9.5 Solve Applications of Quadratic Equations

Learning objectives

Learning objectives

• Solve applications modeled by quadratic equations

Solve Applications Modeled by Quadratic Equations

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.

Let’s first summarize the methods we now have to solve quadratic equations.

As you solve each equation, choose the method that is most convenient for you to work the problem. As a reminder, we will copy our usual Problem-Solving Strategy here so we can follow the steps.

We have solved number applications that involved consecutive even and 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.

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

Example 9.35

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. · $\,\begin{array}{rll} n(n+2) & = & 195 \\ {n}^{2}+2n & = & 195 \\ \end{array}$ Step 5. Solve the equation. Distribute. Write the equation in standard form. Factor. · $\begin{array}{rll} {n}^{2}+2n-195 & = & 0 \\ (n+15)(n-13) & = & 0 \\ \end{array}$ Use the Zero Product Property. Solve each equation. · $\begin{array}{l} n+15=0\,n-13=0 \\ n=-15,\,n=13 \\ \end{array}$ There are two values of n that are solutions. This will give us two pairs of consecutive odd integers for our solution.

$\,\begin{array}{cccc} \mathrm{First odd integer}\,n=13 & & & \mathrm{First odd integer}\,n=-15 \\ \mathrm{next odd integer}\,n+2 & & & \mathrm{next odd integer}\,n+2 \\ \,13+2 & & & \,-15+2 \\ \,15 & & & \,-13 \\ \end{array}$

Step 6. Check the answer.

Do these pairs work?

Are they consecutive odd integers?

$\begin{array}{rccl} 13,15 & & & \mathrm{yes} \\ -13,-15 & & & \mathrm{yes} \\ \end{array}$

Is their product 195?

$\begin{array}{rllccl} 13\cdot{}15 & = & 195 & & & \mathrm{yes} \\ -13(-15) & = & 195 & & & \mathrm{yes} \\ \end{array}$

Step 7. Answer the question. · Two consecutive odd integers whose product is 195 are 13, 15 and −13, −15.

We will use the formula for the area of a triangle to solve the next example.

Recall that when we solve geometric applications, it is helpful to draw the figure.

Example 9.36

Solution

In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.

We will use the formula for the area of a rectangle to solve the next example.

Example 9.37

Solution

The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.

Example 9.38

Solution

The height of a projectile shot upward from the ground is modeled by a quadratic equation. The initial velocity, v${}_{0}$, propels the object up until gravity causes the object to fall back down.

We can use this formula to find how many seconds it will take for a firework to reach a specific height.

Example 9.39

Solution

We have solved uniform motion problems using the formula D = rt in previous chapters. We used a table like the one below to organize the information and lead us to the equation.

The formula D = rt assumes we know r and t and use them to find D. If we know D and r and need to find t, we would solve the equation for t and get the formula $t=\frac{D}{r}.$

Some uniform motion problems are also modeled by quadratic equations.

Example 9.40

Solution

This is a uniform motion situation. A diagram will help us visualize the situation.

Work applications can also be modeled by quadratic equations. We will set them up using the same methods we used when we solved them with rational equations.We’ll use a similar scenario now.

Example 9.41

Solution

This is a work problem. A chart will help us organize the information.

We are looking for how many hours it would take each press separately to complete the job.

Section 9.5 Exercises

Section 9.5 Exercises

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve using any method.

In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

1. The product of two consecutive odd integers is 99. Find the integers.

2. The product of two consecutive even integers is 168. Find the integers.

1. Find the base and height of a triangle whose base is four inches more than six times its height and has an area of 456 square inches.

2. If a triangle that has an area of 110 square feet has a base that is two feet less than twice the height, what is the length of its base and height?

1. The length of a 200 square foot rectangular vegetable garden is four feet less than twice the width. Find the length and width of the garden, to the nearest tenth of a foot.

2. A rectangular tablecloth has an area of 80 square feet. The width is 5 feet shorter than the length.What are the length and width of the tablecloth to the nearest tenth of a foot.?

1. The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is 20 feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth.

2. The distance between opposite corners of a rectangular field is four more than the width of the field. The length of the field is twice its width. Find the distance between the opposite corners. Round to the nearest tenth.

