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### Course: Digital SAT Math>Unit 12

Lesson 10: Quadratic and exponential word problems: advanced

# 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:
1. Interpret quadratic and exponential functions
2. Solve quadratic and exponential word problems
3. 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

Khan Academy video wrapper
Interpret quadratic models: Factored formSee video transcript

### Area of a rectangle

The formula for $A$, the area of a rectangle with length $\ell$ and width $w$ is:
$A=\ell w$
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:
$\begin{array}{rl}A& =\ell w\\ \\ & =x\left(x-2\right)\end{array}$

### Height versus time

The quadratic function $h$ for the height of an object at time $t$ looks like a quadratic function in standard form:
$h\left(t\right)=-a{t}^{2}+bt+c$
Where:
• $t$ is the input variable. It usually represents time in seconds.
• $c$ describes the 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 $h\left(t\right)=-16{t}^{2}+144t+32$, where $t$ is time in seconds:
• The initial height of the projectile is $32$ feet because the constant term is equal to $32$.

### Try it!

Try: find the length of a rectangle
$A=\left(x\right)\left(3x-2\right)$
The equation above models $A$, the area of a rectangle with a width of $x$ meters.
represents the length of the rectangle in meters.
According to the model, the length of the rectangle is
.

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

### Interpreting exponential expression word problem

Khan Academy video wrapper
Interpreting exponential expression word problemSee video transcript

### Population growth and decline

The exponential function $P$ for population looks like the following:
$P\left(t\right)={P}_{0}{r}^{t}$
Where:
• $t$ is the input variable representing the number of time periods elapsed.
• ${P}_{0}$ is the initial population, or the population when $t=0$.
• $r$ describes how the population is changing.
For example, if $P\left(t\right)=75\left(1.04{\right)}^{t}$ describes the population of a village $t$ years after $2010$:
• $t$ represents the number of years after $2010$. In the year $2011$, $t=1$; in the year $2020$, $t=10$.
• The initial population of the village is $75$.
• $1.04$ tells us that for each year after $2010$, the population of the town is $1.04$ times the population in the previous year.
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, then the population is declining.

### Compounding interest

The exponential function $P$ for an amount of money accruing compounding interest looks like the following:
$P\left(t\right)={P}_{0}\left(1+r{\right)}^{t}$
Where:
• $t$ is the input variable representing the number of time periods elapsed.
• ${P}_{0}$ is the initial amount of money, or the amount of money before any interest is accrued.
• $r$ is the interest rate applied for each time period expressed as a decimal.
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 $P\left(t\right)=500\left(1.01{\right)}^{t}$ models the amount of money, in dollars, in a savings account after $t$ years:
• $t$ represents the number of years after the initial deposit.
• $500$ represents the initial amount put into the savings account: $500$ dollars.
• $1.01=1+0.01$, which means $r=0.01$ and the (annual) interest rate is $1\mathrm{%}$.

### Try it!

Try: interpret population change
$P\left(t\right)=680\left(0.94{\right)}^{t}$
The function above models the population of a rural village $t$ years after $1996$.
Based on the function, what was the village's population in $1996$ ?
By what percent is the village's population declining each year? (Ignore the $\mathrm{%}$ symbol when entering your answer. For example, if the answer is $12\mathrm{%}$, enter $12$.)

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

### Quadratic word problem: ball

Khan Academy video wrapper
Quadratic word problem: ballSee video transcript

### Exponential expressions word problems (algebraic)

Khan Academy video wrapper
Exponential expressions word problems (algebraic)See video transcript

### 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:
$h\left(t\right)=-16{t}^{2}+64t$
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 $f\left(t\right)=a\left(b{\right)}^{t}$, where $a$ is the initial value, $b$ describes
, and $t$ is the variable representing time.

Example:
$P\left(t\right)=227\left(1.03{\right)}^{t}$
The function above models $P$, the population of Pallet Town, $t$ years after $1996$. To the nearest whole number, what was the net increase of Pallet Town's population from $1996$ to $2002$ ?

