Solved Examples and Worksheet for Application of Quadratic Equations

Q1"The University U offers dorm facilities for at most 900 students." Which of the following inequalities represent the situation, if S is the number of students that the dorm an accommodate?
A. S < 900
B. S ≥ 900
C. S ≤ 900
D. S > 900

Solutions
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Step: 1
900 is also included in the solution because the dorm can accommodate at most 900 students.
Step: 2
The inequality for the situation is S ≤ 900.
Correct Answer is :   S ≤ 900
Q2Which of the following inequalities represents the statement "In order to continue an account in a bank, the minimum amount in the bank should be $500", if M represents the minimum amount?
A. M > 500
B. M < 500
C. M ≥ 500
D. M ≤ 500

Solutions
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Step: 1
In order to continue the account in the bank, the amount should be greater than or equal to 500.
Step: 2
Among the choices, the inequality M ≥ 500 satisfies the statement.
Correct Answer is :   M ≥ 500
Q3Which of the following inequalities describes the statement "Dr. Mike needs at least 250 beds for his new hospital", if H represents the number of beds.

A. H ≥ 250
B. H ≤ 250
C. H > 250
D. H < 250

Solutions
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Step: 1
The statement says that Dr. Mike needs at least 250 beds for his new hospital. So, 250 is also included in the solution.
Step: 2
The inequality for the statement is H ≥ 250.
Correct Answer is :   H ≥ 250
Q4"American soccer team needs to score no less than 2 goals against Germany to win the match." Which of the following inequalities represents this statement, if G stands for the number of goals?
A. G > 2
B. G ≥ 2
C. G < 2
D. G ≤ 2

Solutions
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Step: 1
The statement says the American soccer team needs to score no less than 2 goals against Germany to win the match. So, 2 is also included in the solution.
Step: 2
The inequality for the statement is G ≥ 2.
Correct Answer is :   G ≥ 2
Q5"A parking lot can accommodate no more than 250 cars." Identify the inequality that represents the capacity of the parking lot, if N stands for the number of cars.

A. N ≤ 250
B. N ≥ 250
C. N < 250
D. N > 250

Solutions
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Step: 1
A parking lot can accommodate no more than 250 cars and 250 is included in the solution.
  [Given statement.]
Step: 2
Among the choices, the inequality N ≤ 250 satisfies the statement.
Correct Answer is :   N ≤ 250
Q6A rectangular fountain in a park has dimensions of a and a + 16. If the area of the fountain is 192 square meters, find the dimensions of the fountain in meters.
A. 9, 25
B. 11, 27
C. 8, 24
D. 10, 26

Solutions
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Step: 1
The area of a rectangle = length × width.
Step: 2
The area of the rectangular fountain = (a) × (a + 16) square meters.
Step: 3
192 = (a) × (a + 16)
  [Original equation.]
Step: 4
192 = a2 + 16a
  [Use distributive property.]
Step: 5
192 + 82 = a2 + 16a + 82
  [Add (162)2 = 82 = 64 to each side.]
Step: 6
256 = (a + 8)2
  [Write the right hand side as a perfect square and simplify.]
Step: 7
± 16 = a + 8
  [Evaluate square roots on both sides.]
Step: 8
± 16 - 8 = a + 8 - 8
  [Subtract 8 from each side.]
Step: 9
a = + 8 or - 24
  [Simplify.]
Step: 10
Width = a = 8 meters
  [The dimensions cannot be negative.]
Step: 11
Length = (a + 16) = (16 + 8) = 24 meters.
  [Substitute 8 for a and add.]
Step: 12
The dimensions of the fountain are 8 meters wide and 24 meters long.
Correct Answer is :   8, 24
Q7The monitor screen of a computer is rectangular with a and a + 24 dimensions. Find the dimensions of the monitor, if its area is 112 square inches.

