Unit 2: Logic & Proof

Homework 2: Compound Statements

The relationship between 30 hours and 15 hours is that the number of hours spent using electronic devices is twice the number of hours of time spent playing sports, i.e. 30 hours = 2 x 15 hours.

There are different ways to approach this question, but one possible relationship between 30 hours and 15 hours to complete the statement "the number of hours students spent using electronic devices is times the number of hours spent playing sports" is:

If a student spends 30 hours using electronic devices and 15 hours playing sports, then the number of hours spent using electronic devices is twice the number of hours spent playing sports.

We can express this relationship using variables as follows:

Let E be the number of hours spent using electronic devices, and let S be the number of hours spent playing sports. Then, we can write:

E = 2S

If we substitute 30 for E and 15 for S in this equation, we get

30 = 2(15)

This equation is true, which means that the relationship holds for these particular values of E and S.

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The value of the 3rd term in the expansion of (x+6) ^6 will be as follows:

540x^4

What is expansion?

Expanding brackets, also known as multiplying out, seeks to eliminate the set of brackets by multiplying each phrase inside a bracket by the term on the outside and subsequently accumulating similar phrases. When solving equations, extending brackets, which is the opposite of factorization, is frequently an essential step.

Here in the question,

We have,

(x+6) ^6

Expanding it we get:

= x^6 + 36x^5 + 540x^4 +4320x³ + 19440x² + 46656x + 46656

So, the 3rd term of the expansion is 540x^4.

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The value of the 3rd term in the expansion of [tex](x+6) ^6[/tex] will be as follows: [tex]540x^4[/tex]. The correct answer is option (c).  [tex](6\ \ 4)36x^4[/tex]

By multiplying each phrase inside a bracket by the word on the outside and then accumulating similar phrases, expanding brackets, also known as multiplying out, aims to eliminate the set of brackets. Extending brackets, which is the opposite of factorization, is frequently a crucial stage in the solution of equations.

An affine transformation termed expansion, in which the scale is expanded, is also referred to as an enlargement or dilation. It is also sometimes referred to as an enlargement and is the polar opposite of a geometric constriction.

Here in the question,

We have,

[tex](x+6) ^6[/tex]

Expanding it we get:

[tex]= x^6 + 36x^5 + 540x^4 +4320x^3 + 19440x^2 + 46656x + 46656[/tex]

So, the 3rd term of the expansion is [tex]540x^4[/tex].

The correct answer is option (c). [tex](6\ \ 4)36x^4[/tex]

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The value of x in the given right triangle is 22.55 units.

Simply put, trigonometric functions—also referred to as circular functions—are the functions of a triangle's angle. This means that these trig functions provide the connection between the angles and sides of a triangle. The ratio of the lengths of the adjacent and opposing sides is known as the tangent function. It should be noted that the ratio of sine and cosine to the tan may also be used to express the tan.

For the given triangle the given sides are opposite and adjacent to the given angle.

The trigonometric function that relates the two sides are:

tan (64) = x/11

2.05(11) = x

x = 22.55

Hence, the value of x in the given right triangle is 22.55 units.

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Answer: 270 miles

Step-by-step explanation:

an hour contains 60 minutes and 3 hours contain 180 minutes so all you have to do is 180 divided by 30 which equals 6 and multiply 6 by 45 and that's your answer on how far she went.

Answer:

Isabella would have gone 270 miles in 180 minutes.

Step-by-step explanation:

60 times 3 = 180

There are 60 minutes in one hour, and there are three hours.

180 divided by 30 = 6

180 is divided by 30 because the rate of speed we know is 45 miles in 30 minutes.

45 times 6 = 270

There were 6 30s in 180, so 45 is multiplied by 6.

hope this helps

Answer:

It depends what x is equal to.

Step-by-step explanation:

For example, the expressions 2x+3 and 3x-6 are equal when x=9.

They can be expressed by 2x+3=3x-6

(2×9)+3=21

(3×9)-6=21

The speed of the current is approximately 39.05 miles per hour.

Distance equals rate times time in the equation for distance, rate, and time. This equation shows how far an item moves over a specific amount of time at a specific pace by relating the three variables in a linear equation. Any of the three variables can be solved for by rearranging the formula.

Let us suppose the speed of current = c.

