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Section 5

Problem Solving

Fill in the blanks.Integers that follow one another, such as 7 and $8,$ are called ______ integers.

Fill in the blanks.An _____ triangle is a triangle with two sides of the same length.

Fill in the blanks.The equal sides of an isosceles triangle meet to form the _____ angle. The angles opposite the equal sides are called _____ angles, and they have equal measures.

Fill in the blanks.When asked to find the dimensions of a rectangle, we are to find its _____ and ______.

A 17 -foot pipe is cut into three sections. The longest section is three times as long as the shortest, and the middle-sized section is 2 feet longer than the shortest. Complete the diagram.(FIGURE CANNOT COPY)

It costs $$ 28$ per hour to rent a trailer. Write an expression that represents the cost to rent the trailer for $x$ hours.

A realtor is paid a $3 \%$ commission on the sale of a house. Write an expression that represents the amount of the commission if a house sells for $\mathrm{Sx}$.

The perimeter of the rectangle below is 15 feet. Fill in the blanks: (2 ) $+2 x=$(FIGURE CANNOT COPY)

What is the sum of the measures of the angles of any triangle?

Refer to the isosceles triangle on the right.(FIGURE CANNOT COPY)a. Find the missing angle measure.b. Find the missing side length.

a. If $x$ represents an integer, write an expression for the next largest integer.b. If $x$ represents an odd integer, write an expression for the next largest odd integer.c. If $x$ represents an even integer, write expressions for the next two largest even integers.

What does $45^{\circ}$ mean?

A 12 -foot board has been cut into two sections, one twice as long as the other. How long is each section?(FIGURE CANNOT COPY)

The robotic arm will extend a total distance of 18 feet. Find the length of each section.(FIGURE CANNOT COPY)

National Parks. The Natchez Trace Parkway is a historical 444 mile route from Natchez, Mississippi, to Nashville, Tennessce. A couple drove the Trace in four days. Each day they drove 6 miles more than the previous day. How many miles did they drive each day?(FIGURE CANNOT COPY)

Touring. A rock group plans to travel for a total of 38 weeks, making three concert stops. They will be in Japan for 4 more weeks than they will be in Australia. Their stay in Sweden will be 2 wecks shorter than that in Australia. How many weeks will they be in each country?

Solar Heating. Two solar panels were installed side-by-side on a roof, as shown below. One panel is 3.4 feet wider than the other. Find the width of each panel.( figure cannot copy)

Accounting. Determine the 2010 income of Aeropostale Inc. for each quarter from the data in the graph below.(image cannot copy)

iPhone Apps. A student spent a total of $$ 52.97$ on the purchase of three applications in the $Lambda$ pps Store on his iPhone. Call of Duty: World at War: Zombies II cost $\$ 7$ more than Guitar Hero, and Tom Tom USA cost $$ 4.11$ more than twelvetimes Guitar Hero. Find the cost of each application.

Water Usage. It takes about 3.8 times more gallons of water to produce one pound of yrain-fed beef than it does to produce one pound of grain-fed chicken. If a combined total of $3,135$ gallons of water are needed to produce one pound of each meat, how many gallons does it take to produce one pound of chicken? How many gallons does it take to produce one pound of beef?

Counting Calories. A slice of pie with a scoop of ice cream has 850 calories. The calories in the pie alone are 100 more than twice the calories in the ice cream alone. How many calories are in each food?

Waste Disposal. Two tanks hold a total of 45 yallons of a toxic solvent. One tank holds 6 gallons more than twice the amount in the other. How many gallons does each tank hold?

Concerts. The fee to rent a concert hall is $\$ 2,250$ plus $\$ 150$ per hour to pay for the support staff. For how many hours can an orchestra rent the hall and stay within a budget of $\$ 3,300 ?$

Truck Mechanics. An engine repair cost a truck owner $$ 1,185$ in parts and labor. If the parts were $\$ 690$ and the mechanic charged $$ 45$ per hour, how many hours did the repair take?

