Keeping the Meaning in Mathematics: The Craft of Word Problems

Appendix B - Collection of Word Problems

Problem Suite A: Projectile Motion

Dimension 1A: Write the equation

• Avery throws a football straight up in the air with an upward velocity of 27 m/s from a height of 1.5 m. Write the equation describing the height of the football as a function of time.
• A soccer player sets up a free kick by putting the ball on the ground near the referee. If she kicks it with an initial upward velocity of 68 ft/s, what equation describes the height of the ball as a function of time?
• If a golf ball is hit with an initial upward velocity of 20 m/s, write the equation describing the height of the golf ball t seconds after it is hit.

Dimension 2A: Evaluate the equation

• A basketball player launched a shot from beyond midcourt just 3 seconds before the final buzzer. If the ball was launched from a height of 8 feet with an initial upward velocity of 41 ft/s, the equation describing height off the ground as a function of time would be h(t) = -16t ² + 41t + 8. How high would the ball be 2.5 seconds after the shot was launched?
• A boat in distress launches a flare straight up with a velocity of 190 ft/s. If the path of the flare is modeled by h(t) = -16t ² + 190t + 20, how high is the flare 10 seconds after it was launched?
• The height h in feet of a person on a waterslide can be modeled by the function h(t) = -0.025t ² - 0.5t + 50, where t is the time in seconds. At the bottom of the slide, the person lands in a swimming pool. How high is the person after 1 second on the slide?

Dimension 3A: h ₀ = 0; find the time it takes an object to return to the ground

• A soccer goalie kicks the ball from the ground at an initial upward velocity of 40 ft/s. How long will it take the ball to hit the ground?
• A golf ball is hit from ground level with an initial upward velocity of 62 ft/s. After how many seconds will the ball hit the ground?

Dimension 4A: h ₀ = 0; find the time it takes an object to reach its maximum height

• Suppose a baseball is thrown straight up from a height of 4.5 ft with an initial upward velocity of 60 ft/s. At what time will the maximum height be attained?
• A football player attempts a field goal. The quarterback holds the ball on the ground as the kicker kicks with an upward velocity of 50 ft/s. How long does it take the ball to reach its maximum height? It its horizontal velocity is 18 ft/s, how far has it gone?

Dimension 5A: h ₀ = 0; find the maximum height reached by an object

• Suppose a baseball is shot straight up from a height of 4.5 ft with an initial velocity of 60 ft/s. What is the maximum height reached by the ball?
• A golf ball leaves the tee with an initial upward velocity of 18 m/s. What is the ball's maximum height? If its horizontal velocity is 6.5 m/s, how far has it gone?

Dimension 6A: h ₀ ¹ 0; find the max, find the time to reach max or ground

• Brandon threw a baseball with an upward velocity of 50 ft/s from a height of 6 ft. How long will it take the ball to reach its maximum height? What is the maximum height the ball reaches?
• A player bumps a volleyball when it is 4 ft above the ground with an initial vertical velocity of 20 ft/s (equation would be h = -16t ² + 20t + 4). What is the maximum height of the ball? If the volleyball were hit under the same conditions, but with an initial velocity of 32 ft/s, how much higher would the ball go?
• In a volleyball game, a player on one team spikes the ball over the net when the ball is 10 ft above the court. The spike drives the ball downward with an initial velocity of -55 ft/s. Players on the opposing team must hit the ball before it touches the court. How much time do the opposing players have to hit the spiked ball?
• Jason lobbed (hit) a tennis ball upward with a velocity of 48 ft/s from a height of 4 ft above the ground. How long does his opponent have to get to the ball before it hits the ground?

Dimension 7A: Find the time(s) to reach specified height, h(t) ¹ 0

• A baseball player hits a high pop-up with an initial upward velocity of 98 ft/s, 4.5 ft above the ground. How long does a player on the opposing team have to catch the ball if he catches it 5.6 ft above the ground?
• A basketball player passes the ball to a teammate who catches it 11 ft above the court, just above the rim of the basket, and slam-dunks it through the hoop (an "alley-oop" play). The first player releases the ball 5 ft above the court with an initial upward velocity of 21 ft/s. How long is the ball in the air before being caught, assuming it is caught as it rises?
• A baton twirler tosses a baton into the air. The baton leaves the twirler's hand 6 ft above the ground and has an initial upward velocity of 45 ft/s. The twirler catches the baton when it falls back to a height if 5 ft. For how long is the baton in the air?

Dimension 8A: Find the initial upward velocity

• A tennis ball hits a winner from 0.5 m above the ground that hits the sideline 1.8 sec later. What was the initial upward velocity of the ball?
• A golfer hits his second shot from the ground. It reaches a maximum height of 100 ft in 2.5 sec. What was its initial upward velocity?
• A football punt reaches a maximum height of 68 ft in 2 sec. What was the initial upward velocity of the football?

Dimension 9A: Find the initial height

• A baseball line drive was hit with an initial upward velocity of 3 m/s. It was caught by the 3 ^rd baseman 0.1sec later at a height of 1.1m. What was the initial height of the ball when it was hit?
• A diving volleyball player bumped the ball with an initial upward velocity of 18 ft/s. Another player was able to set the ball 1 sec later at a height of 5 ft. What was the height of the volleyball when it was bumped?

