Rewrite the logarithmic expression as a single logarithm with the same base. Assume all expressions exist and are well-defined. Simplify any fractions. 9log_(4)2-log_(4)40

Answer:

See below for proof.

Step-by-step explanation:

Given equations for x and y:

  [tex]x=v_0 \cdot t \cdot \cos \theta[/tex]

  [tex]y=-16 \cdot t^2 + v_0 \cdot t \cdot \sin \theta[/tex]

Rearrange the equation for x to isolate t by dividing both sides of the equation by v₀ · cos θ:

[tex]\implies t=\dfrac{x}{v_0 \cdot \cos \theta}[/tex]

Square the equation for x:

[tex]\implies x^2=(v_0 \cdot t \cdot \cos \theta)^2[/tex]

[tex]\implies x^2={v_0}^2 \cdot t^2 \cdot \cos^2 \theta[/tex]

Rearrange to isolate t² by dividing both sides of the equation by v₀² · cos²θ:

[tex]\implies t^2= \dfrac{x^2}{{v_0}^2 \cdot \cos^2 \theta}[/tex]

Now we have created expressions for t and t² in terms of x.

Substitute these into the equation for y:

[tex]\implies y=-16 \cdot t^2 + v_0 \cdot t \cdot \sin \theta[/tex]

[tex]\implies y =-16 \cdot \left(\dfrac{x^2}{{v_0}^2 \cdot \cos^2 \theta}\right)+ v_0 \cdot \left(\dfrac{x}{v_0 \cdot \cos \theta}\right) \cdot \sin \theta[/tex]

Simplify:

[tex]\implies y =-\dfrac{16}{{v_0}^2} \cdot \left(\dfrac{x^2}{\cos^2 \theta}\right)+\left(\dfrac{x}{\cos \theta}\right) \cdot \sin \theta[/tex]

[tex]\implies y =-\dfrac{16}{{v_0}^2} \cdot \left(\dfrac{1}{\cos^2 \theta}\right) \cdot x^2+x \cdot \left(\dfrac{\sin \theta}{\cos \theta}\right)[/tex]

[tex]\textsf{Use\;the\;identities\;\;$\sec^2\theta=\dfrac{1}{\cos^2\theta}$\;\;and\;\;$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$}:[/tex]

[tex]\implies y=-\dfrac{16}{{v_0}^2} \cdot \sec^2 \theta \cdot x^2+x \cdot \tan \theta[/tex]

[tex]\textsf{Hence\;proving\;that\;\;$y=-\dfrac{16}{{v_0}^2} \cdot \sec^2 \theta \cdot x^2+x \cdot \tan \theta$}.[/tex]

The equation can be written as,

[tex]y = \dfrac{-15x^2} { (v_o^2) + (\dfrac{1} { v_o^2})tan^{-1}(\dfrac{y} { x})}[/tex].

An object or particle that is projected in a gravitational field, such as from the surface of the Earth, and moves along a curved path while only being affected by gravity is said to be in projectile motion.

To begin with, we know that the horizontal displacement of the projectile is given by [tex]x = v_ot cos(\theta)[/tex], where vo is the initial velocity of the projectile and θ is the angle it makes with the positive x-axis. Therefore, we can rearrange this equation to solve for time t:

[tex]t = \dfrac{x} { (v_o cos(\theta))}[/tex]

Next, we can use the equation for vertical displacement under constant acceleration due to gravity: [tex]y = -16t^2+ v_ot sin(\theta)[/tex]. Substituting the expression for t derived above, we obtain:

[tex]y = -16[\dfrac{x} { (vo cos(\theta)}]^2 + v_o([\dfrac{x} { (v_o cos(\theta)}] sin(\theta)[/tex]

Simplifying this expression, we get:

[tex]y = \dfrac{-16x^2} { (v_o^2 cos^2(\theta)) + x tan(\theta)}[/tex]

Now, we need to eliminate the angle θ in terms of x and y. We can do this by using the fact that tan(θ) = y / x, which gives:

[tex]\theta= tan^{-1}(\dfrac{y} { x})[/tex]

Substituting this expression for θ into our previous equation, we obtain:

[tex]y =\dfrac{ -16x^2 }{(v_o^2 cos^2(tan^{-1}(\dfrac{y} { x}))} + x\ tan(tan^{-1}(\dfrac{y} { x})[/tex]

Since tan(tan⁻¹(z)) = z, we can simplify this expression further:

[tex]y = \dfrac{-16x^2} { (v_o^2 (1 + (\dfrac{y} { x})^2))} + y[/tex]

Finally, by rearranging terms, we get the desired equation:

[tex]y = \dfrac{-15x^2} { (v_o^2) + (\dfrac{1} { v_o^2})tan^{-1}(\dfrac{y} { x})}[/tex]

To know more about projectile motion follow

https://brainly.com/question/24216590

#SPJ1

A(n) = 3 + 7(n-1)

Arithmetic sequences have common differences and change by the same amount between each term.