Then the formula for distance is given as:

distance = rate x time

For upstream:

distance = 48 miles

rate = 20 - c miles per hour

time = distance / rate

= 48 / (20 - c) hours

For downstream:

distance = 48 miles

rate = 20 + c

time = distance / rate

= 48 / (20 + c) hours

The total time for the trip is 5 hours thus,

48 / (20 - c) + 48 / (20 + c) = 5

Taking the LCM:

48(20 + c) + 48(20 - c) = 5(20 - c)(20 + c)

1920 = 400 - c²

c² = 1520

c ≈ 39.05 miles per hour

Hence, the speed of the current is approximately 39.05 miles per hour.

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Answer: 72

Step-by-step explanation: Each angle in a pentagon is 108 degrees. Since there is a line making the angle supplementary just subtract 180 from 108 and the answer is 72.

The statement illustrates the transitive property of congruence, which is a fundamental concept in geometry.

This property states that if two geometric figures are congruent to a third figure, then they are congruent to each other.

In the given statement, ΔABC is congruent to ΔDEF, and ΔDEF is congruent to ΔXYZ. By the transitive property, we can conclude that ΔABC is also congruent to ΔXYZ.

This property is important because it allows us to establish relationships between geometric figures based on their congruence. It is used in many geometric proofs and applications, such as proving theorems, solving problems involving similar triangles, and determining the congruence of geometric shapes.

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The required value of the function at x=-2 is 10.

A function in mathematics from a set X to a set Y assigns precisely one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively. Initially, functions represented the idealized relationship between two variable quantities.

We have

f(x) = -x + 3 for x≤-3

     = -3x - 4  for -3 ≤ 1

     = -(x- 2)² + 5  for x > 1

To find F(-2) we have to take

f(x) = -3x + 4

f(-2) = -3(-2) + 4

f(-2) = 6 + 4

f(-2) = 10.

Thus, required value of the function is 10.

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Answer:

To use the FOIL method to simplify the expression (2x - 1/2)^2, follow these steps:

F: Multiply the first terms in each set of parentheses:

(2x) * (2x) = 4x^2

O: Multiply the outer terms in each set of parentheses:

(2x) * (-1/2) = -x

I: Multiply the inner terms in each set of parentheses:

(-1/2) * (2x) = -x

L: Multiply the last terms in each set of parentheses:

(-1/2) * (-1/2) = 1/4

Now, combine the like terms:

4x^2 - x - x + 1/4

Simplify by combining like terms:

4x^2 - 2x + 1/4

Therefore, (2x - 1/2)^2 = 4x^2 - 2x + 1/4.

The measure of angle b is which is a supplement of angle a is 162 degrees.

If angles a and b are supplementary, that means they add up to 180 degrees.

Given that;

Since angle a and angle b are supplementary, So, we can set up the equation:

a + b = 180

We know that angle a measures 18 degrees, so we can substitute this value into the equation:

18 + b = 180

Solving for b, we can subtract 18 from both sides:

b = 180 - 18

b = 162 degrees

Therefore, angle b measure 162 degrees.

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The length of the rectangle is 15 dm and the width is 13 dm.

Let's assume that the length of the rectangle is "L" and the width is "W".

From the problem statement, we have two pieces of information:

The area of the rectangle is 195 dm²:

Area = Length x Width

195 dm² = L x W

The width is two less than the length:

W = L - 2

Now, we can substitute the second equation into the first equation to eliminate W and get an equation with only one variable:

195 dm² = L x (L - 2)

Simplifying the equation:

195 dm² = L² - 2L

L² - 2L - 195 dm² = 0

To solve for L, we can use the quadratic formula:

L = (-b ± √(b² - 4ac)) / 2a

Where a = 1, b = -2, and c = -195.

L = (2 ± √(2² + 4 x 1 x 195)) / 2 x 1

L = (2 ± √4 + 780) / 2

L = (2 ± √784) / 2

L = (2 ± 28) / 2

L = 15 or L = -13

Since the length can't be negative, the length of the rectangle is L = 15 dm.

Now we can use the equation W = L - 2 to find the width:

W = 15 dm - 2 dm

W = 13 dm

Therefore, the length of the rectangle is 15 dm and the width is 13 dm.

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Angle B is 50 degrees in measurement.

