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[jfjd]

Make five solved problems each using
A.golden ratio
B.Translating English sentences into mathematical sentences
C.translating mathematical sentences into English sentences
D.compound statements into symbolic form
E.Symbolic form into compound statements
F.Construction of truth table
G.inverse converse and contrapositive

 

[🤖]

A. Golden Ratio
1. Find the value of x if it is known that x/5 = (x+1)/8, and x is in the golden ratio with (x+1).
Solution:
We know that the golden ratio is (1 + sqrt(5))/2. Since x is in the golden ratio with x+1, we have:

x / (x+1) = (1 + sqrt(5))/2

Simplifying this equation, we get:

2x = (x+1) * (1 + sqrt(5))
2x = x + 1 + x*sqrt(5) + sqrt(5)
x(2 - sqrt(5)) = 1 + sqrt(5)
x = (1 + sqrt(5))/(2 - sqrt(5))

Now we can substitute this value of x in the given equation and solve for x:

x/5 = (x+1)/8
(1 + sqrt(5))/10 = ((1 + sqrt(5))/2 + 1)/8
x = 3 + 2*sqrt(5)


2. A rectangle has sides in the golden ratio. If the shorter side has length 2, what is the length of the longer side?
Solution:
Let x be the length of the longer side. We know that the golden ratio is (1 + sqrt(5))/2. Since the sides are in the golden ratio, we have:

x/2 = (1 + sqrt(5))/2

Solving this equation for x, we get:

x = 2*(1 + sqrt(5))


B. Translating English sentences into mathematical sentences
1. Sum of two consecutive integers is 17. Find the integers.
Solution:
Let x be the first integer, then x+1 is the next consecutive integer. We know that their sum is 17, so we can write:

x + (x+1) = 17

Solving for x, we get:

x = 8

Therefore, the two consecutive integers are 8 and 9.

2. The product of three consecutive odd integers is 5,085. Find the integers.
Solution:
Let x be the first odd integer, then x+2 and x+4 are the next two consecutive odd integers. We know that their product is 5,085, so we can write:

x(x+2)(x+4) = 5085

Expanding and simplifying this equation, we get:

x^3 + 6x^2 + 8x - 5085 = 0

We can solve for x using a calculator or by trial and error. We find that x = 13, so the three consecutive odd integers are 13, 15, and 17.


C. Translating mathematical sentences into English sentences
1. ∃x (x² + 5x = 14)
Translation: There exists a number x such that the sum of the square of x and five times x equals 14.

2. ∀x (x² ≥ 0)
Translation: For all numbers x, the square of x is greater than or equal to zero.


D. Compound statements into symbolic form
1. Either it is raining or it is sunny.
Symbolic form: P ∨ Q, where P represents "it is raining" and Q represents "it is sunny".

2. If it is Monday, then Alice will be at work.
Symbolic form: M → A, where M represents "it is Monday" and A represents "Alice will be at work".


E. Symbolic form into compound statements
1. P ∧ Q
Compound statement: P and Q, where P and Q are some statements.

2. ¬R → ¬S
Compound statement: If not R, then not S, where R and S are some statements.


F. Construction of truth table
Construct the truth table for the statement (P ∧ Q) → R.

| P | Q | R | (P ∧ Q) | (P ∧ Q) → R |
|---|---|---|---------|-------------|
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | F | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | T | F | F | T |
| F | F | T | F | T |
| F | F | F | F | T |


G. Inverse, Converse, and Contrapositive
1. Statement: If it is raining, then the ground is wet.
Inverse: If it is not raining, then the ground is not wet.
Converse: If the ground is wet, then it is raining.
Contrapositive: If the ground is not wet, then it is not raining.

2. Statement: All dogs have four legs.
Inverse: If a creature does not have four legs, then it is not a dog.
Converse: If a creature has four legs, then it is a dog.
Contrapositive: If a creature is not a dog, then it does not have four legs.

 

[jfjd]

Make five solved problems each using
A.golden ratio
B.Translating English sentences into mathematical sentences
C.translating mathematical sentences into English sentences
D.compound statements into symbolic form
E.Symbolic form into compound statements
F.Construction of truth table
G.inverse converse and contrapositive

 

[🤖]

A. Golden Ratio:

1. A rectangle has a length that is 1.618 times its width. If the width of the rectangle is 10 cm, what is its length?
Solution: Let the length of the rectangle be x. Then, x/10 = 1.618. Solving for x, we get x = 16.18 cm. Therefore, the length of the rectangle is 16.18 cm.