A. 3 in. and 27 in.
B. 6 in. and 30 in.
C. 5 in. and 29 in.
D. 4 in. and 28 in.

Solutions
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Step: 1
The area of a rectangle = length × width.
Step: 2
The front view area of the monitor = (a) × (a + 24) square in.
Step: 3
112 = (a) × (a + 24)
  [Original equation.]
Step: 4
112 = a2 + 24a
  [Use distributive property.]
Step: 5
112 + 122 = a2 + 24a + 122
  [Add (242)2 = 122 = 144 to each side.]
Step: 6
256 = (a + 12)2
  [Write the right hand side as a perfect square and simplify.]
Step: 7
±16 = (a + 12)
  [Evaluate square roots on both sides.]
Step: 8
±16 - 12 = a + 12 - 12
  [Subtract 12 from each side.]
Step: 9
a = + 4 or - 28
  [Simplify.]
Step: 10
Width = a = 4 in.
  [Dimensions cannot be negative.]
Step: 11
Length = (a + 24) = (4 + 24) = 28 in.
  [Substitute 4 for a and add.]
Step: 12
The dimensions of the front view of the monitor are 4 in. wide and 28 in. long.
Correct Answer is :   4 in. and 28 in.
Q8The area of a rectangular book is 364 square centimeters. If its dimensions are a and a + 12, find the length and width of the book.
A. 16 cm, 28 cm
B. 14 cm, 26 cm
C. 15 cm, 27 cm
D. 13 cm, 25 cm

Solutions
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Step: 1
The area of a rectangle = length × width.
Step: 2
The area of the rectangular book = (a) × (a + 12) square centimeters.
Step: 3
364 = (a) × (a + 12)
  [Original equation.]
Step: 4
364 = a2 + 12a
  [Use distributive property.]
Step: 5
364 + 62 = a2 + 12a + 62
  [Add (122)2 = 62 = 36 to each side.]
Step: 6
400 = (a + 6)2
  [Write the right hand side as a perfect square and simplify.]
Step: 7
± 20 = (a + 6)
  [Evaluate square roots on both sides.]
Step: 8
± 20 - 6 = a + 6 - 6
  [Subtract 6 from each side.]
Step: 9
a = + 14 or - 26
  [Simplify.]
Step: 10
Width = a = 14 centimeters
  [Dimensions cannot be negative.]
Step: 11
Length = (a + 12) = (14 + 12) = 26 centimeters
  [Substitute 14 for a and add.]
Step: 12
The book is 14 centimeters wide and 26 centimeters long.
Correct Answer is :   14 cm, 26 cm
Q9The notice board of a school measures a feet wide and (a - 6) feet long. What are the dimensions of the board, if its area is 27 square feet?

A. 11 feet by 5 feet
B. 9 feet by 3 feet
C. 8 feet by 2 feet
D. 10 feet by 4 feet

Solutions
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Step: 1
The area of a rectangle = Length × Width.
Step: 2
The area of the rectangular notice board = (a) × (a - 6) square feet.
Step: 3
27 = (a) × (a - 6)
  [Original equation.]
Step: 4
27 = a2 - 6a
  [Use distributive property.]
Step: 5
27 + (- 3)2 = a2 - 6a + (- 3)2
  [Add (- 62)2 = (- 3)2 = 9 to each side.]
Step: 6
36 = (a - 3)2
  [Write the right side of the equation as a perfect square and simplify.]
Step: 7
± 6 = (a - 3)
  [Evaluate square roots on both sides.]
Step: 8
± 6 + 3 = a - 3 + 3
  [Subtract 3 from each side.]
Step: 9
a = - 3 or 9
  [Simplify.]
Step: 10
Width of the rectangular notice board is a = 9 feet.
  [Dimensions cannot be negative.]
Step: 11
Length of the rectangular notice board is (a - 6) = (9 - 6) = 3 feet.
  [Repalce a with 9 and add.]
Step: 12
The rectangular notice board is 9 feet by 3 feet.
Correct Answer is :   9 feet by 3 feet
Q10A rectangular carpet measures a feet long and (a - 10) feet wide. What are the dimensions of the carpet, if its area is 56 square feet?
A. 15 feet by 5 feet
B. 13 feet by 3 feet
C. 14 feet by 4 feet
D. 16 feet by 6 feet