Field Trips. It costs a school $ 65$ a day plus $$ 0.25$ per mile to rent a 15 -passenger van. If the van is rented for two days, how many miles can be driven on a $$ 275$ budget?

Decorations. A party supply store charges a set-up fee of $$ 80$ plus $35 mathfrak{k}$ per balloon to make a balloon arch. A business has $$ 150$ to spend on decorations for their grand opening. How many balloons can they have in the arch?

Tutoring. High school students enrolling in a private tutoring program must first take a placement test (cost $$ 25$ ) before receiving tutoring (cost $18.75 per hour ). If a family has set aside $$400$ to get their child extra help, how many hours of tutoring can they afford?

Data Conversion. The Books 2 Bytes service converts old print books to Microsoft Word electronic files for $$ 20$ per book plus $2.25$ per page. If it cost $1,201.25$ to convert a novel, how many pages did the novel have?

Cattle Auctions. A cattle rancher is going to sell one of his prize bulls at an auction and would like to make $$ 45,500$ after paying a $9 \%$ commission to the auctioneer. For what selling price will the rancher make this amount of money?

Listing Price. At what price should a home be listed if the owner wants to make $$ 567,000$ on its sale after paying a $5.5 \%$ real estate commission?

Selling Used Clothing. A consignment shop accepts an item of clothing that no longer fits (or one you have grown tired of) and sells it for you. The shop then charges you an agreed on percent of the selling price as their profit. Suppose the owner of a designer wool coat would like to make $$ 210$ on its sale at a consignment shop. If there is a $12, \frac{1}{2} \%$ consignment charge, for what price must the coat be sold?

Finder's Fees. A finder $$$ fee is an amount of money that is paid to someone who brings people together for business purposes. Suppose the owner of a software company needs to sell it and make $ 9.950,000$ to pay back creditors. If he expects to pay a finder's fee of $frac{1}{2} %$ of the selling price to find a qualified buyer, for what price must the company be sold?

Savings Accounts. The balance in a savings account grew by $5 \%$ in one year, to $\$ 5,512.50 .$ What was the balance at the beginning of the year?

Aluminum Cans. Today's aluminum cans are much thinner and lighter than those of the past. From 1972 to 2010 , the number of empty cans produced from one pound of aluminum has increased by about $45 \% .$ If 32 cans could be produced from one pound of aluminum in 2010 , how many cans could be produced from one pound of aluminum in $1972 ?$ Round to the nearest can.

Soccer. Ronaldo of Brazil and Gerd Mueller of Germany rank 1 and 2 , respectively, with the most goals scored in World Cup play. The number of goals Ronaldo and Mueller have scored are consecutive integers that total $29 .$ Find the number of goals scored by each man.

Dictionaries. The definitions of the words job and join are on back-to-back pages in a dictionary. If the sum of those page numbers is $1,411$, on what page can the definition of $j o b$ be found?

TV History. Friends and Leave It to Beaver are two of the most popular television shows of all time. The number of episodes of each show are consecutive even integers whose sum is $470 .$ If there are more episodes of Friends, how many episodes of each were there?

Time Oft. The table shows the average number of days off an employed adult receives for selected countries. Complete the table. The numbers of days are listed in descending order.(table cannot copy)

Celebrity Birthdays. Selena Gomez, Jennifer Lopez, and Sandra Bullock have birthdays (in that order) on consecutive even-numbered days in July. The sum of the calendar dates of their birthdays is $72 .$ Find each birthday.

Locks. The three numbers of the combination for a lock are consecutive integers, and their sum is $81 .$ Find the combination.(image cannot copy)

Tennis. The perimeter of a regulation singles tennis court is 210 feet and the length is 3 feet less than three times the width. What are the dimensions of the court?

