Match the numbers in standard notation with the corresponding numbers in scientific notation.

Tiles
864,000,000,000

8,640,000,000

86,400,000,000,000

864,000,000,000,000

Boxes
8.64E9

8.64E14

8.64E11

8.64E13

Answer:

Step-by-step explanation:

a) The proportion remaining (p) after d days can be described by ...

  p = (1 -0.73)^(d/180) = 0.27^(d/180)

Then p=1/2 when ...

  0.50 = 0.27^(d/180)

  log(0.50) = (d/180)log(0.27)

  180(log(0.50)/log(0.27) = d ≈ 95.3

The half-life is about 95.3 days.

__

b) For the proportion remaining to be 60/100, we can use the same solution process. In the end, 0.50 will be replaced by 0.60, and we have ...

  d = 180(log(0.60)/log(0.27) ≈ 70.2 . . . days

60 mg will remain of a 100 mg sample after 70.2 days.

The half-life of the unknown radioactive element, which decays and reduces radioactivity by 73% in 180 days, is approximately 64.44 days. For a sample of 100 mg of this element, it would take about the same duration of one half-life (64.44 days) to decay to a mass just below 60 mg.

The subject of this question relates to radioactive decay and half-life, part of physics in nuclear chemistry. In this scenario, we have an unknown element that decays and diminishes in radioactivity by 73% over the course of 180 days.

We can find the half-life using the formula: t(half-life) = t(total time)/log2(1/% remaining). Substituting the given values into this formula, we get: t(half-life) = 180/log2(1/0.27) = 64.44 days. So, the half-life of the unknown radioactive element is approximately 64.44 days.

For the second part of the question, we first need to ascertain how many half-lives it would take for a 100mg sample to decay to 60mg. Each half-life reduces the substance's amount by half. So, we need to find out the number of half-lives where halving still results in a mass greater than 60mg. This occurs after one half-life (100mg to 50mg), so approximately one half-life (64.44 days) is needed for the sample to decay from 100mg to just below 60mg.

https://brainly.com/question/1770619

#SPJ3

Answer:

The ratio of chocolate covered caramels to solid chocolates is 1 to 2. Or 1 : 2

Step-by-step explanation:

Answer: 1/2

Step-by-step explanation: We can write a ratio using the word "to," using a colon, or using a fraction bar. I would personally write the ratio using a fraction bar since it's easier to write in lowest terms.

So we need to compare the number of chocolate covered caramels to the number of solid chocolates.

We know that we have 4 chocolate covered caramels so we write 4 in the numerator of our ratio. We know that we have 8 solid chocolates so we put an 8 in the denominator of our ratio.

Now we have the fraction 4/8.

Notice however that 4/8 is not in lowest terms so we need to divide the numerator  and denominator by the greatest common factor of 4 and 8 which is 4 to get 1/2. So the ratio of the number of chocolate covered caramels to the number of solid chocolates is 1/2.

Answer:

The total volume of the pieces of fruit in the can is [tex]640\ cm^{3}[/tex]

Step-by-step explanation:

step 1

Find the volume of the cylindrical can

The volume of the can  is equal to

[tex]V=BH[/tex]

where

B is the area of the base of the can

H is the height of the can

we have

[tex]B=75\ cm^{2}[/tex]

[tex]H=10\ cm[/tex]

substitute

[tex]V=(75)(10)=750\ cm^{3}[/tex]

step 2

Find the volume of the pieces of fruit in the can

The volume of the pieces of fruit in the can is equal to subtract  the volume of syrup from the volume of the can

[tex]750\ cm^{3}-110\ cm^{3}=640\ cm^{3}[/tex]

The volume of the cylinder is defined as the product of the base or height.

The total volume of the pieces of fruit in the can is 640 cubic cm.

The base of the can has an area of 75 cm2, and the height of the can is 10cm.

If 110 cm3 of syrup is needed to fill the can to the top then the total volume of the pieces of fruit.

The volume of the cylinder is defined as the product of the base or height.

The volume is the cylinder is given by;

[tex]\rm Volume \ of \ the \ cylinder = Base \times Height[/tex]

Substitute all the values in the formula;

[tex]\rm Volume \ of \ the \ cylinder = Base \times Height\\\\\rm Volume \ of \ the \ cylinder = 75 \times 10\\\\\rm Volume \ of \ the \ cylinder = 750[/tex]

Therefore,

The total volume of the pieces of fruit in the can is,

[tex]= 750 -110\\\rm \\=640 \ cm^3[/tex]

Hence, the total volume of the pieces of fruit in the can is 640 cubic cm.