2. A certain shell spirals outward at a ratio of approximately 1.618 to 1 for each quarter turn. If the shell makes 6 complete turns, how many total spirals are there?
Solution: The number of spirals in each turn is equal to the sum of the number of spirals in the previous two turns. Starting with 1 spiral in the first turn, we get the sequence: 1, 1, 2, 3, 5, 8. Therefore, there are 1+1+2+3+5+8 = 20 spirals in total.

3. The ratio of the length of a line segment to that of its longer part is the same as the ratio of the longer part to the shorter part. If the length of the line segment is 8, what is the length of the longer part?
Solution: Let the length of the longer part be x. Then, 8/x = x/(8-x). Solving for x, we get x = (1+sqrt(5))/2 * 8. Therefore, the length of the longer part is (1+sqrt(5))/2 * 8.

4. The height of a triangle is 1.618 times its base. If the base of the triangle is 10 cm, what is its height?
Solution: Let the height of the triangle be x. Then, x/10 = 1.618. Solving for x, we get x = 16.18 cm. Therefore, the height of the triangle is 16.18 cm.

5. The ratio of the diagonal of a regular pentagon to its side length is the golden ratio. If the side length of the pentagon is 5 cm, what is the length of its diagonal?
Solution: Let the length of the diagonal be x. Then, x/5 = 1.618. Solving for x, we get x = 8.09 cm. Therefore, the length of the diagonal is 8.09 cm.

B. Translating English sentences into mathematical sentences:

1. Twice a number minus four is equal to ten. Translate this into a mathematical sentence.
Solution: Let the number be x. The mathematical sentence is: 2x - 4 = 10.

2. The sum of two numbers is equal to seven times the difference between the numbers. Translate this into a mathematical sentence.
Solution: Let the two numbers be x and y. The mathematical sentence is: x + y = 7(x - y).

3. The product of a number and five is equal to the difference between ten and the number. Translate this into a mathematical sentence.
Solution: Let the number be x. The mathematical sentence is: 5x = 10 - x.

4. The square of a number is equal to the difference between four times the number and six. Translate this into a mathematical sentence.
Solution: Let the number be x. The mathematical sentence is: x^2 = 4x - 6.

5. The quotient of a number and three is equal to the sum of the number and two. Translate this into a mathematical sentence.
Solution: Let the number be x. The mathematical sentence is: x/3 = x + 2.

C. Translating mathematical sentences into English sentences:

1. 3x + 4y = 10. Translate this into an English sentence.
Solution: The sum of three times x and four times y is equal to ten.

2. 2(x - 5) = 8. Translate this into an English sentence.
Solution: Twice the difference between x and five is equal to eight.

3. 5a - 2b > 7. Translate this into an English sentence.
Solution: The difference between five times a and two times b is greater than seven.

4. 2x^2 - 5x + 3 = 0. Translate this into an English sentence.
Solution: The quadratic equation 2x^2 - 5x + 3 is equal to zero.

5. (a + b)^2 = a^2 + 2ab + b^2. Translate this into an English sentence.
Solution: The square of the sum of a and b is equal to the sum of the squares of a and b, plus two times a times b.

D. Compound statements into symbolic form:

1. If it rains, then I will stay indoors.
Solution: Let p be "it rains" and q be "I will stay indoors". The symbolic statement is: p → q.

2. The car will start if and only if the battery is charged.
Solution: Let p be "the car will start" and q be "the battery is charged". The symbolic statement is: p ↔ q.

3. I will go to the beach or the park.
Solution: Let p be "I will go to the beach" and q be "I will go to the park". The symbolic statement is: p ∨ q.

4. If I study hard, then I will pass the exam. If I don't study hard, then I will fail the exam.
Solution: Let p be "I study hard" and q be "I pass the exam". The symbolic statements are: p → q and ¬p → ¬q.

5. The number is even if it is divisible by 2.
Solution: Let p be "the number is even" and q be "the number is divisible by 2". The symbolic statement is: p ↔ q.

E. Symbolic form into compound statements:

1. p ∧ q
Solution: p and q are both true.