1. An arrow is shot from the ground into the air at an initial speed of 108 ft/s. Use the formula h = −16*t*${}^{2}$ + v${}_{0}$t to determine when the arrow will be 180 feet from the ground. Round the nearest tenth.

2. A man throws a ball into the air with a velocity of 96 ft/s. Use the formula h = −16*t*${}^{2}$ + v${}_{0}$t to determine when the height of the ball will be 48 feet. Round to the nearest tenth.

1. MaryAnne just returned from a visit with her grandchildren back east . The trip was 2400 miles from her home and her total time in the airplane for the round trip was 10 hours. If the plane was flying at a rate of 500 miles per hour, what was the speed of the jet stream?

2. Gerry just returned from a cross country trip. The trip was 3000 miles from his home and his total time in the airplane for the round trip was 11 hours. If the plane was flying at a rate of 550 miles per hour, what was the speed of the jet stream?

1. The weekly news magazine has a big story naming the Person of the Year and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes 6 hours more than Press #2 to do the job and when both presses are running they can print the job in 4 hours. How long does it take for each press to print the job alone?

2. Erlinda is having a party and wants to fill her hot tub. If she only uses the red hose it takes 3 hours more than if she only uses the green hose. If she uses both hoses together, the hot tub fills in 2 hours. How long does it take for each hose to fill the hot tub?

1. The product of two consecutive odd numbers is 255. Find the numbers.

2. The product of two consecutive even numbers is 624. Find the numbers.

1. The product of two consecutive odd numbers is 483. Find the numbers.

2. A triangle with area 45 square inches has a height that is two less than four times the base Find the base and height of the triangle.

1. The area of a triangular flower bed in the park has an area of 120 square feet. The base is 4 feet longer that twice the height. What are the base and height of the triangle?

2. The length of a rectangular driveway is five feet more than three times the width. The area is 50 square feet. Find the length and width of the driveway.

1. A rectangular table for the dining room has a surface area of 24 square feet. The length is two more feet than twice the width of the table. Find the length and width of the table.

2. The hypotenuse of a right triangle is twice the length of one of its legs. The length of the other leg is three feet. Find the lengths of the three sides of the triangle.

1. A rectangular garden will be divided into two plots by fencing it on the diagonal. The diagonal distance from one corner of the garden to the opposite corner is five yards longer than the width of the garden. The length of the garden is three times the width. Find the length of the diagonal of the garden.

2. Gerry plans to place a 25-foot ladder against the side of his house to clean his gutters. The bottom of the ladder will be 5 feet from the house.How far up the side of the house will the ladder reach?

1. A firework rocket is shot upward at a rate of 640 ft/sec. Use the projectile formula h = −16*t*${}^{2}$ + v${}_{0}$t to determine when the height of the firework rocket will be 1200 feet.

2. A bullet is fired straight up from a BB gun with initial velocity 1120 feet per second at an initial height of 8 feet. Use the formula h = −16*t*${}^{2}$ + v${}_{0}$t + 8 to determine how many seconds it will take for the bullet to hit the ground. (That is, when will h = 0?)

1. Two painters can paint a room in 2 hours if they work together. The less experienced painter takes 3 hours more than the more experienced painter to finish the job. How long does it take for each painter to paint the room individually?

2. It takes two hours for two machines to manufacture 10,000 parts. If Machine #1 can do the job alone in one hour less than Machine #2 can do the job, how long does it take for each machine to manufacture 10,000 parts alone?

1. The businessman took a small airplane for a quick flight up the coast for a lunch meeting and then returned home. The plane flew a total of 4 hours and each way the trip was 200 miles. What was the speed of the wind that affected the plane which was flying at a speed of 120 mph?

2. Roy kayaked up the river and then back in a total time of 6 hours. The trip was 4 miles each way and the current was difficult. If Roy kayaked at a speed of 5 mph, what was the speed of the current?

1. Make up a problem involving the product of two consecutive odd integers.

ⓐ Start by choosing two consecutive odd integers. What are your integers?

ⓑ What is the product of your integers?

ⓒ Solve the equation n(n + 2) = p, where p is the product you found in part (b).

ⓓ Did you get the numbers you started with?