Table of Contents

### Warmup Questions II

*These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.*

What is the area of a rectangle of length $\ell$ and width $w$?

The area is $\ell w$.

Translate the phrase into an algebraic expression:

8 more than five times a number

$5x+8$

Translate the phrase into an algebraic expression:

5 less than a number $x$

$x-5$

### Review

*If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.*

Need a review on solving quadratic equations? Check

### Quick Intro II

*This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.*

**A Quick Intro** **to Applications of Quadratic Equations**

** **

**Key Words.** Quadratic equations, applications.

Here are some suggestions when solving a verbal problem.

- Read the problem several times.
- Identify what the question is asking for. That is the unknown.
- Write down the information that is given.
- Draw a picture.
- Find relationships.
- Is the problem related to areas, volumes etc? Is it about a square, rectangle, circle etc? Write down the related formula.
- Plug all the pieces of information that are given.
- Identify the equation you need to solve.
- Solve and check.

Recall that we have seen three methods for solving a quadratic equation.

- Factor and use the Zero Product Rule
- Use the Square Root Property; complete the square first, if necessary
- Use the Quadratic Formula

Do you know when to use each method?

### Video Lesson II

*Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!*

**Video Lesson 2**

**A description of the video**

In the video you will see the following problems.

- Suppose you have a 100 ft of fencing to make a rectangular enclosure, using a house as one side. Suppose you want the area to be 1200 $\mathrm{ft}^2$. What are the dimensions of the enclosure?
- Suppose in the above example, the area is 100 $\mathrm{ft}^2$.
- Try: suppose area is 5000 sqft. What are the dimensions of the enclosure then?

**Video Lesson 3**

**A description of the video**

In this video you will see the following problem.

- Suppose Tasha throws a ball in the air and suppose the height in feet of the ball in terms of time in seconds. $$h(t)=-16t^2+32t+2$$ When does the ball hit the ground if Tasha doesn’t catch it?

### Try Questions II

*Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.*

The width of a rectangle is 5 feet less than its length. If the area of the rectangle is 8 square feet, find the dimensions of the rectangle.

Let $w$ be the width and $\ell$ be the length of the rectangle. Then $w=\ell-5$. The area is

$$(\ell-5)\cdot \ell= 8 $$

$$\ell^2-5\ell = 8 $$

$$\ell^2-5\ell-8 = 0$$

Using the quadratic formula, we obtain:

$$\ell = \dfrac{5\pm\sqrt{(-5)^2-4\cdot (-8)}}{2} $$

$$\ell = \dfrac{5\pm\sqrt{57}}{2} $$

The length cannot be negative, so it cannot be $\dfrac{5-\sqrt{57}}{2} \approx -1.27$. So

$$\ell = \dfrac{5+\sqrt{57}}{2}$$

and $$w= \ell-5=\dfrac{5+\sqrt{57}}{2} -5 =\dfrac{\sqrt{57}-5}{2}.$$

### WeBWorK II

*You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.*

**WeBWork** **II**

There is no WeBWorK assignment on today’s material, but you can do the homework from the textbook.

When you are done, come back to this page for the Exit Questions.

### Exit Questions II

*After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!*

- When solving a word problem, why might it be important to draw a picture representing the situation?
- What aspects of the drawing are important?

$\bigstar$ The length of a rectangular photograph is $7\text{ in}$. more than the width. If the area is $78\text{ in}^2$, what are the dimensions?

We have that $\ell=w+7$ where $\ell$ is the length and $w$ is the width of the photograph. The area is $\ell\cdot w=78$, and so $(w+7)w=78$, i.e. $w^2+7w-78=0$ or $(w+13)(w-6)=0$, and so $w=-13$ or $w=6$. Since the width $w$ cannot be negative, we have that $w=6$ in which case $\ell = w+7=6+7=13$. So the dimensions are $6$ inches and $13$ inches.