10.4 Solve applications modeled by quadratic equations

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By the end of this section, you will be able to:
• Solve applications modeled by Quadratic Equations

Before you get started, take this readiness quiz.

1. The sum of two consecutive odd numbers is $-100$ . Find the numbers.
If you missed this problem, review [link] .
2. The area of triangular mural is 64 square feet. The base is 16 feet. Find the height.
If you missed this problem, review [link] .
3. Find the length of the hypotenuse of a right triangle with legs 5 inches and 12 inches.
If you missed this problem, review [link] .

Solve applications of the quadratic formula

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications. To get us started, we will copy our usual Problem Solving Strategy here so we can follow the steps.

Use the problem solving strategy.

1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

We have solved number applications that involved consecutive even integers    and consecutive odd integers    by modeling the situation with linear equations. Remember, we noticed each even integer is 2 more than the number preceding it. If we call the first one n, then the next one is $n+2$ . The next one would be $n+2+2$ or $n+4$ . This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.

$\begin{array}{cccc}\hfill \mathbf{\text{Consecutive even integers}}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\mathbf{\text{Consecutive odd integers}}\hfill \\ \hfill 64,66,68\hfill & & & \hfill \phantom{\rule{4em}{0ex}}77,79,81\hfill \\ \hfill \begin{array}{cccc}n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{even integer}\hfill \\ n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer}\hfill \\ n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\begin{array}{cccc}n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{odd integer}\hfill \\ n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive odd integer}\hfill \\ n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive odd integer}\hfill \end{array}\hfill \end{array}$

Some applications of consecutive odd integers    or consecutive even integers    are modeled by quadratic equations. The notation above will be helpful as you name the variables.

The product of two consecutive odd integers is 195. Find the integers.

Solution

 Step 1. Read the problem. Step 2. Identify what we are looking for. We are looking for two consecutive odd integers. Step 3. Name what we are looking for. Let $n=$ the first odd integer. $n+2=$ the next odd integer Step 4. Translate into an equation. State the problem in one sentence. "The product of two consecutive odd integers is 195." The product of the first odd integer and the second odd integer is 195. Translate into an equation Step 5. Solve the equation. Distribute. Subtract 195 to get the equation in standard form. Identify the a, b, c values. Write the quadratic equation. Then substitute in the values of a, b, c. . Simplify. Simplify the radical. Rewrite to show two solutions. Solve each equation. There are two values of n that are solutions. This will give us two pairs of consecutive odd integers for our solution. First odd integer $n=13$ next odd integer $n+2$ $\phantom{\rule{6.8em}{0ex}}13+2$ $\phantom{\rule{8em}{0ex}}15$ First odd integer $n=-15$ next odd integer $n+2$ $\phantom{\rule{6.1em}{0ex}}-15+2$ $\phantom{\rule{7em}{0ex}}-13$ Step 6. Check the answer. Do these pairs work? Are they consecutive odd integers? Is their product 195? $\begin{array}{cccc}\hfill 13,15,\phantom{\rule{0.2em}{0ex}}\text{yes}& & & -13,-15,\phantom{\rule{0.2em}{0ex}}\text{yes}\hfill \\ \hfill 13\cdot 15=195,\phantom{\rule{0.2em}{0ex}}\text{yes}& & & -13\left(-15\right)=195,\phantom{\rule{0.2em}{0ex}}\text{yes}\hfill \end{array}$ Step 7. Answer the question. The two consecutive odd integers whose product is 195 are 13, 15, and −13, −15.

DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned$1600 more than Jim earned. How much did they earned
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
how to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
What is observation
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him$10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
35 min
Debra