Arithmetic Sequences

Arithmetic sequences change by the same amount each term. In the sequence above, each term increases by 7. This means that 7 is added to the previous term to make the new term. Using this information we can write a function to represent this sequence.

Explicit Rule

The function that describes an arithmetic sequence is known as an explicit rule. Explicit rules are written as A(n) = A(1) + d(n-1). In this equation, A(1) represents the first term in a sequence and d represents the common difference. As you can see, the first term is 3, so A(1) = 3. The common difference is the change between terms. The previous paragraph shows that the common difference for this sequence is 7.

This means the explicit rule for this sequence is A(n) = 3 + 7(n-1).

As we proved that there exists a δ > 0 such that the function f(x) > 0 for all x ∈ (p − δ, p + δ).

To prove this, we will use the definition of continuity of a function. According to this definition, for a function to be continuous at a point p, the limit of the function as x approaches p must be equal to the value of the function at p.

In this case, we know that f(p) is greater than 0. So, if we can find a δ such that for all x within δ units of p, f(x) is also greater than 0, then we can show that f is continuous at p.

Let's begin by assuming that there is no such δ, i.e., for any δ > 0, there exists an x within δ units of p such that f(x) is less than or equal to 0. This means that we can find a sequence of points xn that approach p such that f(xn) is less than or equal to 0 for all n.

Since f is continuous, we know that lim xn → p f(xn) = f(p) (by the definition of continuity). But we also know that f(p) is greater than 0, so lim xn → p f(xn) cannot be less than or equal to 0. This is a contradiction, which means our assumption that there is no δ such that f(x) is greater than 0 for all x within δ units of p must be false.

Therefore, there must exist a δ > 0 such that f(x) is greater than 0 for all x within δ units of p, which proves our original statement.

To know more about function here

https://brainly.com/question/13088885

#SPJ4

Complete Question:

Let f : (a, b) → R be continuous such that for some p ∈ (a, b), f(p) > 0. Show that there exists a δ > 0 such that f(x) > 0 for all x ∈ (p − δ, p + δ).

The value of the machine after 4 years can be calculated using the formula:
V = P * (1 - r) ^ n
Where:
V = value after n years
P = initial purchase price
r = annual depreciation rate
n = number of years

In this case, P = 3,000.00, r = 0.10 (10%), and n = 4.
Plugging these values into the formula, we get:
V = 3,000.00 * (1 - 0.10) ^ 4
V = 3,000.00 * 0.90 ^ 4
V = 3,000.00 * 0.6561
V = 1,968.30

Therefore, the value of the machine after 4 years is $1,968.30.

To know more about it refer here:

https://brainly.com/question/30045916

#SPJ11

104 consumers bought at least one of the three labels, 12 consumers bought labels A and B but not C 44 consumers bought label A and 91 consumers bought none of these labels because of the equations.

To find the number of consumers who buy the product sold under at least one of the three labels, we need to add up the number of consumers who buy the product under each label individually and those who buy the product under multiple labels.

However, we need to be careful not to double count those who buy the product under all three labels. Therefore, the formula for this calculation is:

At least one of the three labels = (A only) + (B only) + (C only) + (A and B only) + (A and C only) + (B and C only) + (A, B, and C)

Plugging in the given values, we get:

At least one of the three labels = 15 + 20 + 28 + 12 + 9 + 12 + 8 = 104 consumers

To find the number of consumers who buy the product sold under labels A and B but not C, we simply use the value given for those who buy only those sold under labels A and B:

Labels A and B but not C = 12 consumers

To find the number of consumers who buy the product sold under label A, we need to add up the number of consumers who buy the product under label A individually and those who buy the product under label A and one or both of the other labels. Therefore, the formula for this calculation is:

Label A = (A only) + (A and B only) + (A and C only) + (A, B, and C)

Plugging in the given values, we get:

Label A = 15 + 12 + 9 + 8 = 44 consumers

Finally, to find the number of consumers who buy the product sold under none of these labels, we subtract the number of consumers who buy the product under at least one of the three labels from the total number of consumers surveyed:

None of these labels = Total consumers - At least one of the three labels = 195 - 104 = 91 consumers

To know more about equation, refer here:

https://brainly.com/question/29657983#
#SPJ11

Answer:

Step-by-step explanation:

I'm assuming you mean is the number 90 a rational number. The answer is that yes, it is a rational number.

