Equations & Inequalities The value of a baseball card is increasing at a rate of $40 per year. If Tom purchased the card for $300, how many years will it take for the card to at least triple in value?

Answer:

[tex]2 \tan (6x)+2 \sec (6x)+\text{C}[/tex]

Step-by-step explanation:

Fundamental Theorem of Calculus

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

[tex]\displaystyle \int \dfrac{12}{1-\sin (6x)}\:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int a\:\text{f}(x)\:\text{d}x=a \int \text{f}(x) \:\text{d}x$\end{minipage}}[/tex]

If the terms are multiplied by constants, take them outside the integral:

[tex]\implies 12\displaystyle \int \dfrac{1}{1-\sin (6x)}\:\:\text{d}x[/tex]

Multiply by the conjugate of 1 - sin(6x) :

[tex]\implies 12\displaystyle \int \dfrac{1}{1-\sin (6x)} \cdot \dfrac{1+\sin(6x)}{1+\sin(6x)}\:\:\text{d}x[/tex]

[tex]\implies 12\displaystyle \int \dfrac{1+\sin(6x)}{1-\sin^2(6x)} \:\:\text{d}x[/tex]

[tex]\textsf{Use the identity} \quad \sin^2 x+ \cos^2 x=1:[/tex]

[tex]\implies \sin^2 (6x) + \cos^2 (6x)=1[/tex]

[tex]\implies \cos^2 (6x)=1- \sin^2 (6x)[/tex]

[tex]\implies 12\displaystyle \int \dfrac{1+\sin(6x)}{\cos^2(6x)} \:\:\text{d}x[/tex]

Expand:

[tex]\implies 12\displaystyle \int \dfrac{1}{\cos^2(6x)}+\dfrac{\sin(6x)}{\cos^2(6x)} \:\:\text{d}x[/tex]

[tex]\textsf{Use the identities }\:\: \sec \theta=\dfrac{1}{\cos \theta} \textsf{ and } \tan\theta=\dfrac{\sin \theta}{\cos \theta}:[/tex]

[tex]\implies 12\displaystyle \int \sec^2(6x)+\dfrac{\tan(6x)}{\cos(6x)} \:\:\text{d}x[/tex]

[tex]\implies 12\displaystyle \int \sec^2(6x)+\tan(6x)\sec(6x) \:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating $\sec^2 kx$}\\\\$\displaystyle \int \sec^2 kx\:\text{d}x=\dfrac{1}{k} \tan kx\:\:(+\text{C})$\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{6 cm}\underline{Integrating $ \sec kx \tan kx$}\\\\$\displaystyle \int \sec kx \tan kx\:\text{d}x= \dfrac{1}{k}\sec kx\:\:(+\text{C})$\end{minipage}}[/tex]

[tex]\implies 12 \left[\dfrac{1}{6} \tan (6x)+\dfrac{1}{6} \sec (6x) \right]+\text{C}[/tex]

Simplify:

[tex]\implies \dfrac{12}{6} \tan (6x)+\dfrac{12}{6} \sec (6x)+\text{C}[/tex]

[tex]\implies 2 \tan (6x)+2 \sec (6x)+\text{C}[/tex]

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Substitute [tex]y=6x[/tex] and [tex]dy=6\,dx[/tex] to transform the integral to

[tex]\displaystyle \int \frac{12}{1-\sin(6x)} \, dx = 2 \int \frac{dy}{1 - \sin(y)}[/tex]

Now substitute [tex]t=\tan\left(\frac y2\right)[/tex] and [tex]dt=\frac12 \sec^2\left(\frac y2\right) \, dy[/tex] to transform this to

[tex]\displaystyle 2 \int \frac{dy}{1 - \sin(y)} = 2 \int \frac1{1-\frac{2t}{1+t^2}}\cdot\frac{2\,dt}{1+t^2} = 4 \int \frac{dt}{(t-1)^2}[/tex]

Finally, substitute [tex]s = t-1[/tex] and [tex]ds=dt[/tex] to get

[tex]\displaystyle 4 \int \frac{dt}{(t-1)^2} = 4 \int \frac{ds}{s^2} = -\dfrac4s + C[/tex]

Now recover the antiderivative in terms of [tex]x[/tex].

[tex]\displaystyle \int \frac{12}{1-\sin(6x)} \, dx = -\frac4s + C \\\\ ~~~~~~~~ = -\frac4{t-1} + C \\\\ ~~~~~~~~ = -\frac4{\tan\left(\frac y2\right) - 1} + C \\\\ ~~~~~~~~ = \boxed{-\frac4{\tan(3x) - 1} + C}[/tex]

Mathematical logic is important as it's a way to learn new experience through continuous self assessment.

