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# Quadratic and exponential word problems | Lesson

## What are quadratic and exponential word problems, and how frequently do they appear on the test?

Quadratic and exponential word problems ask us to solve equations or evaluate functions that model real-world scenarios using quadratic and exponential expressions.
On your official SAT, you'll likely see 1 to 2 questions that are quadratic or exponential word problems.
This lesson builds on an understanding of the following skills:
You can learn anything. Let's do this!

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

Quadratic word problem: ballSee video transcript

### Exponential expressions word problems (algebraic)

Exponential expressions word problems (algebraic)See video transcript

### Common word problem scenarios on the SAT

The Interpreting nonlinear expressions lesson details how to interpret quadratic and exponential expressions modeling common scenarios that appear on the SAT. The four common word problem scenarios are:
• Area of a rectangle (quadratic)
• Population growth and decay (exponential)
• Compounding interest (exponential)

#### Area of a rectangle

Because the formula for the area of a rectangle, A, equals, 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 parenthesis, t, right parenthesis, equals, minus, 16, t, squared, plus, 64, t
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 parenthesis, t, right parenthesis, equals, a, left parenthesis, b, right parenthesis, start superscript, t, end superscript, where a is the initial value, b describes
, and t is the variable representing time.

Example:
P, left parenthesis, t, right parenthesis, equals, 227, left parenthesis, 1, point, 03, right parenthesis, start superscript, t, end superscript
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!

The length of a rectangular plot of land is 3 times its width. The area of the plot is 1, point, 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 dollar sign, 500 into a savings account that yields 1, percent interest annually.
The initial amount of money in the savings account is dollar sign
.
After one year, the amount of money in the account is dollar sign
, or 1, point, 01 times the initial amount.
After two years, the amount of money in the account is dollar sign
, or 1, point, 01 times 1, point, 01 times the initial amount.
Write an expression that describes the amount of money in the account after t years.

practice: calculate compounding interest
Janice uses the expression 2300, left parenthesis, 1, point, 058, right parenthesis, start superscript, t, end superscript 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 dollar sign when entering your answer)

Practice: write population function
A gym currently has 5, comma, 500 members. Due to an economic downturn, an analyst predicts that the gym will lose 2, point, 5, percent of its members each month for the foreseeable future. Which of the following functions models P, left parenthesis, t, right parenthesis, the number of gym members t months from now?

Practice: solve a word problem representing an uncommon scenario
p, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 0, point, 6, right parenthesis, left parenthesis, 300, minus, 75, x, right parenthesis
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?

## Want to join the conversation?

• Uhm this is probs a dumb question but why on the population growth/decay example the p(t) becomes p(4) instead of p(6) (the amount of years since 1996)?
• Looks like just a simple typo to me, good catch! If you want, you can tell the folks at KA to fix it by clicking the "Report a mistake" link right below where you go to ask a question. This way, other learners won't be confused either.
• In the ball qns (video), how does the 50 Ft above ground affect the equation? Why has Sal not taken that into account?
• The 50 feet that the ball has for an initial height does affect the equation, you're right. However, based on the way the question is worded, we can trust that the given equation will represent that specific ball's height as a function of time, when it started up 50 feet and was shot with an initial velocity of 20 feet/sec.
The "+50" term in the equation basically accounts for the added height. If you're curious about how the equation is derived in general, the Khan Academy videos on projectile motion in the physics playlist offer a great starting point.
• In the first video at , why can you divide it all by -2? I never would have thought to do that.