Dimension 10A: Interpret the result/compare result to information given

• A baseball is popped up into foul territory with an upward velocity of 42 ft/s from a height of 3.5 ft above the ground. If the left fielder is 100 ft away and runs at an average speed of 18 ft/s, will he be able to reach the ball before it hits the ground?
• A player throws the ball home from a height of 5.5 ft with an initial upward velocity of 28 ft/s. The ball is caught at home plate at a height of 5 ft. Three seconds before the ball is thrown, a runner on third base starts toward home plate, 90 ft away, at a speed of 25 ft/s. Does the runner reach home plate before the ball does?

Dimension 11A: Including the x and y components of velocity

• A golf ball leaves the tee with an initial velocity of 30m/s at an angle of 37° to the horizontal. At what time(s) will the golf ball be at 10m above the ground? What is the maximum height reached by the ball? What is its range (horizontal distance traveled by the ball)?
• A quarterback passes a football with a velocity of 50ft/s at an angle of 40° to the horizontal toward an intended receiver 30 yd downfield. The pass is released 5ft above the ground. Assume that the receiver is stationary and that he will catch the ball if it comes to him. Will the pass be completed?

Problem Suite B: Geometry

Dimension 1B: Find the maximum area, given the perimeter

• You have a 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard, and what is the largest area?
• Steve has 120 ft of fence to make a rectangular kennel for his dogs. What dimensions produce a kennel with the greatest area?
• Joe has 30 ft of fence to make a rectangular kennel for his dogs, but plans to use his garage as one side. What dimensions produce the greatest area?
• A roll of aluminum with a width of 32cm is to be bent into rain gutters by folding up two sides at 90°angles. A rain gutter's greatest capacity, or volume, is determined by the gutter's greatest cross-sectional area. Find the length of aluminum that should be folded up on each side to maximize the cross-sectional area.
• Suppose a stream borders our land, and we want to make a right-triangular garden with the stream as the hypotenuse. If we have only 80 feet of fencing, what is the maximum area of our garden?
• To create a temporary grazing area, a farmer is using 1800 ft of electric fence to enclose a rectangular field and then to subdivide the field into two equal plots. What is the largest area of the field the farmer can enclose?

Dimension 2B: Find the dimensions, given the area and perimeter

• An ecology center wants to set up an experimental garden using 300m of fencing to enclose a rectangular area of 5000 m ². Find the dimensions of the garden.
• A student environmental group wants to build a rectangular ecology garden. The area of the garden should be 800 square feet to accommodate all the species of plants the group wants to grow. A construction company has donated 120 feet of iron fencing to enclose he garden. What should the dimensions of the garden be? If additional plants are donated that require 110 ft ² of space, will the 120 ft of fencing be enough for the enlarged garden?
• A kennel owner has 164 ft of fencing with which to enclose a rectangular region. He wants to subdivide this region into 3 smaller rectangles of equal length. If the total area must be 575 sq ft, find the dimensions of the entire enclosed region.

Dimension 3B: Borders

• Tonya wants to buy a mat for a photograph that measures 14 in. by 20 in. She wants to have an even border around the picture when it is mounted on the mat. If the area of the mat she chooses (before it is cut) is 352 in ², how wide will the border be?
• A landscape architect has included a rectangular flowerbed measuring 9ft by 5ft in her plans for a new building. She wants to use two colors of flowers in the bed, one in the center and the other for a border of the same width on all four sides. If she has enough plants to cover 24 ft ² for the border, how wide can the border be?
• A family has a round swimming pool in their back yard with a diameter of 48 ft, and they want to build a circular deck around it. If the space available for the pool and deck is 2300 ft ², and they want the deck to be a uniform width, how wide can the deck be?
• A ring of grass with an area of 314 yd ² surrounds a circular flowerbed, which has a radius of 10 yd. Find the width of the ring of grass.

Dimension 4B: Volume

• A square piece of cardboard has 3 in squares cut from its corners and then has the flaps folded up to form an open-top box. What original length would yield a box with volume 432 in ³?
• You are designing the ventilation hood for a restaurant's stove. The hood is to be made by cutting squares from the corners of a piece of sheet metal, then folding the corners and welding them together. The piece of sheet metal is 5 ft wide. The length of the finished hood should be 9 ft, and its volume must be 22 ft ³. The height of the hood should not exceed 1 ft. What will be the height of the completed ventilation hood?

Dimension 5B: Pythagorean Theorem

• A nature conservancy group decides to construct a raised wooden walkway through a wetland area. To enclose the most interesting part of the wetlands, the walkway will have the shape of a right triangle with one leg 700 yd longer than the other and the hypotenuse 100 yd longer than the longer leg. Find the total length of the walkway.
• The perimeter of a TV screen is 88 in. Find the least possible value of the length of the diagonal. What are the dimensions of the TV screen?
• A kite is flying on 50 ft of string. Its vertical distance from the ground is 10 ft more than its horizontal distance from the person flying it. Assuming that the string is being held at ground level, find its horizontal distance from the person and its vertical distance from the ground.