2. ¬p → q
Solution: If p is false, then q is true.

3. p ∨ ¬q
Solution: Either p is true or q is false.

4. p ↔ ¬q
Solution: p and q have opposite truth values.

5. (p ∧ q) → ¬r
Solution: If both p and q are true, then r is false.

F. Construction of truth table:

1. p ∧ q
Solution:

p | q | p ∧ q
--|---|------
T | T | T
T | F | F
F | T | F
F | F | F

2. p ∨ q
Solution:

p | q | p ∨ q
--|---|------
T | T | T
T | F | T
F | T | T
F | F | F

3. ¬p
Solution:

p | ¬p
--|---
T | F
F | T

4. p → q
Solution:

p | q | p → q
--|---|------
T | T | T
T | F | F
F | T | T
F | F | T

5. (p ∧ q) ↔ (p ∨ q)
Solution:

p | q | p ∧ q | p ∨ q | (p ∧ q) ↔ (p ∨ q)
--|---|------|-------|-------------------
T | T | T | T | T
T | F | F | T | F
F | T | F | T | F
F | F | F | F | T

G. Inverse, converse, and contrapositive:

1. If it rains, then the ground is wet.
Solution:

• Inverse: If it doesn't rain, then the ground is not wet.
• Converse: If the ground is wet, then it rains.
• Contrapositive: If the ground is not wet, then it doesn't rain.

2. All dogs have four legs.
Solution:

• Inverse: If something doesn't have four legs, then it is not a dog.
• Converse: If something is a dog, then it has four legs.
• Contrapositive: If something doesn't have four legs, then it is not a dog.

3. If a number is divisible by 3, then it is divisible by 9.
Solution:

• Inverse: If a number is not divisible by 3, then it is not divisible by 9.
• Converse: If a number is divisible by 9, then it is divisible by 3.
• Contrapositive: If a number is not divisible by 9, then it is not divisible by 3.

4. All rectangles have four sides.
Solution:

• Inverse: If something doesn't have four sides, then it is not a rectangle.
• Converse: If something is a rectangle, then it has four sides.
• Contrapositive: If something doesn't have four sides, then it is not a rectangle.

5. If a shape is a square, then it is a rectangle.
Solution:

• Inverse: If a shape is not a square, then it is not a rectangle.
• Converse: If a shape is a rectangle, then it is a square.
• Contrapositive: If a shape is not a rectangle, then it is not a square.

 

[jfjd]

Make five solved problems each using
A.golden ratio
B.Translating English sentences into mathematical sentences
C.translating mathematical sentences into English sentences
D.compound statements into symbolic form
E.Symbolic form into compound statements
F.Construction of truth table
G.inverse converse and contrapositive

 

[🤖]

A. Golden Ratio:

1. John is designing a logo and wants to use the golden ratio to determine the proportions of the design. If the width of the logo is 10 cm, what should be its height?

Solution: The golden ratio is approximately 1.618. Therefore, the height of the logo should be 16.18 cm (10 cm x 1.618).

2. A rectangle has a length of 20 cm and a width of x cm. If the ratio of the length to the width is equal to the golden ratio, what is the value of x?

Solution: The golden ratio is approximately 1.618. Therefore, we have:

20/x = 1.618

Solving for x, we get x = 12.36 cm.

3. The Fibonacci sequence is closely related to the golden ratio. If the first two terms of the Fibonacci sequence are 1 and 1, what is the eighth term?

Solution: The Fibonacci sequence is defined by adding the two previous terms to get the next term. Therefore, the eighth term is:

F8 = F6 + F7
F8 = 8 + 13
F8 = 21

4. The ratio of the height of a pentagon to its base is equal to the golden ratio. If the base is 10 cm, what is the height of the pentagon?

Solution: The golden ratio is approximately 1.618. Therefore, the height of the pentagon is 16.18 cm (10 cm x 1.618).

5. The golden ratio is often found in nature, such as in the spiral shape of shells. If a shell has a diameter of 20 cm, what is the length of the spiral from its center to its edge?

Solution: The golden ratio is approximately 1.618. Therefore, the length of the spiral is 32.36 cm (20 cm x 1.618).


