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Interpret parts of quadratic expressions: word problems

For additional income, Franklin plans to build craft rocking chairs and sell them at the fair. He has found that people are willing to pay less per rocking chair if he builds more rocking chairs. By building qq rocking chairs to sell, he can price each chair at 144−8q144-8q dollars, which means his revenue will be q(144−8q)q(144-8q) dollars. It will cost Franklin $16\$16 in supplies to build each rocking chair, and he just bought a new power drill and saw for $312\$312. So, his costs will be 16q+31216q+312 dollars. Therefore, if he builds and sells qq rocking chairs, his profit in dollars will be q(144−8q)−(16q+312)q(144-8q)-(16q+312). This expression can be simplified to −8q2+128q−312-8q^2+128q-312 and then written in factored form as −8(q−3)(q−13)-8(q-3)(q-13). What do the numbers 144−8q144-8q00 and 144−8q144-8q11 represent in the expression?
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Hector, a rug maker, is working on a custom rectangular rug for a client's living room. To fit the room, the rug needs to be 22 times as long as it is wide. In addition to the interior pattern, which Hector will design, the rug will have a floral border that is 11 foot wide on each side.\newlineThe area of the interior pattern of the rug in square feet can be modeled by the expression (2x−2)(x−2)(2x - 2)(x - 2), where xx is the width of the entire rug in feet.\newlineWhat does the quantity 2x−22x - 2 represent in the expression?\newlineChoices:\newline(A)the area of the border in square feet\newline(B)the perimeter around the interior pattern of the rug in feet\newline(C)the area of the interior pattern of the rug in square feet\newline(D)the length of the interior pattern of the rug in feet
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Pam is installing a new window in her backyard workshop, to let in more natural light. She wants the window to be square, since she thinks that will look best, and she will include a 1.51.5-inch-wide window frame on all four sides of the window.\newlineThe total area of the window and frame in square inches can be modeled by the expression (x+3)(x+3)(x + 3)(x + 3), where xx is the width of the window in inches. This expression can also be written in the form x2+6x+9x^2 + 6x + 9.\newlineWhat does the quantity 6x+96x + 9 represent in the expression?\newlineChoices:\newline(A)the perimeter around the outside of the frame in inches\newline(B)the width of the window and frame together in inches\newline(C)the area of the frame in square inches\newline(D)the area of the window in square inches\newline
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Tim has decided to add a wraparound porch to his beach cottage so that he can spend more time outside. His cottage is 4545 feet long and 2525 feet wide. The porch will go around three sides of the cottage and be the same width on all sides.\newlineThe total area of the cottage and porch together in square feet can be modeled by the expression (45+2x)(25+x)(45 + 2x)(25 + x), where xx is the width of the porch in feet. This expression can be also written in the form 2x2+95x+1,1252x^2 + 95x + 1,125.\newlineWhat does the quantity 2x2+95x2x^2 + 95x represent in the expression?\newlineChoices:\newline(A)the perimeter around the outside of the porch in feet\newline(B)the area of the cottage in square feet\newline(C)the area of the porch in square feet\newline(D)the length of the cottage and porch together in feet
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When Zamzu Electronics released its first video game, the game's value initially decreased. Once the game became a collector's item, however, its value started to increase. The value of the video game in dollars can be modeled by the expression 0.5t2−6t+320.5t^2 - 6t + 32, where tt is the time in years after it was released.\newlineWhat does the number 3232 represent in the expression?\newlineChoices:\newline(A)the video game's value in dollars when it was worth the least\newline(B)the time in years from when the video game was released until its value was the same as when it was first released\newline(C)the video game's value in dollars when it was first released\newline(D)the time in years from when the video game was released until its value was the least\newline
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Every year, the Gambo Toy Company releases a collector's set of miniature stuffed animals. The company has found that when it charges more per set, it sells fewer sets that year. Its revenue from selling the stuffed-animal sets, in dollars, can be modeled by the expression p(2,400−30p)p(2,400 - 30p), where pp is the price per set in dollars. This expression can be written in factored form as −30p(p−80)-30p(p - 80).\newlineWhat does the number 8080 represent in the expression?\newlineChoices:\newline(A)the price per stuffed-animal set in dollars so that Gambo's revenue is zero\newline(B)Gambo's minimum revenue in dollars\newline(C)Gambo's maximum revenue in dollars\newline(D)the price per stuffed-animal set in dollars that maximizes Gambo's revenue