Solutions
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Step: 1
The area of a rectangle = Length × Width
Step: 2
The area of the rectangular carpet = (a) × (a - 10) square feet
Step: 3
56 = (a) × (a - 10)
  [Original equation.]
Step: 4
56 = a2 - 10a
  [Use distributive property.]
Step: 5
56 + (- 5)2 = a2 - 10a + (- 5)2
  [Add (- 102)2 = (- 5)2 = 25 to each side.]
Step: 6
81 = (a - 5)2
  [Write the right side of the equation as a perfect square and simplify.]
Step: 7
± 9 = (a - 5)
  [Evaluate square roots on both sides.]
Step: 8
± 9 + 5 = a - 5 + 5
  [Add 5 on each side.]
Step: 9
a = - 4 or 14
  [Simplify.]
Step: 10
Length of the rectangular carpet is a = 14 feet.
  [Dimensions cannot be negative.]
Step: 11
Width of the rectangular carpet is (a - 10) = (14 - 10) = 4 feet.
  [Repalce a with 14 and add.]
Step: 12
The dimensions of the carpet are 14 feet by 4 feet.
Correct Answer is :   14 feet by 4 feet
Q11Andy throws a pen from the top of a 124 feet tall building with an initial downward velocity of - 30 feet per second. If the equation to model the height of the pen is h = - 16t2 - 30t + 124, how long will the pen take to reach the ground?

A. 2 seconds
B. 124 seconds
C. 30 seconds
D. 3 seconds

Solutions
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Step: 1
h = - 16t2 - 30t + 124
  [Original equation.]
Step: 2
0 = - 16t2 - 30t + 124
  [Height = 0, when the pen is on the ground.]
Step: 3
Compare the original equation with the standard form to get the values of a, b and c.
Step: 4
t = [-(-30)±[(-30)2-4(-16)(124)]][2(-16)]
  [Substitute the values in the quadratic formula.]
Step: 5
t = [30±(900+7936)](-32)
  [Evaluate power and multiply.]
Step: 6
= [30±94](-32)
  [Simplify the radical.]
Step: 7
= - 3.875, 2
  [Simplify.]
Step: 8
The ball reaches the ground after 2 seconds.
  [Consider positive value as t represents time.]
Correct Answer is :   2 seconds
Q12Laura dives into a pool from the diving board, that is 16 feet high from the water. She dives with an initial downward velocity of - 24 feet per second. If the equation to model the height of the dive is h = - 16t2 + (- 24)t + 16, then find the time in seconds it takes Laura to reach the water.
A. 1
B. 1.5
C. 5.50
D. 0.50

Solutions
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Step: 1
h = - 16t2 + (- 24)t + 16
  [Original equation.]
Step: 2
0 = - 16t2 + (- 24t) + 16
  [Replace h with 0, as the height is zero at the water level.]
Step: 3
t = {-(-24)±[(-24)2-4(-16)(16)]}[2(-16)]
  [Substitute a = - 16, b = - 24 and c = 16 in the quadratic formula.]
Step: 4
t = 24±576+1024-32
  [Simplify.]
Step: 5
t = 24±1600-32
  [Simplify inside the radical.]
Step: 6
t = 24±40-32 = -2, 0.50
  [Simplify the radical.]
Step: 7
t = 0.50
  [Since t represents time, consider the positive integer.]
Correct Answer is :   0.50
Q13Frank stands on a bridge 73.5 feet above the ground holding an apple. He throws it with an initial downward velocity of - 25 feet per second. How long will it take for the apple to reach the ground, if the vertical motion is given by the equation h = - 16t2 + vt + s?
(s = 73.5 feet)

A. 1.5 seconds
B. 3.06 seconds
C. 2 seconds
D. 2.5 seconds

Solutions
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Step: 1
h = - 16t2 + vt + s
  [Original equation.]
Step: 2
0 = - 16t2 + vt + s
  [h = 0 for ground level.]
Step: 3
0 = - 16t2 - 25t + 73.5
  [Replace v with - 25 and s = 73.5.]
Step: 4
t = [-(-25)±[(-25)2-4(-16)(73.5)]]2(-16)
  [Substitute the values of a = - 16, b = - 25 and c = 73.5 in the quadratic formula.]
Step: 5
= [25±(625+4704)]-32
  [Evaluate the power and multiply.]
Step: 6
= 25±5329-32
  [Add within the grouping symbols.]
Step: 7
= 25±73-32
  [Find the square root.]
Step: 8
t = - 3.0625 or 1.5
  [Simplify.]
Step: 9
The apple will reach the ground about 1.5 seconds after it was thrown.
Correct Answer is :   1.5 seconds

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