Swimming Pools. The seawater Orthlieb Pool in Casablanca, Morocco, is the largest swimming pool in the world. With a perimeter of $1,110$ meters, this rectangular-shaped pool is 30 meters longer than 6 times its width. Find its dimensions.

Art. The Mona Lisa was completed by Leonardo da Vinci in $1506 .$ The length of the picture is 11.75 inches shorter than twice the width. If the perimeter of the picture is 102.5 inches, find its dimensions.(image cannot copy)

New York City. Central Park, which lies in the middle of Manhattan, is rectangular-shaped and has a 6 -mile perimeter. The length is 5 times the width. What are the dimensions of the park?

Engineering. A truss is in the form of an isosceles triangle. Fach of the two equal sides is 4 feet shorter than the third side. If the perimeter is 25 feet, find the lengths of the sides.(figure cannot copy)

First Aid. A sling is in the shape of an isosceles triangle with a perimeter of 144 inches. The longest side of the sling is 18 inches longer than either of the other two sides. Find the lengths of each side.

TV Towers. A TV tower is supported by several guy wires. Two of the guy wires are attached to the top of the tower to form an isosceles triangle with the ground, as shown on the right. The measure of each of the base angles of the triangle is 4 times the third angle (the vertex angle). Find the measure of the vertex angle.

Clotheslines. A pair of damp jeans are hung in the middle of a clothesline to dry. Find $x,$ the angle that the clothesline makes with the horizontal.(image cannot copy)

Mountain Bicycles. For the bicycle frame shown, the angle that the horizontal crossbar makes with the seat support is $15^{\circ}$ less than twice the angle at the steering column. The angle at the pedal gear is $25^{\circ}$ more than the angle at the steering column. Find these three angle measures.(figure cannot copy)

Triangles. The measure of $\angle 1$ (read as angle 1 ) of a triangle is one-half that of $\angle 2 .$ The measure of $\angle 3$ is equal to the sum of the measures of $\angle 1$ and $\angle 2 .$ Find each angle measure.

Angles. Two angles are called complementary angles when the sum oftheir measures is $90^{\circ} .$ Refer to the figure on the right. Find $x$. Then find the measures of the complementary angles.(figure cannot copy)

Angles. Two angles are called supplementary angles when the sum of their measures is $180^{\circ} .$ Refer to the figure below. Find $x$. Then find the measures of the supplementary angles.(FIGURE CANNOT COPY)

"Lightning Bolt " In 2010, Usain Bolt of Jamaica held the world record for the 100 meters and the 200 meters sprints. His maximum stride angle shown below is $5^{\circ}$ less than 1.5 times its supplement. Find his maximum stride angle. You may need to refer to problem 52 to review the geometry involved. (IMAGE CANNOT COPY)

The sweep angle of a windshield wiper arm is $115^{\circ}$ as shown below. Find $x$ so that the area that is cleared by the wiper is centered on the car's windshield.(IMAGE CANNOT COPY)

Create a geometry problem that could be answered by solving the equation $2 w+2(w+5)=26$

What information do you need to know to answer the following question?

A business rented a copy machine for $$ 85$ per month plus $4 â‚¬$ for every copy made. How many copies can be made each month?Make a list of words and phrases that translate to an equal eymbol =.

Define the word strategy.

Solve.$$\frac{5}{8} x=-15$$

Solve.$$\frac{12 x+24}{13}=36$$

Solve.$$\frac{3}{4} y=\frac{2}{5} y-\frac{3}{2} y-2$$

Solve.$$4.2(y-4)-0.6 y=-13.2$$

What concept discussed in this section is illustrated by the following day and time? Two minutes and three seconds past 1 A.M. on the 5 th day of April, 2006

Manufacturing. A company has two machines that make widgets. The production costs are listed below. Machine 1: Setup cost $$ 400$ and $ 1.70$ per widget Machine 2: Setup cost $ 500$ and $$ 1.20$ per widget Find the number of widgets for which the cost to manufacture them on either machine is the same.