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On a family vacation at the beach, Harold throws a beach ball upward as high as he can. When the ball reaches a height of 3030 feet, it starts to fall back down. The ball's height above the ground in feet can be modeled by the expression 30−16t230 - 16t^2, where tt is the time in seconds after the ball starts to fall back down.\newlineWhat does the quantity 16t216t^2 represent in the expression?\newlineChoices:\newline(A)the time in seconds it takes for the ball to fall tt feet\newline(B)the time in seconds it takes for the ball to reach a height of tt feet\newline(C)the distance in feet the ball has fallen after tt seconds\newline(D)the height in feet of the ball above the ground after tt seconds\newline
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Weber's Windows installs custom-made windows in new homes. For one home, they will install a window in a wall opening that has a height of 4040 inches and a width of 2828 inches. A wood frame will go around the window, with an equal width on all sides.\newlineThe area of the window inside the frame in square inches can be modeled by the expression (40−2x)(28−2x)(40 - 2x)(28 - 2x), where xx is the width of the frame in inches.\newlineWhat does the quantity 40−2x40 - 2x represent in the expression?\newlineChoices:\newline(A)the height of the window inside the frame in inches\newline(B)the perimeter around the window inside the frame in inches\newline(C)the area of the frame in square inches\newline(D)the area of the window inside the frame in square inches
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Trevor and his friends are making a four-square court that will be 1010 feet long and 1010 feet wide. To protect spectators from the action, they will include a safety zone around the outside of the court that will be the same width around each edge.\newlineThe total area of the court and safety zone together in square feet can be modeled by the expression (10+2x)(10+2x)(10 + 2x)(10 + 2x), where xx is the width of the safety zone in feet. This expression can also be written in the form 4x2+40x+1004x^2 + 40x + 100.\newlineWhat does the quantity 4x2+40x4x^2 + 40x represent in the expression?\newline(A)the perimeter around the outside of the safety zone in feet\newline(B)the area of the court in square feet\newline(C)the length of the court and safety zone together in feet\newline(D)the area of the safety zone in square feet
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Mitchell, a photographer, recently returned from a photo shoot in the Himalayas. Now in the editing phase, he decides that one photo of a climber needs an added visual effect. To give it the right look, he will place a solid border of equal width around the entire 2424-by-3636-inch photo. The total area of the photo and border in square inches can be modeled by the expression (24+2x)(36+2x)(24 + 2x)(36 + 2x), where xx is the width of the border in inches.\newlineWhat does the quantity 36+2x36 + 2x represent in the expression?\newline(A)the length of the photo and border together in inches\newline(B)the area of the border in square inches\newline(C)the total area of the photo and border in square inches\newline(D)the perimeter around the outside of the border in inches
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The Beech Grove town council is organizing this year's downtown fair. The council members have found that when they charge more for each entrance ticket, they sell fewer tickets overall. The fair's total revenue from selling entrance tickets, in dollars, can be modeled by the expression p(840−35p)p(840 - 35p), where pp is the price per ticket in dollars. This expression can be written in factored form as −35p(p−24)-35p(p - 24).\newlineWhat does the number 2424 represent in the expression?\newline(A)the fair's minimum total revenue in dollars\newline(B)the price per entrance ticket in dollars that maximizes the fair's total revenue\newline(C)the price per entrance ticket in dollars so that the fair's total revenue is zero\newline(D)the fair's maximum total revenue in dollars
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To score a field goal in football, a player kicks the ball up into the air with an initial upward velocity of 5656 feet per second. Therefore, the football's height above the ground in feet, tt seconds after it is kicked, can be modeled by the expression −16t2+56t-16t^2 + 56t. This expression can be written in factored form as −16t(t−3.5)-16t(t - 3.5).\newlineWhat does the number 3.53.5 represent in the expression?\newlineChoices:\newline(A)the height in feet of the football when the player kicks it\newline(B)the time in seconds from when the player kicks the football until it lands on the ground\newline(C)the height in feet of the football when it reaches its highest point\newline(D)the time in seconds from when the player kicks the football until it reaches its highest point
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To supplement her income, Ms. McCormick grows lucky bamboo in her backyard garden and sells ceramic pots of bamboo to customers online. She has found that when she decreases the price per pot, the number of pots she sells each month increases. Her monthly revenue can be modeled by the expression p(96−4p)p(96 - 4p), where pp is the price Ms. McCormick charges for each pot of bamboo.\newlineWhat does the quantity 96−4p96 - 4p represent in the expression?\newlineChoices:\newline(A)the number of pots Ms. McCormick sells each month if she charges pp dollars per pot\newline(B)Ms. McCormick's monthly revenue in dollars if she charges pp dollars per pot\newline(C)the price per pot in dollars if Ms. McCormick sells pp pots of bamboo\newline(D)Ms. McCormick's monthly revenue in dollars if she sells pp pots of bamboo\newline