Logic is important as it enables us to form sound judgements and beliefs.

The study of logic can help as it can help us to understand disagreement and ambiguity.

It also helps us in making a reasonable emotional life.

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The angle m∠MLP in the rhombus is 56 degrees.

A rhombus has all its sides equal to each other.  The diagonals of a rhombus are angle bisectors.

Therefore,

m∠MLP = m∠MLO / 2

m∠MLO = 112°

m∠MLP = 112 / 2

Therefore,

m∠MLP = 56 degrees.

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Answer:

B. 56

Step-by-step explanation:

trust

Answer:

  7

Step-by-step explanation:

A 2-way table can be useful for recording the given information and for finding the missing numbers.

The attached table shows the numbers of families booking in July and August, and also counted by destination. Totals are at the right and bottom, and the grand total is the number 100 at the lower right.

Underlined numbers are those in the problem statement. The remaining numbers are computed so as to make the totals be correct.

Overall, 30+35 = 65 went to Portugal or France, so 100-65 = 35 went to Spain. Of those, 20 booked in August, so 15 booked to Spain in July.

59 booked in August, so a total of 41 booked in July.

Now we know 19 booked for France and 15 booked for Spain in July, so the remaining 7 booked for Portugal in July.

Answer:

50 minutes

Step-by-step explanation:

3:20 pm - 4:50 = 90 minutes

90 = 2a + 15 + a

90-15= 75

75= 2a + a

75 = 3a

75/3 = 25

25 x 2 = 50 minutes

The probability of picking a card with an even number is [tex]\frac{3}{10}[/tex].

To find the probability of picking a card with an even number:

Therefore, the probability of picking a card with an even number is [tex]\frac{3}{10}[/tex].

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COMPLETE QUESTION:

The numbers 7, 11, 12, 13, 14, 18, 21, 23, 27, and 29 are written on separate cards, and the cards are placed on a table with the numbers facing down. The probability of picking a card with an even number is _____.

The portion of the mixture that is fruit to the nearest hundredth is 10.00 ounces

12 ounce juice:

= 70/100 × 12

= 0.7 × 12

= 8.4 ounces

Water = 30%

16 ounces;

= 10/100 × 16

= 0.1 × 16

= 1.6 ounces

Portion of mixture that is fruits = 8.4 ounces + 1.6 ounces

= 10.00 ounces

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Answer:53 mph

Step-by-step explanation:106 miles per hour would get him home in 1 hour but it takes 2 hours so you can divide 106 by 2 to 53 miles per hour.

Answer:

Step-by-step explanation:

Joe needs to get to his house, which is 106 miles away in 2 hours. How fast does he need to drive?

106 miles is 2 hours = 53 miles in 1 hour (106 : 2)

so your answer is 53 mph

The probability that emission of both CO and hydrocarbon is in excess is 0.8.

The proportion of favorable cases to all possible cases used to determine how likely an event is to occur.

Let, A be the event in which cars are emitting excess CO:

P(A) = 0.12

Let, B be the event in which cars are emitting excess hydrocarbons:

P(B) = 0.15

The cars which emit both CO and Hydrocarbons in:

P(A∩B) = 0.8

The probability that emission of both CO and hydrocarbon is in excess is 0.8.

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Answer:

Θ ≈ 37° , x ≈ 13 cm

Step-by-step explanation:

(a)

using the cosine ratio in the right triangle

cosΘ = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{BC}{AC}[/tex] = [tex]\frac{8}{10}[/tex] , then

Θ = [tex]cos^{-1}[/tex] ([tex]\frac{8}{10}[/tex] ) ≈ 37° ( to the nearest whole number )

(b)

using the sine ratio in the right triangle and the exact value

sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , then

sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{x}{15}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )

2x = 15[tex]\sqrt{3}[/tex] ( divide both sides by 2 )

x = 7.5[tex]\sqrt{3}[/tex] ≈ 13 cm ( to the nearest whole number )

Answer:

54 degrees

Step-by-step explanation:

The actual angles of the triangle are not changing since this is just a reflection. The angles inside a triangle all add up to 180, so we can do 180-59-67 to get 54.

The expression that is not equivalent to (1 - sin²(x)) tan(-x) is D. (cos²x - 1)(cot -x).

It should be noted that (cos²x - 1)(cot -x). is not equivalent to the given expression.

This is illustrated as:

(cos²x - 1)(cot -x)

= (-sin²x) × (-cot x)

= sin²x × cosx/sin x = sinxcosx

In conclusion, the correct option is D.