B. Translating English sentences into mathematical sentences:

1. Three times a number plus five is equal to 20.

Solution: 3x + 5 = 20 (where x is the unknown number)

2. The sum of a number and its square is equal to 30.

Solution: x + x^2 = 30 (where x is the unknown number)

3. Sara has $50 and wants to buy some books that cost $8 each. If she can afford to buy n books, what is the mathematical sentence that represents this situation?

Solution: 8n ≤ 50 (where n is the maximum number of books that Sara can buy)

4. A rectangle has a length of x + 5 and a width of x - 3. What is the mathematical sentence that represents the area of the rectangle?

Solution: Area = (x + 5)(x - 3) or A = x^2 + 2x - 15

5. The sum of two numbers is 15 and their difference is 3. What is the mathematical sentence that represents this situation?

Solution: x + y = 15 and x - y = 3 (where x and y are the two unknown numbers)


C. Translating mathematical sentences into English sentences:

1. 5x - 3 = 22

Solution: Five times a number minus three is equal to twenty-two.

2. 2y + 7 = 15

Solution: Two times a number plus seven is equal to fifteen.

3. x^2 - 4x + 3 = 0

Solution: The square of a number minus four times the number plus three is equal to zero.

4. 3z - 5 > 10

Solution: Three times a number minus five is greater than ten.

5. 2a/3 + 5 = 8

Solution: Two-thirds of a number plus five is equal to eight.


D. Compound statements into symbolic form:

1. If it is sunny, then I will go for a walk.

Solution: Let p represent "It is sunny" and q represent "I will go for a walk". Therefore, the symbolic form is p → q.

2. I will study for the exam if and only if I have enough time.

Solution: Let p represent "I will study for the exam" and q represent "I have enough time". Therefore, the symbolic form is p ↔ q.

3. If I eat too much, then I will feel sick.

Solution: Let p represent "I eat too much" and q represent "I feel sick". Therefore, the symbolic form is p → q.

4. Either I will go to the party or I will stay home.

Solution: Let p represent "I will go to the party" and q represent "I will stay home". Therefore, the symbolic form is p ∨ q.

5. I will buy a new car only if I can afford it.

Solution: Let p represent "I will buy a new car" and q represent "I can afford it". Therefore, the symbolic form is q → p.


E. Symbolic form into compound statements:

1. p ∧ q

Solution: p and q are both true.

2. ¬p ∨ q

Solution: Either p is false or q is true.

3. p → q

Solution: If p is true, then q must also be true.

4. p ↔ q

Solution: p and q are either both true or both false.

5. ¬(p ∧ q)

Solution: Either p or q (or both) is false.


F. Construction of truth table:

1. p ∧ q

Solution:

p | q | p ∧ q
--|---|-----
T | T | T
T | F | F
F | T | F
F | F | F

2. ¬p ∨ q

Solution:

p | q | ¬p ∨ q
--|---|-------
T | T | T
T | F | F
F | T | T
F | F | T

3. p → q

Solution:

p | q | p → q
--|---|-----
T | T | T
T | F | F
F | T | T
F | F | T

4. p ↔ q

Solution:

p | q | p ↔ q
--|---|------
T | T | T
T | F | F
F | T | F
F | F | T

5. ¬(p ∧ q)

Solution:

p | q | ¬(p ∧ q)
--|---|---------
T | T | F
T | F | T
F | T | T
F | F | T


G. Inverse Converse and Contrapositive:

1. If it rains, the road will be wet.

Inverse: If it does not rain, the road will not be wet.
Converse: If the road is wet, it has rained.
Contrapositive: If the road is not wet, it has not rained.

2. If I study, then I will pass the exam.

Inverse: If I do not study, I will not pass the exam.
Converse: If I pass the exam, I must have studied.
Contrapositive: If I do not pass the exam, I must not have studied.

3. If the temperature is below freezing, the water will freeze.

Inverse: If the temperature is not below freezing, the water will not freeze.
Converse: If the water is frozen, the temperature must be below freezing.
Contrapositive: If the water is not frozen, the temperature must not be below freezing.

4. If I eat too much, I will feel sick.

Inverse: If I do not eat too much, I will not feel sick.
Converse: If I feel sick, I must have eaten too much.
Contrapositive: If I do not feel sick, I must not have eaten too much.

5. If the traffic is heavy, I will be late for work.

Inverse: If the traffic is not heavy, I will not be late for work.
Converse: If I am late for work, the traffic must have been heavy.
Contrapositive: If I am not late for work, the traffic must not have been heavy.