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A geyser erupts, shooting water into the air with an initial upward velocity of 29.4 meters per second29.4\,\text{meters per second}. Therefore, the height of a single water droplet above the ground in meters, tt seconds after it comes out of the geyser, can be modeled by the expression −4.9t2+29.4t-4.9t^2 + 29.4t. This expression can be written in factored form as −4.9t(t−6)-4.9t(t - 6). What does the number 66 represent in the expression?\newline(A)the height in meters of a water droplet when it comes out of the geyser\newline(B)the height in meters of a water droplet when it reaches its highest point\newline(C)the time in seconds from when a water droplet comes out of the geyser until it hits the ground\newline(D)the time in seconds from when a water droplet comes out of the geyser until it reaches its highest point
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Carson wants to frame his diploma, which has the dimensions 1414 inches by 1111 inches. He is considering putting a border around the diploma that has the same width all around.\newlineThe total area of the diploma and border in square inches can be modeled by the expression (11+2x)(14+2x)(11 + 2x)(14 + 2x), where xx is the width of the border in inches.\newlineWhat does the quantity 14+2x14 + 2x represent in the expression?\newlineChoices:\newline(A)the perimeter around the outside of the border in inches\newline(B)the area of the border in square inches\newline(C)the total area of the diploma and border in square inches\newline(D)the length of the diploma and border in inches
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Mrs. Thomas is building a deck for her mountain cabin, where she plans to host summer cookouts for family and friends. Her cabin covers a rectangular area that is 4040 feet by 3030 feet. The deck will go around two sides of the cabin and be the same width on both sides.\newlineThe total area of the cabin and deck in square feet can be modeled by the expression (30+x)(40+x)(30 + x)(40 + x), where xx is the width of the deck in feet. This expression can also be written in the form x2+70x+1,200x^2 + 70x + 1,200. What does the quantity x2+70xx^2 + 70x represent in the expression?\newlineChoices:\newline(A)the area of the deck in square feet\newline(B)the length of the cabin and deck together in feet\newline(C)the area of the cabin in square feet\newline(D)the perimeter around the outside of the cabin and deck in feet
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The value of the first edition of the Detective Kluey graphic novel series decreased after it was released. Over time, as the series became more popular, the value of the first edition started to increase. The value of the first edition in dollars can be modeled by the expression 0.25t2−5t+320.25t^2 - 5t + 32, where tt is the time in years after it was released.\newlineWhat does the number 3232 represent in the expression?\newline(A)the time in years from when the first edition was released until its value was the least\newline(B)the time in years from when the first edition was released until its value was the same as when it was first released\newline(C)the first edition's value in dollars when it was worth the least\newline(D)the first edition's value in dollars when it was first released
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Blue-Green Lagoons is a company that designs and installs wading pools. The standard design includes a rectangular pool that is 22 times as long as it is wide. The wading pool has an 88-foot-wide patio on all sides.\newlineThe total area of the pool and patio in square feet can be modeled by the expression (2x+16)(x+16)(2x + 16)(x + 16), where xx is the width of the pool in feet. This expression can also be written in the form 2x2+48x+2562x^2 + 48x + 256.\newlineWhat does the quantity 48x+25648x + 256 represent in the expression?\newlineChoices:\newline(A)the area of the patio in square feet\newline(B)the length of the pool and patio together in feet\newline(C)the perimeter around the outside of the patio in feet\newline(D)the area of the pool in square feet
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Montox Timepieces manufactures and sells a special-edition wristwatch each year. The company has noted that when it charges more for each wristwatch, customers buy fewer of them that year. Its revenue from selling the wristwatches, in dollars, can be modeled by the expression p(1,080−6p)p(1,080 - 6p), where pp is the price per special-edition wristwatch in dollars. This expression can be written in factored form as −6p(p−180)-6p(p - 180).\newlineWhat does the number 180180 represent in the expression?\newline(A)Montox's maximum revenue in dollars\newline(B)Montox's minimum revenue in dollars\newline(C)the price per special-edition wristwatch in dollars that maximizes Montox's revenue\newline(D)the price per special-edition wristwatch in dollars so that Montox's revenue is zero\newline