Dimension 6B: Surface Area

• The surface area of a box with open top has a square base and a height of 4 in. If the surface area of the box is 161 in ², find the dimensions of the base.
• A manufacturing firm wants to package its product in a cylindrical container 3 ft. high with surface area 8p ft ³. What should the radius of the circular top and bottom of the container be?

Dimension 7B: Dilations

For each problem,

• a. predict the answer,
• b. calculate the answer,
• c. compare your calculation to your prediction, and
• d. reason why your prediction was right or wrong.

• OFFICE/WORK SPACE: A company bought office space measuring 14 m by 20 m. They want to create cubicles or work areas in the center, surrounded by a hallway that is the same width all the way around. In the first design, the area of the cubicles is equal to the area of the hallways. What is the width of the hallways? If the width of the hallways is cut in half to provide more work area, what is the corresponding area remaining for the cubicles?
• WORK SPACE: The manager of an auto body shop wants to expand his business and enlarge the work area of his garage. If the original garage area is 30 ft by 80 ft. and he plans to double the work area, what are the new dimensions of the enlarged work area if it is enlarged by the same amount in each direction?
• WORK SPACE: The manager of an auto body shop wants to expand his business and enlarge the work area of his garage. If the original garage area is 50 ft by 60 ft. and he plans to double both the length and width, what is the increase in work area?
• DRAFTING: A house plan shows a center entranceway with rooms off of it on three sides (left, right and back). The homeowner wants to cut the area of the entranceway in half by moving the 3 walls in by the same amount to give each of the surrounding rooms more space. If the original entranceway was 18 ft by 18 ft, how far should each wall be moved?
• LANDSCAPING: A student environmental group wants to build a rectangular ecology garden. How many feet of fencing does the group need if the maximum area they expect to plant is 500 ft ²?
• If the group decides to double the maximum area, what is the increased length of fence needed?
• If the group is given twice as much fencing as they need, how much additional area could they plant?
• CARPENTRY: A builder found 80 ft of "vintage" crown molding to use for a custom home. What is the area of the largest room he can design to display all of the molding? If he chooses to split the molding evenly between two rooms, what is the maximum area of each room?
• CARPENTRY: Suppose the builder chooses to use 80 ft of "vintage" crown molding in a 12 ft by 15 ft room with a tray ceiling (the ceiling has a rectangular recessed area surrounded by a uniform border on all sides like a picture frame). What are the dimensions of the "tray" if the molding is used for the perimeter of the room AND the perimeter of the tray?
• Students in the Early Childhood class were assigned the task of designing a new fenced playground. They had a total of 120 ft of fencing to work with. What are the dimensions of the largest possible play area? If they were given twice as much fencing, what are the new dimensions and area for the playground?
• MASONRY: A homeowner wants to double the area of his 15 ft by 25 ft brick patio by adding a different-color-brick border on 3 sides (one of the 25 ft sides is against the house). If the border has a uniform width, how wide should the border be? What are the dimensions of the enlarged patio?
• AUTO: The specifications for a Ford F150 truck show it's a 6-cylinder, 4.2 L engine. Each cylinder has a bore (diameter) of 9.68 cm and a stroke (assume it's the height) of 9.5 cm. If the design engineer decided to cut the diameter of each cylinder in half, but maintain the same displacement (volume per cylinder), how much change would there be in the height of each cylinder?
• CULINARY: A cake batter fills two 9-inch (diameter) round cake pans to a level of 1.5 in. What radius would be needed for all of the batter to fit in one round pan filled to the same level?
• ELECTRICAL: For every six increases in gauge numbers, wire diameter is cut in half. No. 18 AWG has a diameter of 0.403 in and No. 24 AWG has a diameter of 0.0201 in. What is the change in cross-sectional area from No. 18 to No. 24?
• HVAC: Although it usually over-sizes them, one rule of thumb used by some contractors to calculate the size for a cooling unit is 1 ton of air conditioning for each 600 ft ² in the house.
• According to this rule of thumb, what size unit (in tons) would be needed to cool a 1-story house that measures 40 ft by 35 ft?
• If the original house is doubled in both dimensions to 80 ft by 70 ft, what size cooling unit would be needed?
• If the family can afford a cooling unit twice the original size, and if the original house must be enlarged by the same amount in each direction, what are the new dimensions of the house?
• umbing Suppliers lists the following specifications:
• peSize | Outer Diameter
• " | 0.840" o.d.
• " | 1.050" o.d.
• | 1.315" o.d.
• 1/4" | 1.660" o.d.
• /2" | 1.900" o.d.
• | 2.375" o.d.
• /2" | 2.875" o.d.
• | 3.500" o.d.
• | 4.500" o.d.
• What is the volume of PVC used to make a 1½" pipe that is 8 ft long?
• What is the volume of PVC needed to make a 3" pipe that is 8 ft long?
• A building site plan originally called for ½-inch pipe to be used. However, the plans needed to be changed so that the pipe could carry twice the amount of flow from the site. What is the change in pipe diameter required to allow for twice the flow volume?