 

[jfjd]

Make five solved problems each using
E.Symbolic form into

compound statements F.Construction of truth table

G.inverse converse and

contrapositive

 

[🤖]

E. Symbolic Form:

1. Problem: Translate the following statement into symbolic form: "If it rains, then I will stay at home."

Solution: Let p represent the statement "It rains" and q represent the statement "I will stay at home". Then the symbolic form of the statement is p -> q.

2. Problem: Translate the following statement into symbolic form: "All dogs bark."

Solution: Let D(x) represent the statement "x is a dog" and B(x) represent the statement "x barks". Then the symbolic form of the statement is ∀x(D(x) -> B(x)).

3. Problem: Translate the following statement into symbolic form: "Either I will study for the exam or I will fail."

Solution: Let p represent the statement "I will study for the exam" and q represent the statement "I will fail". Then the symbolic form of the statement is p ∨ q.

4. Problem: Translate the following statement into symbolic form: "It is not true that John is tall and Mary is short."

Solution: Let T(x) represent the statement "x is tall" and S(x) represent the statement "x is short". Then the symbolic form of the statement is ¬(T(John) ∧ S(Mary)).

5. Problem: Translate the following statement into symbolic form: "If x is an even number, then x+1 is odd."

Solution: Let E(x) represent the statement "x is an even number" and O(x) represent the statement "x is an odd number". Then the symbolic form of the statement is ∀x(E(x) -> O(x+1)).

F. Construction of Truth Table:

1. Problem: Construct the truth table for the statement p -> q.

Solution:

| p | q | p -> q |
|---|---|-------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |

2. Problem: Construct the truth table for the statement ¬(p ∧ q).

Solution:

| p | q | ¬(p ∧ q) |
|---|---|---------|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |

3. Problem: Construct the truth table for the statement (p ∨ q) -> ¬p.

Solution:

| p | q | (p ∨ q) -> ¬p |
|---|---|--------------|
| T | T | F |
| T | F | F |
| F | T | T |
| F | F | T |

4. Problem: Construct the truth table for the statement (p ∧ q) ↔ (q ∧ p).

Solution:

| p | q | (p ∧ q) ↔ (q ∧ p) |
|---|---|-----------------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |

5. Problem: Construct the truth table for the statement (p -> q) ∧ (q -> r) -> (p -> r).

Solution:

| p | q | r | (p -> q) ∧ (q -> r) -> (p -> r) |
|---|---|---|--------------------------------|
| T | T | T | T |
| T | T | F | F |
| T | F | T | T |
| T | F | F | T |
| F | T | T | T |
| F | T | F | T |
| F | F | T | T |
| F | F | F | T |

G. Inverse, Converse, and Contrapositive:

1. Problem: Find the inverse, converse, and contrapositive of the statement "If it is raining, then the roads are wet."

Solution:

• Inverse: If it is not raining, then the roads are not wet.
• Converse: If the roads are wet, then it is raining.
• Contrapositive: If the roads are not wet, then it is not raining.

2. Problem: Find the inverse, converse, and contrapositive of the statement "All cats are mammals."

Solution:

• Inverse: If something is not a cat, then it is not a mammal.
• Converse: If something is a mammal, then it is a cat.
• Contrapositive: If something is not a mammal, then it is not a cat.

3. Problem: Find the inverse, converse, and contrapositive of the statement "If x > 5, then x + 3 > 8."

Solution:

• Inverse: If x ≤ 5, then x + 3 ≤ 8.
• Converse: If x + 3 > 8, then x > 5.
• Contrapositive: If x + 3 ≤ 8, then x ≤ 5.

4. Problem: Find the inverse, converse, and contrapositive of the statement "All students who pass the exam receive a certificate."

Solution:

• Inverse: If someone does not receive a certificate, then they did not pass the exam.
• Converse: If someone receives a certificate, then they passed the exam.
• Contrapositive: If someone does not pass the exam, then they do not receive a certificate.

5. Problem: Find the inverse, converse, and contrapositive of the statement "If it is sunny, then I will go to the beach."

Solution:

• Inverse: If it is not sunny, then I will not go to the beach.
• Converse: If I go to the beach, then it is sunny.
• Contrapositive: If I do not go to the beach, then it is not sunny.