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Mr. Ballard builds birdhouses in his free time and sells them at a roadside market on the weekends. He notices that when he charges less for the birdhouses he builds, more customers buy his birdhouses. His total revenue in dollars can be modeled by the expression p(120−3p)p(120 - 3p), where pp is the price per birdhouse in dollars.\newlineWhat does the quantity 120−3p120 - 3p represent in the expression?\newlineChoices:\newline(A) Mr. Ballard's total revenue in dollars if he charges pp dollars per birdhouse\newline(B) the number of birdhouses Mr. Ballard sells if he charges pp dollars per birdhouse\newline(C) the price per birdhouse in dollars if Mr. Ballard sells pp birdhouses\newline(D) Mr. Ballard's total revenue in dollars if he sells pp birdhouses
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Fernando, an interior decorator, is designing an entryway for a client's new home. The entryway will include a set of double doors and hardwood moulding. The moulding will be an equal width around the sides and top of the double doors, which will be 6060 inches wide and 8080 inches tall. The total area of the double doors and moulding in square inches can be modeled by the expression (2x+60)(x+80)(2x + 60)(x + 80), where xx is the width of the moulding in inches. This expression can also be written in the form 2x2+220x+4,8002x^2 + 220x + 4,800. What does the quantity 2x2+220x2x^2 + 220x represent in the expression?\newline(A)the perimeter around the outside of the double doors and moulding in inches\newline(B)the area of the double doors in square inches\newline(C)the height of the double doors and moulding together in inches\newline(D)the area of the moulding in square inches
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Ed is planning to put a square fish pond in the corner of his backyard. He is also planning to add a 55-foot-wide walkway around two sides of the fish pond, since the backyard fence will go along the other two sides.\newlineThe total area of the fish pond and walkway in square feet can be modeled by the expression (x+5)(x+5)(x + 5)(x + 5), where xx is the width of the fish pond in feet. This expression can also be written in the form x2+10x+25x^2 + 10x + 25.\newlineWhat does the quantity 10x+2510x + 25 represent in the expression?\newline(A)the length of the fish pond and walkway together in feet\newline(B)the perimeter around the outside of the fish pond and walkway together in feet\newline(C)the area of the fish pond in square feet\newline(D)the area of the walkway in square feet
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The Last Laugh, a comedy club, hosts the No Funny Business Comedy Festival every year. The club owners have observed that when they increase the price for festival tickets, fewer people buy tickets overall. The club's festival-ticket revenue in dollars can be modeled by the expression p(1,400−20p)p(1,400 - 20p), where pp is the price per ticket in dollars. This expression can be written in factored form as −20p(p−70)-20p(p - 70).\newlineWhat does the number 7070 represent in the expression?\newlineChoices:\newline(A)the price per festival ticket in dollars that maximizes the club's total revenue\newline(B)the club's maximum total revenue in dollars\newline(C)the club's minimum total revenue in dollars\newline(D)the price per festival ticket in dollars so that the club's total revenue is zero\newline
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Addison, a carpenter, is installing a new set of closet doors with a simple moulding for a client's bedroom. The closet doors are 7575 inches wide and 8585 inches tall, and the moulding will be an equal width around the sides and top of the doors. The total area of the closet doors and moulding in square inches can be modeled by the expression (75+2x)(85+x)(75 + 2x)(85 + x), where xx is the width of the moulding in inches. What does the quantity 85+x85 + x represent in the expression?\newline(A)the height of the closet doors and moulding together in inches\newline(B)the perimeter around the outside of the closet doors and moulding in inches\newline(C)the total area of the closet doors and moulding in square inches\newline(D)the area of the moulding in square inches
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Kylie enjoys arts and crafts. For her current project, she is making envelopes out of recycled paper. The dimensions will be the standard 66 inches by 99 inches. To give her envelopes a little flair, she will design a playful border, the same width all around, to line the edges of each. The area of the blank space inside the border in square inches can be modeled by the expression (6−2x)(9−2x)(6 - 2x)(9 - 2x), where xx is the width of the border in inches. What does the quantity 9−2x9 - 2x represent in the expression?\newline(A)the perimeter around the blank space inside the border in inches\newline(B)the length of the blank space inside the border in inches\newline(C)the area of the blank space inside the border in square inches\newline(D)the area of the border in square inches
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