Let me explain:

Under all the possible numbers, there is a taxonomy to organize these types of numbers. It goes down a list from:

- all numbers, which is divided into two groups; complex and real

Under real numbers, there's rational and irrational:

Under rational, there's integers and fractions

Under integers, there's whole numbers and negative

Under whole numbers, there's natural numbers and 0 (yes, zero).

90 falls under natural numbers, which means it is also a rational number. Hope this helps!

15. The student got 18 questions correct.

16. The weight of vitamin C in that tablet is 14%.

17. The secretary's salary has been increased by 60%.

18. The typist's original typing speed was 21 words per minute.

19. The salesperson made $1,022.40 in commission.

20. The salesperson's monthly earnings will be $19,546.

15. On a 45 questions exam, a student gets 40%.

The number of correct questions that the student got is
= 40% × 45 questions

= 18 questions

16. A vitamin tablet, which weighs 250 milligrams, contains 35 milligrams of vitamin C.

The percent of the weight of this tablet is vitamin C is
= 35mg/250mg × 100%

= 14%

17. Five years ago, a secretary made $11,200 annually. The secretary now makes $17,920 annually.

The percentage the secretary's salary has been increased by
= ($17,920/$11,200 × 100%) - 100%

= 160% - 100%

= 60%

18. A typist was able to increase his speed by 120% to 42 words per minute.

Her original typing speed was
= 42/120%  × 100%

= 42/1.2

= 21 words per minute

19. A salesperson makes a commission of 12% on the total amount of each sale. In one month, she makes a total of $8,520 in sales.

In that month, she made a commission of
= $8,520 × 12%

= $1,022.40

20. A salesperson receives a salary of $850 per month plus a commission of 82% of her sales. In a particular month, she sells $22,800 worth of merchandise.

Her monthly earnings will be
= $850 salary + 82% × $22,800

= $19,546

Learn more about percentage here:

brainly.com/question/29832925

#SPJ11

the total money will be spent is 47.4 ∈.

What is trapezoid?

A trapezoid, commonly referred to as a trapezium, is a quadrilateral or polygon with four sides. It has a set of parallel opposite sides as well as a set of non-parallel sides. The bases and legs of the trapezoid are referred to as parallel and non-parallel sides, respectively. A trapezoid is a four-sided closed 2D figure with a perimeter and an area. The bases of the trapezoid are two of the shape's sides that are parallel to one another. The legs or lateral sides of a trapezoid are its non-parallel sides. The altitude is the shortest distance between any two parallel sides.

Here the parallel sides are given 12m and 20m.

The perpendicular line will be the fence

so the other line is given 17m.

The area of the trapezoid is (a+b/2)*h

h =√17²-8² =15

So the area will be 12+20/2 * 15

A= 16*15 = 240m²

For 100 m² cost 19.75

For 240 it will be 19.75/100 * 240= 47.4

Hence the total money will be spent is 47.4 ∈.

Learn more about trapezoid, by the following link.

https://brainly.com/question/1410008

#SPJ1

(a) The permutations can be written as a product of disjoint cycles as follows:
α = (1 5 4 3 2)(6 7 8)
β = (1)(2 3 8 4 7)(5 6)
γ = (1 2 3 5)(4 5 6 8)

(b) The order of each permutation is the least common multiple of the lengths of the disjoint cycles. Therefore:
Order of α = lcm(5, 3) = 15
Order of β = lcm(1, 5, 2) = 10
Order of γ = lcm(4, 4) = 4

(c) The permutations can be written as a product of 2-cycles as follows:
α = (1 2)(2 3)(3 4)(4 5)(5 1)(6 7)(7 8)(8 6)
β = (1 1)(2 3)(3 8)(8 4)(4 7)(7 1)(5 6)(6 5)
γ = (1 2)(2 3)(3 5)(5 1)(4 5)(5 6)(6 8)(8 4)
Since α and β have an even number of 2-cycles, they are even permutations. γ has an odd number of 2-cycles, so it is an odd permutation.

(d) The inverse of each permutation can be found by reversing the order of the elements in each cycle. The inverses can be written as a product of disjoint cycles as follows:
[tex]α^-1 = (1 2 3 4 5)(6 8 7)β^-1 = (1)(2 7 4 8 3)(5 6)γ^-1 = (1 5 3 2)(4 8 6 5)[/tex]

(e) The products αβ and βα can be found by performing the permutations in the given order. The products can be written as a product of disjoint cycles as follows:
αβ = (1 6 8 4 7 2 3 5)(5 1)
βα = (1 7 4 8 6 5 3 2)(2 1)
The order of αβ is lcm(8, 2) = 8, and the order of βα is lcm(8, 2) = 8.