### Try it!

try: write a quadratic equation
The length of a rectangular plot of land is $3$ times its width. The area of the plot is $1.08$ square miles.
If we use $x$ to represent the width of the plot in miles, then the length of the plot is
miles.
Write an equation that gives us the width of the plot when solved.

try: write an exponential expression
Yuji put $\mathrm{}500$ into a savings account that yields $1\mathrm{%}$ interest annually.
The initial amount of money in the savings account is $\mathrm{}$
.
After one year, the amount of money in the account is $\mathrm{}$
, or $1.01$ times the initial amount.
After two years, the amount of money in the account is $\mathrm{}$
, or $1.01$ times $1.01$ times the initial amount.
Write an expression that describes the amount of money in the account after $t$ years.

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

### Interpreting time in exponential models

Khan Academy video wrapper
Interpreting time in exponential modelsSee video transcript

### 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 $M=100\left(1.62{\right)}^{t}$, 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 exponent of $1.62$ is $1$ when $m=12$ and $2$ when $m=24$ (two years), we need to divide $m$ by $12$ in the exponent:
$M=100\left(1.62{\right)}^{{}^{\frac{m}{12}}}$
When changing the time units in an exponential expression:
• Identify the unit equivalence used for the question, e.g., $1\phantom{\rule{0.167em}{0ex}}\text{year}=12\phantom{\rule{0.167em}{0ex}}\text{months}$.
• 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., $1$ year and $12$ months, $2$ years and $24$ months, etc.

### Try it!

Try: convert from hours to days
$P=50\left(1.1{\right)}^{h}$
The equation above models the population, $P$, of a bacteria culture after $h$ hours of incubation.
There are $24$ hours in a day. Since a day is
than an hour, if we want to write an equation that models the population of the bacteria culture after $d$ days of incubation, we must
the exponent by $24$.
Which of the following equations models the population of the bacteria culture after after $d$ days of incubation?
Choose 1 answer:

## Your turn!

Practice: interpret the meaning of a constant
$h\left(t\right)=-4.9{t}^{2}+7.7t+0.5$
The function above models the height $h$, in meters, of a soccer ball above ground $t$ seconds after being kicked by a soccer player. What does the number $0.5$ represents in the function?
Choose 1 answer:

Practice: determine the value of a constant
The value of a rare baseball card was $\mathrm{}80$ the year it was released. Pepe estimates that the value of the card increases by $7\mathrm{%}$ each year and uses the expression $80\left(x{\right)}^{t}$ to estimate the value of the card $t$ years after its release. What is the value of $x$ in the expression?

practice: calculate compounding interest
Janice uses the expression $2300\left(1.058{\right)}^{t}$ to model the amount of money in her investment account after $t$ years. To the nearest whole dollar, how much money is in Janice's investment account after $10$ years? (Ignore the $\mathrm{}$ when entering your answer)

Practice: write population function
A gym currently has $5,500$ members. Due to an economic downturn, an analyst predicts that the gym will lose $2.5\mathrm{%}$ of its members each month for the foreseeable future. Which of the following functions models $P\left(t\right)$, the number of gym members $t$ months from now?
Choose 1 answer:

Practice: solve a word problem representing an uncommon scenario
$p\left(x\right)=\left(x-0.6\right)\left(300-75x\right)$
Roy uses the function above to model $p$, the daily profit in dollars, of his taco truck from selling tacos at $x$ dollars each. Based on the model, at which of the following prices would Roy be able to earn the most profit?
Choose 1 answer:

Practice: change the time unit in an exponential model
$D=200\left(1.16{\right)}^{m}$
The equation above models the number of total downloads, $D$, for an app Clara created $m$ months after its launch. Of the following, which equation models the number of total downloads $y$ years after launch?
Choose 1 answer:

## Want to join the conversation?

• Sal=g.o.a.t
(123 votes)
• yes. i agree
(9 votes)
• My brain is melting down
(91 votes)
• cannot agree more
(10 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)
• me neither
(4 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)
• 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)