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The expression _______ is not equivalent to (1 − sin2(x)) tan(-x).

a. (1 - cos^2(x)) cot(-x)

b. (cos^2(x) - 1) cot(x)

c. (sin^(x) - 1) tan(x)

d. (cos^2(x) - 1) cot(-x)

Answer:

See attached image

Step-by-step explanation:

Hello,

2x² - 8x = 0

2x × x - 2x × 4 = 0

2x(x - 4) = 0

2x = 0 or x - 4 = 0

x = 0 or x = 4

The expected value for the given Five possibilities is $6.

We have,

Payoffs of $2, $4, $6, $8, and $10.

Now,

We know that,

The expected value  [tex]=\Sigma (x*P(x))[/tex]

i.e. The sum of product of possible outcome and each outcome.

Here, x = Each outcome

And

P(x) = Possible outcomes

So,

Probability of x (Px) [tex]=\frac{1}{5}[/tex],

Now,

According to the above mentioned formula,

i.e.

The expected value  [tex]=\Sigma (x*P(x))[/tex]

We get,

[tex]=\Sigma\ (\frac{1}{5} * 2) + (\frac{1}{5} * 4) + (\frac{1}{5} * 6) +(\frac{1}{5} * 8) +(\frac{1}{5} * 10)[/tex]

On solving we get,

[tex]=\Sigma\ (0.4 + 0.8 + 1.2 + 1.6 + 2)[/tex]

i.e.

The expected value = $6

So,

The expected value for given possibilities is $6.

Hence we can say that the expected value for the given Five possibilities is $6.

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The equation of directrix is D. y=± 432/13.

Definition of hyperbola -

1. The vertex i at (0, 36) and a focus at (0, 39), then you have:

  a=36

a²=1296

2. The equation of directrix is:

y =  a² =c

c  =  39

y =  1296/39

y  =  432/13

Therefore, the answer is the option D, which is: D. y=± 432/13

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The complete question is -

A hyperbola centered at the origin has a vertex at (0, 36) and a focus at (0, 39)

Which are the equations of the directrices ?

A. x= ±12/13

B. y=±12/13

C. x=±432/13

D. y=± 432/13

Answer: Correct

Step-by-step explanation:

Answer:

2061.9[tex]mm^{3}[/tex]

Step-by-step explanation:

We can see the original shape of the gold block is a cuboid.

Volume of Cuboid = Length x Width x Height

Volume of Original Gold Block = 27mm x 8mm x 12mm = [tex]2592mm^{3}[/tex]

Next, we can see the shape of the cut-out hole is a cylinder.

Volume of Cylinder = [tex]\pi r^{2} h[/tex]

We know that r = 0.5 x diameter = 0.5 x 5 = 2.5mm

Volume of cut-out hole = [tex]\pi (2.5)^{2} (27)\\[/tex] = [tex]168.75\pi mm^{3}[/tex]

Volume of Remaining Gold Block = Volume of Original Gold block - Volume of cut-out hole

= [tex]2592mm^{3} -168.75\pi mm^{3}[/tex]

= 2061.9[tex]mm^{3}[/tex]

The discriminant of the quadratic equation is -20 and there are two non real solutions.

The discriminant of a quadratic equation uses the equation:

[tex]b^{2} -4ac[/tex]

Where the value of this calculation can tell you what solutions there are, plug known values in:

[tex]x^{2} +6x+14[/tex]

a=1

b=6

c=14

[tex]b^{2} -4ac[/tex]

which is equal to -20 on putting the values of a,b and c in the equation

As -20 < 0, this means that there is not a real solution, resulting in the first option being correct.

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The function that describes the fund that was raised is 0.05³ - 30x - 12425

The revenue is  R (z) = 0.05x³ + 75

the cost = C(z) = 30x + 12,500.

Please note that the equation for revenue missed a sign in the question so I made use of the plus sign

We would have revenue - cost

Hence ( 0.05x³ + 75) - (30x + 12,500)

We would open the bracket

0.05x³ + 75 -30x -12500

We would take the like term to

0.05³ - 30x -12500 + 75

0.05³ - 30x - 12425

The function that describes the fund that was raised is 0.05³ - 30x - 12425

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Answer: F(x)=0.05x^3-30x-12,575

Step-by-step explanation:

Correct on test.

The circumference of the circle and round to the nearest tenth is 25.1 yds

The formula for calculating the circumference of a circle is expressed as:

C = 2πr

Given the following

radius r = 4yds

Substitute

C = 2(3.14)(4)

C = 25.12 yds

Hence the circumference of the circle and round to the nearest tenth is 25.1 yds

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Answer:

180 - 92 - 41 = 47

There are not two similar congruent angle in the triangles.

