the Pallet Town exercise. I am recharged again :D(8 votes)
- Sal does Interpreting-time-in-exponential-models question wrong. He solves for 9/5, while he must do so for 4/5.(1 vote)
- he solved for 9/5 because the question is when the tree gains 4/5 of the initial amount of branches. lets take the initial amount of branches as 1 or 5/5. so, when the tree gains 4/5 of the initial amount of branches we have to add the initial amount and the amount of branches gained to get the current amount that is 9/5. so, when the tree gains 4/5 branches the total amount of branches would be 9/5. so, when solving for 9/5 we can find when the tree gained 4/5 branches.(13 votes)

- i don't still understand these time units conversion in exponential expression(6 votes)
- I also don't, try to take them as they are.(2 votes)