To know more about the Volume of Cylinder click the link given below.

https://brainly.com/question/14013015

On the left side, you can condense the logarithms into one:

[tex]\ln(1-x)-\ln(3x+5)=\ln\dfrac{1-x}{3x+5}[/tex]

Then

[tex]\ln\dfrac{1-x}{3x+5}=\ln(1-6x)\implies e^{\ln((1-x)/(3+5))}=e^{\ln(1-6x)}\implies\dfrac{1-x}{3x+5}=1-6x[/tex]

From here it's a purely algebraic equation. Multiply both sides by [tex]3x+5[/tex] to get

[tex]1-x=(1-6x)(3x+5)[/tex]

[tex]1-x=5-27x-18x^2[/tex]

[tex]18x^2+26x-4=0[/tex]

[tex]9x^2+13x-2=0[/tex]

By the quadratic formula,

[tex]x=\dfrac{-13\pm\sqrt{241}}{18}[/tex]

or about [tex]x\approx-1.5847[/tex] and [tex]x\approx0.14023[/tex].

Before we finish, first note that in order for the original equation to make sense, we need [tex]x[/tex] to satisfy 3 conditions:

[tex]-x+1>0\implies x<1[/tex]

[tex]3x+5>0\implies x>-\dfrac53\approx-1.67[/tex]

[tex]-6x+1>0\implies x<\dfrac16\approx0.17[/tex]

or taken together,

[tex]-\dfrac53<x<\dfrac16[/tex]

so both solutions found above are valid.

To solve the logarithmic equation, combine the logarithms, set the arguments equal to each other, solve for x, and check the solution.

To solve the given equation ln(–x + 1) – ln(3x + 5) = ln(–6x + 1), we can use the properties of logarithms to simplify it.

1. Combine the logarithms using the quotient rule: ln((–x + 1)/(3x + 5)) = ln(–6x + 1).

2. Set the arguments equal to each other: (–x + 1)/(3x + 5) = –6x + 1.

3. Solve for x by cross-multiplying and simplifying the equation.

4. Check the solution in the original equation for validity.

The solution to the equation is x = -1.

https://brainly.com/question/36026737

#SPJ2

Answer:

(B) The correct answer is B: m = n, so y = am / bn is the horizontal asymptote.

The second part is The horizontal asymptote is y = 5

The horizontal asymptote is a horizontal line that guides the graph for values of x, but is not part of the graph

The correct option that defines the horizontal asymptote is the option;

Reason:

The possible function of the question is [tex]f(x) = \dfrac{20 + 5 \cdot x}{x}[/tex]

The general form of the rational function is presented as follows;

[tex]f(x) = \dfrac{x^m+...+ a \cdot x + c}{x^n + ...+b\cdot x + d}[/tex]

The power or degree of the numerator and denominator of a rational function  determine the nature of the horizontal asymptote

Where highest power in numerator is less than the highest power or degree of the denominator, the horizontal asymptote is at y = 0

Therefore;

m < n the horizontal asymptote is y = 0

Where the power of the numerator is larger than the power of the denominator by one, the asymptote is slant, and the graph has no asymptote

m > n, there is no horizontal asymptote

In a rational function where the power of the numerator is equal to the power of the denominator, the horizontal asymptote occurs at the ratio of the leading zeros, [tex]y = \dfrac{a_m}{b_n}[/tex]

m = n, the horizontal asymptote is [tex]y = \dfrac{a_m}{b_n}[/tex]

Therefore;

In the given function, [tex]f(x) = \dfrac{20 + 5 \cdot x}{x}[/tex], the power of the numerator is equal to the power of the denominator, therefore, we have;

The horizontal asymptote of the function is [tex]y = \dfrac{5}{1} = 5[/tex]

Learn more here:

https://brainly.com/question/12730051

Answer: OPTION A

Step-by-step explanation:

The formula that is used to calculate the surface area of a cone is:

[tex]SA=\pi rl+\pi r^2[/tex]

Where "r" is the radius and "l" is the slant height.

You can identify in the figure that:

[tex]l=12units\\r=3units[/tex]

Then, you need to substitute these values into the formula [tex]SA=\pi rl+\pi r^2[/tex].

Therefore, the surface area of this right cone is:

[tex]SA=\pi (3units)(12units)+\pi (3units)^2[/tex]

[tex]SA=45\pi\ units^2[/tex]

Answer:

6.05 units

Step-by-step explanation:

We are given that radius,r=3 units

Volume of cone=57 cubic units

We have to find the height of cone.

We know that

Volume of cube=[tex]\frac{1}{3}\pi r^2 h[/tex]

Where [tex]\pi=3.14[/tex]

Using the formula

[tex]57=\frac{1}{3}\times 3.14\times (3)^2\times h[/tex]

[tex]h=\frac{3\times 57}{3.14(3)^2}[/tex]

h=6.05 units

Hence, the height of cone=6.05 units

The triangles are not the same size so a dilation made the original one smaller and a translation moved it to map ABC to A'B'C'.