Hope this helps!

Answer:

96 is the number of original women investors.

Step-by-step explanation:

Let X be the number of young women in the initial group.  They raised 480000, so the average payment per person was (480000/X).  

When 4 pull out, the new average is (480000/(X-4)).  We are told that this new average required the remaining women (X-4) to add another 20000.

The four women therefore had contributed 20000 in total, making their average 20000/4 = 5000 each.  

This would have been the same amount contributed by all X women.  Thus, we can set the average (480000/X) equal to 5000

(480000/X) = 5000

(480000) = 5000X

(480000)/5000 = X

X = 96

The original number of women was 96.

===

Check

(96 Women)(5000/Woman) = 480000  CHECKS

(4 Women pull out)*(5000) = 20000 that needs to be added to stay at 480000.  CHECKS

Answer:

37/70

Step-by-step explanation:

5/8 + 3/10 = (5 × 5)/ (8 × 5) + (3 × 4)/ (10 × 4) =

25/40 + 12/40 = 37/40

Answer:

Step-by-step explanation:

g(x) = [tan( 4x-1)]2

The table that shows a function that is decreasing only over the interval (–1, ∞) is:

A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 1, negative 3, negative 5, negative 2, negative 1, 2.

A function is decreasing if when we increase the value of x, we decrease the value of y, and vice versa.

Decreasing on the (–1, ∞) means that as x increases on the interval, i.e. x = -100, then x = -10, then x = -2, the value of y decreases. The function that follows this pattern only on this interval is:

A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 1, negative 3, negative 5, negative 2, negative 1, 2.

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Answer:

A

Step-by-step explanation:

Taking into account the change of units, the total distance that he biked in his first month of training is 55.8 miles.

In first place, the rule of three is a way of solving problems of proportionality between three known values and an unknown value, establishing a relationship of proportionality between all of them.

That is, what is intended with it is to find the fourth term of a proportion knowing the other three.  

If the relationship between the magnitudes is direct, that is, when one magnitude increases, so does the other (or when one magnitude decreases, so does the other) , the direct rule of three must be applied.

To solve a direct rule of three, the following formula must be followed, being a, b and c known data and x the variable to be calculated:

a ⇒ b

c ⇒ x

So: [tex]x=\frac{cxb}{a}[/tex]

In his first month of training,Landon biked 3.1 miles and ran 0.75 miles each workout. He completed 18 workouts that month.

So you can apply the following rule of three: if for each workout Landon biked 3.1 miles, in 18 workouts he biked how far?

1 workout ⇒ 3.1 miles

18 workouts ⇒ total distance

So: [tex]total distance=\frac{18 workoutsx3.1 miles}{1 workout}[/tex]

Solving:

total distance= 55.8 miles

In summary, the total distance that he biked in his first month of training is 55.8 miles.

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The correct option for the matrix will be:

a) If an n x n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.

It could have repeated eigenvalues as long as the basis of each eigenspace is equal to the multiplicity of that eigenvalue.

b) If A is diagonalizable the A2 is diagonalizable

If A is diagonalizable then there exists an invertible matrix

c) If Rn has a basis of eigenvectors of A, then A is diagonalizable.

d) A is diagonalizable if and only if A has n eigenvalues, counting multiplicity.

e) If A is diagonalizable, then A is invertible.

It’s invertible if it doesn’t have a zero as eigenvalue but this doesn’t affect diagonalizable.

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The solution to the inequality is (–2.5, ∞)

The inequality is given as:

5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)

Open the brackets

15.3 + 11.22x > –14.25 – 10.2x - 24

Collect the like terms

11.22x + 10.2x > -15.3 - 14.25- 24

Evaluate the like terms

21.42x > -53.55

Divide both sides by 21.42

x > -2.5

This can also be represented as (–2.5, ∞)

Hence, the solution to the inequality is (–2.5, ∞)

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Answer:

c

Step-by-step explanation:

on edge

Answer: D. 2x + 4x = 150

Step-by-step explanation:

We know that the section of 2x and 4x together equal 150 degrees because they are vertical angles.  The 150 degrees is vertical to the section with 2x and 4x.  So you would use equation D to solve for x.

In order to create the reverse of the 150 angle, when two angles meet, 2x and 4x are combined.

Thus, the angles formed by 2x, 4x, and 150 are vertical. We know that vertical angles are equivalent, .Therefore, the labeled angles are equivalent.

Answer :-

[tex]\bf D. \: 2 {x}^{o} + 4 {x}^{o} = 15 {0}^{o} [/tex]