One method to find the measure of angle b is to use the properties of angles in a triangle. We know that the sum of the angles in a triangle is 180 degrees. We also know that angles a and c have measures of 50 degrees and 80 degrees respectively. Therefore, we can find the measure of angle b by subtracting the sum of angles a and c from 180 degrees:

angle b = 180 degrees - angle a - angle c

angle b = 180 degrees - 50 degrees - 80 degrees

angle b = 50 degrees

Therefore, angle b has a measure of 50 degrees. Another method to find the measure of angle b is to use trigonometry, such as the sine or cosine rule, depending on the given information.

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What is one method to find the measure of angle B?

A. use the Pythagorean theorem to find BC, then solve the equation tan(B)=8/BC

B. because of the 30-60-90 triangle theorem, you know the measure of angle B is 60

C. solve the equation cos(B)=8/89(square rooted)

The probability of ruin at or before time 3 with initial surplus 3 is 0.6915

The probability of ruin at or before time 3 with initial surplus 3, given the claim severity per period follows a binomial normal distribution with mean 4 and standard deviation 0.2, is calculated as follows:

Determine the z-score from the normal distribution corresponding to a surplus of 3 and a mean of 4.

z-score = (3-4)/0.2 = -0.5

Hence, the z-score result is -0.5

The next step is to use the cumulative probability density function to calculate the probability of ruin.

Probability of ruin = 1 - CDF(-0.5) = 1 - 0.3085 = 0.6915

Therefore, the probability of ruin at or before time 3 with initial surplus 3 is 0.6915.

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Answer:

Y=3

Step-by-step explanation:

Put in x=-1

y-3 = 2(-1+1)

y-3 = 2(0)

y-3=0

add 3 more to both sides

y-3+3 = 0+3

y =3

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Answer:

[tex] \sf \: y = 3[/tex]

Step-by-step explanation:

Now we have to,

→ Find the required value of y.

We have to use,

→ x = -1

The equation is,

→ y - 3 = 2(x + 1)

Then the value of y will be,

→ y - 3 = 2(x + 1)

→ y = 2(x + 1) + 3

→ y = 2((-1) + 1) + 3

→ y = 2(0) + 3

→ y = 0 + 3

→ [ y = 3 ]

Hence, the value of y is 3.

Answer:

Translation 7 units to the right

Step-by-step explanation:

Pick 2 points to compare

A (-4,3)  to  E (3,3)

We see the x increase by 7, so the map translation 7 units to the right.

So, Translate Figure ABCD 7 units right to form figure EFGH.

Answer:

4

Step-by-step explanation:

V = πr²h

h = V/πr² = (314) / π(5)² ≈ 4

a) cos  φ= -4/13`

b) cos φ  -2√13/13`c)

c) inverse tangent function will not give us the angle we are looking for because our angle is in the third quadrant

d) tanφ=32​ => φ ≈ -57.99°

e) coterminal angles to -57.99° in radians is:`(-319.93 + 360n)π/180`, where `n` is an integer.

a) The formula for the tangent of an angle in the third quadrant is, `tan(π + φ) = tan φ` and, hence, we have:`tan(π + φ) = 3/2`Using the fundamental identity for the tangent, we get:`tan(π + φ) = -tan φ``tan φ = -3/2`Then, using the Pythagorean identity `sin^2 φ + cos^2 φ = 1` to solve for `cos φ` in the third quadrant where `cos φ < 0`, we get:`cos φ = -√(1 - sin^2 φ) = -√(1 - (tan^2 φ)/(1 + tan^2 φ)) = -√(1 - (9/13)) = -4/13`b) Since `tan φ = 3/2`, we can construct a right triangle with legs of length `3` and `2` and hypotenuse of length `√(3^2 + 2^2) = √13`.Since the angle is in the third quadrant, the cosine of the angle is negative. Thus:`cos φ = -2/√13 = (-2/√13) * (√13/√13) = -2√13/13`c) The inverse tangent function is only able to give you the value of the angle in the first or fourth quadrant. Therefore, using the inverse tangent function will not give us the angle we are looking for because our angle is in the third quadrant.d) `tanφ=32​ => φ ≈ -57.99°`e) All coterminal angles to -57.99° in radians are given by:`θ = -57.99° + 360n, n ∈ ℤ`Thus, we can convert to radians using the formula `π/180°`:`θ = (-57.99° + 360n)π/180°`Simplifying:`θ = (-319.93 + 360n)π/180`Therefore, the expression for all coterminal angles to -57.99° in radians is:`(-319.93 + 360n)π/180`, where `n` is an integer.

