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Mathematics LibreTexts

9.5E: Exercises

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    30916
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    Practice Makes Perfect

    Exercise \(\PageIndex{15}\) Solve Applications Modeled by Quadratic Equations

    In the following exercises, solve using any method.

    1. The product of two consecutive odd numbers is \(255\). Find the numbers.
    2. The product of two consecutive even numbers is \(360\). Find the numbers.
    3. The product of two consecutive even numbers is \(624\). Find the numbers.
    4. The product of two consecutive odd numbers is \(1,023\). Find the numbers.
    5. The product of two consecutive odd numbers is \(483\). Find the numbers.
    6. The product of two consecutive even numbers is \(528\). Find the numbers.
    Answer

    1. Two consecutive odd numbers whose product is \(255\) are \(15\) and \(17\), and \(−15\) and \(−17\).

    3. The first and second consecutive odd numbers are \(24\) and \(26\), and \(−26\) and \(−24\).

    5. Two consecutive odd numbers whose product is \(483\) are \(21\) and \(23\), and \(−21\) and \(−23\).

    Exercise \(\PageIndex{16}\) Solve Applications Modeled by Quadratic Equations

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

    1. 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.
    2. The base of a triangle is six more than twice the height. The area of the triangle is \(88\) square yards. Find the base and height of the triangle.
    3. 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?
    4. A triangular banner for the basketball championship hangs in the gym. It has an area of \(75\) square feet. What is the length of the base and height , if the base is two-thirds of the height?
    5. 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.
    6. A rectangular lawn has area \(140\) square yards. Its width that is six less than twice the length. What are the length and width of the lawn?
    7. 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.
    8. The new computer has a surface area of \(168\) square inches. If the the width is \(5.5\) inches less that the length, what are the dimensions of the computer?
    9. 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.
    10. The hypotenuse of a right triangle is \(10\) cm long. One of the triangle’s legs is three times as the length of the other leg . Round to the nearest tenth. Find the lengths of the three sides of the triangle.
    11. 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.
    Image shows a rectangular segment of grass with fence around 4 sides and across the diagonal. The vertical side of the rectangle is labeled w and the horizontal side is labeled 3 w. The diagonal fence is labeled w plus 5.
    Figure 9.5.44

    12. Nautical flags are used to represent letters of the alphabet. The flag for the letter, O consists of a yellow right triangle and a red right triangle which are sewn together along their hypotenuse to form a square. The hypotenuse of the two triangles is three inches longer than a side of the flag. Find the length of the side of the flag.

    Image shows a square with side lengths s. The square is divided into two triangles with a diagonal. The top triangle is red and the lower triangle is yellow. The diagonal is labeled s plus 3.
    Figure 9.5.45

    13. 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 for up the side of the house will the ladder reach?

    14. John has a \(10\)-foot piece of rope that he wants to use to support his \(8\)-foot tree. How far from the base of the tree should he secure the rope?

    15. 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.

    16. An arrow is shot vertically upward at a rate of \(220\) feet per second. Use the projectile formula \(h=-16 t^{2}+v_{0} t\), to determine when height of the arrow will be \(400\) feet.

    17. 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 \(h=0\)?)

    18. A stone is dropped from a \(196\)-foot platform. Use the formula \(h=-16 t^{2}+v_{0} t+196\) to determine how many seconds it will take for the stone to hit the ground. (Since the stone is dropped, \(v_{0}=0\).)

    19. 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?

    20. The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of \(5\) hours and each way the trip was \(300\) miles. If the plane was flying at \(125\) mph, what was the speed of the wind that affected the plane?

    21. 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?

    22. Rick paddled up the river, spent the night camping, and then paddled back. He spent \(10\) hours paddling and the campground was \(24\) miles away. If Rick kayaked at a speed of \(5\) mph, what was the speed of the current?

    23. 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?

    24. Two gardeners can do the weekly yard maintenance in \(8\) minutes if they work together. The older gardener takes \(12\) minutes more than the younger gardener to finish the job by himself. How long does it take for each gardener to do the weekly yard maintenance individually?

    25. 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?

    26. Sully is having a party and wants to fill his swimming pool. If he only uses his hose it takes \(2\) hours more than if he only uses his neighbor’s hose. If he uses both hoses together, the pool fills in \(4\) hours. How long does it take for each hose to fill the pool?

    Answer

    1. The width of the triangle is \(5\) inches and the height is \(18\) inches.

    3. The base is \(24\) feet and the height of the triangle is \(10\) feet.

    5. The length of the driveway is \(15.0\) feet and the width is \(3.3\) feet.

    7. The length of table is \(8\) feet and the width is \(3\) feet.

    9. The length of the legs of the right triangle are \(3.2\) and \(9.6\) cm.

    11. The length of the diagonal fencing is \(7.3\) yards.

    13. The ladder will reach \(24.5\) feet on the side of the house.

    15. The arrow will reach \(400\) feet on its way up in \(2.8\) seconds and on the way down in \(11\) seconds.

    17. The bullet will take \(70\) seconds to hit the ground.

    19. The speed of the wind was \(49\) mph.

    21. The speed of the current was \(4.3\) mph.

    23. The less experienced painter takes \(6\) hours and the experienced painter takes \(3\) hours to do the job alone.

    25. Machine #1 takes \(3.6\) hours and Machine #2 takes \(4.6\) hours to do the job alone.

    Exercise \(\PageIndex{17}\) writing exercises

    1. Make up a problem involving the product of two consecutive odd integers.
      1. Start by choosing two consecutive odd integers. What are your integers?
      2. What is the product of your integers?
      3. Solve the equation \(n(n+2)=p\), where \(p\) is the product you found in part (b).
      4. Did you get the numbers you started with?
    2. Make up a problem involving the product of two consecutive even integers.
      1. Start by choosing two consecutive even integers. What are your integers?
      2. What is the product of your integers?
      3. Solve the equation \(n(n+2)=p\), where \(p\) is the product you found in part (b).
      4. Did you get the numbers you started with?
    Answer

    1. Answers will vary.

    Self Check

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

    This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can solve applications of the quadratic formula.” “Confidently,” “with some help,” or “No, I don’t get it.”
    Figure 9.5.46

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