To know more about permutation refer to-

brainly.com/question/30649574#

#SPJ11

Answer:

2.40 per pound

Step-by-step explanation:

hope that's right

The solution to the system of equations is (x, y) = (-2, 8).15)Unfortunately, the question is incomplete, and I cannot provide an answer without knowing the complete question.

The two given equations are as follows:2x + y = 2x = (1/2)y + 6To solve the above system of equations, we will use the substitution method. First, we will substitute the value of x from the second equation to the first equation.2(1/2)y + 6 + y = 22.5y + 6 = 2Subtracting 6 from both sides, we get:2.5y = -4Dividing both sides by 2.5, we get:y = -4/2.5y = -8/5Substituting the value of y in the second equation to get the value of x:x = (1/2)(-8/5) + 6Multiplying and simplifying:x = -4/5 + 30/5x = 26/514)The given system of equations is:x + y = 6-2x + y = -3We will use the elimination method to solve the system of equations. Adding both the equations, we get:2y = 32y = 16y = 8Substituting the value of y in any of the equations to get the value of x:x + 8 = 6x = 6 - 8x = -2Therefore, the solution to the system of equations is (x, y) = (-2, 8).15)Unfortunately, the question is incomplete, and I cannot provide an answer without knowing the complete question.

Learn more about Equations

brainly.com/question/29657988

#SPJ11

We are given the expression √4x^2+100 and the substitution x/5=tan(θ). Our goal is to use the substitution to write the expression as a trigonometric expression and simplify as much as possible.

First, let's substitute x/5=tan(θ) into the expression:

√4(tan(θ)*5)^2+100

Next, let's simplify the expression:

√4(25tan^2(θ))+100

√100tan^2(θ)+100

Now, let's factor out 100 from the expression:

√100(tan^2(θ)+1)

10√tan^2(θ)+1

Finally, let's use the trigonometric identity 1+tan^2(θ)=sec^2(θ) to simplify the expression further:

10√sec^2(θ)

10sec(θ)

Therefore, the expression √4x^2+100 can be written as 10sec(θ) using the substitution x/5=tan(θ).

To learn more about trigonometry here:

https://brainly.com/question/25618616#

#SPJ11

Answer:

C) 84 mm squared

Step-by-step explanation:

The formula for finding area of a triangle is 1/2 times base times height, so substitute and solve for the answer

1/2 times 8 times 21

4 times 21

84

Step-by-step explanation:

for % questions always find 100% and/or 1%. everything else can be calculated based on these.

100% = $115

1% = 100%/100 = 115/100 = $1.15

how many % are $21 ?

as many as times 1.15 (1%) fits into 21 :

21/1.15 = 18.26086957...% ≈ 18.26%

Answer: Let's use the trigonometric ratio of sine to find the measure of the larger acute angle of the right triangle. We have:

sin(theta) = opposite / hypotenuse

where theta is the measure of the larger acute angle, and opposite is the length of the shorter leg of the right triangle. Substituting the given values, we get:

sin(theta) = 2sqrt(6) / 2sqrt(15)

          = sqrt(2/5)

Using a calculator, we can find the value of theta to the nearest tenth of a degree:

theta = arcsin(sqrt(2/5))

     ≈ 38.2°

Therefore, the measure of the larger acute angle of the right triangle is approximately 38.2 degrees.

Step-by-step explanation:

Answer:

8 - w

Step-by-step explanation:

In my opinion, an expression for eight minus w would be 8 - w. Thanks.

Hope it helps.

The solution to the equation (-3/4)(-16 + 8x) by applying the distributive property is 12 - 6x

An equation is an expression containing numbers and variables linked together by mathematical operations such as addition, subtraction, division, multiplication and exponents.

The distributive property states that for three variables a, b and c, the following rule apply:

a(b + c) = ab + ac

Given that:

(-3/4)(-16 + 8x)

Applying the distributive property:

= (-3/4)(-16) + (-3/4)(8x)

= 12 - 6x

The solution is 12 - 6x

Find out more on equation :https://brainly.com/question/13763238

#SPJ1

Answer:

See explanation

Step-by-step explanation:

50 is 5*10, so multiply 3 by 5 also to get 5 and 15/50. First answer is 15

5 and 15/50 is also equal to 4 and 65/50. second answer is 65

carry the numerator on the second line to get 30 for the third answer.

finally, subtract 3 from 4 to get 1, and subtract 30 from 65 to get 35/50.

last two answers are 1, and 35.