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Step-by-step explanation:

We can start by assuming that the price of the meal is x dollars. Then, we know that:

15% of x is equal to $4.89

18% of x is equal to $5.87

20% of x is equal to $6.52

We can set up three equations using these statements:

0.15x = 4.89

0.18x = 5.87

0.20x = 6.52

Solving for x in each equation, we get:

x = 4.89 / 0.15 = 32.60

x = 5.87 / 0.18 = 32.61

x = 6.52 / 0.20 = 32.60

Since all three equations give us a value of x that is very close to 32.60, we can assume that the price of the meal was $32.60, rounded to the nearest cent.

The difference quotient of the function[tex]\(f(x)=x^{2}-1\) is \(2x+h\)[/tex], where [tex]\(h\)[/tex] is the small change in [tex]\(x\)[/tex]

The difference quotient of the function [tex]\(f(x)=x^{2}-1\)[/tex] can be found by using the following formula: [tex]\[\frac{f(x+h)-f(x)}{h}, h\neq0\][/tex]. We can start by substituting the given function into the formula and simplify the expression as follows:[tex]\[\frac{(x+h)^{2}-1-(x^{2}-1)}{h}\],[/tex]

First, let's expand the expression by using the formula for the square of a binomial:[tex][(x+h)^{2}=x^{2}+2hx+h^{2}\][/tex],

Substituting this into the expression above, we get: [tex][\frac{x^{2}+2hx+h^{2}-1-x^{2}+1}{h}\][/tex], Simplifying the expression, we can cancel out the [tex]\(x^{2}\)[/tex] terms, and the [tex](1\)s:\[\frac{2hx+h^{2}}{h}\][/tex]

Next, we can factor out the \(h\) from the numerator: [tex]\[h\cdot\frac{2x+h}{h}\][/tex].

Cancelling out the [tex]\(h\)s[/tex], we get:[tex]\[2x+h\][/tex] ,Therefore, the difference quotient of the function [tex]\(f(x)=x^{2}-1\) is \(2x+h\)[/tex], where [tex]\(h\)[/tex] is the small change in [tex]\(x\)[/tex].

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The most likely data sets to have a negative association and a correlation coefficient between 0 and -1 are: number of minutes spent exercising; number of calories burned, which is option C.

The correlation coefficient, a numerical measure of the strength and direction of the relationship between two variables, is used to describe the association between two data sets. It ranges from -1 to +1, where a correlation coefficient of -1 indicates a negative correlation and +1 indicates a positive correlation.

Number of minutes spent exercising and the number of calories burned while exercising have a negative association. That is, as the number of minutes spent exercising increases, the number of calories burned decreases.

Therefore, the most likely data sets to have a negative association and a correlation coefficient between 0 and -1 are "number of minutes spent exercising; number of calories burned."

Hence, the correct answer is C.

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Answer: Average Annual Temperature in the US; Annual sweater sales by an American retailer

Step-by-step explanation:

Therefore , the solution of the given problem of function comes out to be the range of r(t) is [22 - 483, 22 + 483].

The midterm test questions will cover all of the topics, including fictitious and real places as well as mathematical variable design. a diagram showing the relationships between different elements that cooperate to create the same result. A service is composed of numerous distinctive components that cooperate to create distinctive results for each input. Every mailbox has a particular area that might be used as a haven.

Here,

For all real values of t such that the expression inside the cube root is non-negative, the function r(t) = (t2 - 44t + 1) is specified.

Therefore, in order to determine the scope of r, we must resolve the inequality t2 - 44t + 1  0.(t).

The quadratic method can be used to eliminate this inequality:

=> t = [44 ± √(44² - 4(1)(1))]/(2(1))

=> t = [44 ± √(1936 - 4)]/2

=> t = [44 ± √1932]/2

=> t = [44 ± 2√483]/2

=> t = 22 ± √483

Consequently, the range of r(t) is [22 - 483, 22 + 483].

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The coordinates of B of the diameter of the circle is: (5, -1).

Since AB is a diameter of the circle, its midpoint will be the center of the circle, which is given to be (1, 1). Therefore, we can find the coordinates of point B by using the midpoint formula.

Midpoint formula:

The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is ((x1+x2)/2, (y1+y2)/2).

Let A be (-3, 3), which is one endpoint of the diameter AB. Let B be the other endpoint of the diameter AB.

Since the midpoint of AB is (1, 1), we have:

((x-coordinate of A + x-coordinate of B)/2, (y-coordinate of A + y-coordinate of B)/2) = (1, 1)

Substituting the coordinates of point A, we get:

((-3 + x-coordinate of B)/2, (3 + y-coordinate of B)/2) = (1, 1)

Multiplying both sides of each equation by 2, we get:

(-3 + x-coordinate of B, 3 + y-coordinate of B) = (2, 2)

Adding 3 to both sides of the first equation and subtracting 3 from both sides of the second equation, we get:

(x-coordinate of B, y-coordinate of B) = (2+3, 2-3) = (5, -1)

Therefore, the coordinates of point B are (5, -1).

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Half of the intercepted arc is equals to the inscribed angle. Therefore, the  measure of the arc is 170 degrees.

The arc of a circle is said to be the part or segment of the circumference of a circle.

The degree of an arc is equals to the measure of the central angle that creates the arc.

Therefore, half of the intercepted arc is equals to the inscribed angle. In other words, the inscribed angle theorem states that the angle inscribed inside a circle is always half the measure of the central angle.

Hence,

∠JKL = 1 / 2 arc angle

Therefore,

85 = 1 / 2 x

cross multiply

x = 85(2)

x = 170 degrees

Therefore,

arc angle = 170 degrees

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Answer:

a ≥ 18

p ≤ 275

Step-by-step explanation:

a. The inequality for the minimum age for voting in the United States is:

a ≥ 18

This inequality states that a person's age (represented by 'a') must be greater than or equal to 18 years in order to be eligible to vote in the United States.

b. The inequality for the maximum number of people that can attend a performance in the theater is:

p ≤ 275

This inequality states that the number of people (represented by 'p') attending a performance in the theater must be less than or equal to 275 in order to accommodate all attendees within the seating capacity of the theater.

The logarithm that is equal to the number of years it will take for the population to increase to 15,000 students is log(11.08).

Let t be the number of years it will take for the enrollment to increase to 15,000 students. We can use the formula for continuous growth to set up an equation:

[tex]A = Pe^{(rt)[/tex]

where A is the final amount, P is the initial amount, r is the annual growth rate as a decimal, and t is the time in years.

In this case, we know that P = 12,000, A = 15,000, and r = 0.025 (since the growth rate is 2.5%). Plugging these values into the equation, we get:

[tex]15,000 = 12,000 e^{(0.025t)[/tex]

Dividing both sides by 12,000, we get:

[tex]1.25 = e^{(0.025t)[/tex]

To solve for t, we can take the natural logarithm of both sides:

[tex]ln(1.25) = ln(e^{(0.025t))[/tex]

Using the property of logarithms that [tex]ln(e^x) = x[/tex], we can simplify the right-hand side:

ln(1.25) = 0.025t

Finally, dividing bοth sides by 0.025, we get:

t = ln(1.25)/0.025

Using a calculatοr tο evaluate ln(1.25)/0.025, we get:

t ≈ 11.08

Therefοre, the lοgarithm that is equal tο the number οf years it will take fοr the pοpulatiοn tο increase tο 15,000 students is lοg(11.08)

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The distance between Mrs. Meyer and Dalton would be = 8.5m

The shape that is being formed between the three individuals is the shape of a triangle.

Distance can be defined as the length that is covered by a moving object.

The distance between Mrs Meyer and Leslie =a= 8m(opposite)

The distance between Dalton and Leslie =b = 3 m (adjacent)

Therefore, the hypotenuse = ?

Using the Pythagorean theorem;

c² = a² + b²

C ² = 8²+3²

C = 64 + 9

c² = 73

C = √ 73

C = 8.5m

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Step-by-step explanation:

remember the trigonometric triangle in a circle ?

sine is the up/down leg, cosine is the left/right leg.

all we need to consider in a circle with a radius <> 1, that we need to multiply the trigonometric functions by the radius to get the actual side lengths.

the radius is the Hypotenuse (the side opposite of the 90° angle).

y = cos(30)×8 = sqrt(3)/2 × 8 = 4×sqrt(3)

the number in the green box is therefore 4.