Answer:

1278in^3

Step-by-step explanation:

21×8=168

168×6=1008in^3

5×9=45

45×6=270in^3

1008+270=1278in^3

Check the picture below.

Mika's estimate was 216 and the actual sum of the numbers is 216.27. The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

All real numbers are supposed to be explicable by the four fundamental operations, often known as "arithmetic operations". Quotient, product, sum, and difference are the four operations in mathematics that follow division, multiplication, addition, and subtraction.

We are given numbers as 189.27, 15.8 and 11.2.

Mika's estimate is as follows:

189 + 16 + 11 = 216

Actual sum of numbers is as follows:

189.27 + 15.8 + 11.2 = 216.27

Hence, Mika's estimate was 216 and the actual sum of the numbers is 216.27.

Learn more about arithmetic operations from the given link

https://brainly.com/question/30283549

#SPJ1

1. The value of Z is 1.96 determined from Z-table or a calculator. 2. The confidence interval equation is 161 ± 1.96* (4/√20). 3.The standard error is 0.8944. 4. The margin of error is 1.754. 5. The upper and lower limits are 162.754 and 159.246 respectively. 6. Confidence interval indicate how confident the researchers is that true mean is within the range.

1. For a 95% confidence interval, the value for Z is 1.96. This value is found by looking at a Z-table or using a calculator to find the Z-score that corresponds to a 95% confidence level.

2. The confidence interval equation is:

CI = x ± Z* (σ/√n)

Where x is the sample mean, Z is the Z-score, σ is the sample standard deviation, and n is the sample size.

Plugging in the values from the question, we get:

CI = 161 ± 1.96* (4/√20)

3. The standard error is calculated by dividing the sample standard deviation by the square root of the sample size:

SE = σ/√n = 4/√20 = 0.8944

4. The margin of error is calculated by multiplying the Z-score by the standard error:

ME = Z*SE = 1.96*0.8944 = 1.754

5. The upper and lower limits are calculated by adding and subtracting the margin of error from the sample mean:

Upper limit = x + ME = 161 + 1.754 = 162.754

Lower limit = x - ME = 161 - 1.754 = 159.246

6.  The 95% confidence interval for the mean drug price is between $159.246 and $162.754. This means that we are 95% confident that the true mean price of the drug is within this range.

Learn more about Confidence interval:

https://brainly.com/question/17097944

#SPJ11

The expected lifetime reproductive value for an individual in a population with an intrinsic rate of increase (r) of 0 is 1, the correct option is A.

The calculation for the expected lifetime reproductive value can be expressed as:

R₀ = ∑ lxmx

In a stable population with an intrinsic rate of increase of 0, the survival probability (lx) is constant across all age classes, and the average number of offspring produced by an individual (mx) is also constant.

R₀ = lxm

where l is the survival probability and m is the average number of offspring per individual.

Since the population is stable, each individual replaces itself with one offspring, so m = 1.

Therefore, the calculation for the expected lifetime reproductive value in a population with an intrinsic rate of increase of 0 is:

R₀ = lxm = 1 x 1 = 1, the correct option is A.

To learn more about population follow the link: brainly.com/question/27991860

#SPJ4

Answer:

Below

Step-by-step explanation:

(4x- 9)(4x + 9)        Just by looking.....

Answer:

4v - 9

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

multiply inside to the parentheses and add all commmon numbers getting you to, 5v -18 = 2 then add 18 5v=20 divided to v=4

The profits exist the same (slope) but the y-intercept is more increased than the actual diagram.

A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Therefore, it is possible to write the

change in y coordinate with respect to the change in x coordinate as,

m = Δy/Δx where, m is the slope

Given:

The slope-intercept form of a linear equation is where one side includes only "y"

2x+ 3y= 1470

y= -2/3 x + 490

Hence, the y-intercept exists at 490 and the slope is -2/3x.

As, 2x + 3y = 1593

3y = -2x +1593

y = -2/3x +531

The profits exist the same (slope) but the y-intercept is more increased than the actual diagram.

Learn more about Slope here:

https://brainly.com/question/3605446

#SPJ1

Answer:120

Step-by-step explanation:

Step-by-step explanation:

the height is 17 feet and the distance from the wall is 8 feet

using pythogras theorem

[tex] {h}^{2} + {b}^{2} = {hypotenuse}^{2} \\ {h}^{2} + {8}^{2} = {17}^{2} \\ h = \sqrt{( {17}^{2} - {8}^{2} } \\ h = 15[/tex]