ANSWER

a=53°

b=106°

c=74°

d=37°

EXPLANATION

A semicircle creates a right angle on the circumference.

d+53°=90°

d=90°-53°

d=37°

Angle subtended at the center by an arc is twice the angle subtended at the circumference by the same arc.

c=2d

c=2(37)

c=74°

Adjacent angles on a straight line add up to 180°

c+b=180°

b=180°-c

b=180°-74°

b=106°

The sum of interior angles in an isosceles triangle is 180°

a+a+c=180°

2a+74=180

2a=180-74

2a=106

a=53°

Answer:

Domain is set of all real numbers

Range is set of all real numbers

Step-by-step explanation:

Domain is the set of x values for which the function is defined

to find domain we look at the graph and check if there is any restriction for x

WE have line graph for all x values

So there is no restriction for x. Hence, Domain is set of all real numbers

Range is the set of y values for which the function is defined

to find range we check the continuity on graph

The graph is contininious and there is no break . The graph is continious for all y values

So range is set of all real numbers

Answer:

its d.

i just took it

Step-by-step explanation:

Answer:

Option A

Domain = Range = (-∞,∞)

Step-by-step explanation:

We can easily solve this question by using a graphing calculator or any plotting tool.

The function is

f(x) = -2x + | 3 sin(x) |

Which can be seen in the picture below

We can notice that f(x) is a line with periodical ups and downs thanks to the sinusoidal term, but there are no restrictions over the domaoin or range of the function.

It can take any real value as an input, and can produce any real value as an output

Use the chain rule:

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\cdot\dfrac{\mathrm dt}{\mathrm dx}[/tex]

So we have

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}[/tex]

[tex]x=t^2+4\implies\dfrac{\mathrm dx}{\mathrm dt}=2t[/tex]

[tex]y=t^3+3t\implies\dfrac{\mathrm dy}{\mathrm dt}=3t^2+3[/tex]

[tex]\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{3t^2+3}{2t}[/tex]

Now write [tex]f(t)=\dfrac{\mathrm dy}{\mathrm dx}[/tex]. Then by the chain rule,

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{\mathrm dy}{\mathrm dx}\right]=\dfrac{\mathrm df}{\mathrm dx}=\dfrac{\mathrm df}{\mathrm dt}\cdot\dfrac{\mathrm dt}{\mathrm dx}=\dfrac{\frac{\mathrm df}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}[/tex]

so that

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\frac{\mathrm d}{\mathrm dt}\left[\frac{3t^2+3}{2t}\right]}{2t}=\dfrac{3(t^2-1)}{4t^3}[/tex]

The curve is concave upward when the second derivative is positive:

[tex]\dfrac{3(t^2-1)}{4t^3}>0\implies t^2>1\implies\sqrt{t^2}>\sqrt1\implies|t|>1[/tex]

or equivalently, when [tex]t<-1[/tex] or [tex]t>1[/tex].

The dy/dx is equal to 1 + 3/2t, and the d2y/dx2 is 3/2t. The curve is concave upward for t values in the (0, Infinity) interval.

To answer your question, we first need to take the derivatives of x and y with respect to t. The derivatives of x = t^2 + 4 and y = t^2 + 3t by t yield dx/dt = 2t and dy/dt = 2t + 3. Now, dy/dx = (dy/dt) / (dx/dt) = (2t + 3) / 2t = 1 + 3/2t.

Next the second derivative d2y/dx2 (concavity), is obtained as the derivative of dy/dx = 1 + 3/2t with respect to t, which is 3/2t. Now, the curve will be concave upward whenever d2y/dx2 > 0. Solving 3/2t > 0 gives the interval for t as (0, Infinity), which indicates the values of t for which the curve is concave upward.

https://brainly.com/question/33446764

#SPJ11

Answer:

Option D

Step-by-step explanation:

The equation to predict the amount saved is

[tex]y = 6x ^ 2 + 75x + 200.[/tex]

Assuming that the variable x represents the saving months, then substitute [tex]x = 25[/tex] in the quadratic equation and solve for y.

[tex]y = 6 (25) ^ 2 +75 (25) +200\\\\y = \$\ 5,825[/tex]

The answer is the option D  $5,825

Here is your answer

d) 7y(-4x + 5)

EXPLANATION:

Given,

[tex]-28xy + 35y[/tex]

It can be written as-

[tex]7×(-4)xy + 7×5y[/tex]

The two terms [tex]7×4xy[/tex] and [tex]7×5y[/tex] have 7y common in them.

So, it can be further written as-

[tex]7y(-4x + 5)[/tex]

HOPE IT IS USEFUL

Answer:

27 in^2

Step-by-step explanation:

   To solve this, we must first find the height. We can split the trapezoid into a triangle and rectangle! Since 11-7=4, one of the legs of the triangle is 4, and the height of the trapezoid is 3 because 3,4,5 is a pythagorean triple. Since the area of a trapezoid is the average of the bases times the height, we multiply 9, the average, by 3 and we get 27 in^2.

Answer:

C) The expected value is -$2000, so the detective should not take the job.

Step-by-step explanation:

The expected value is the sum of products of income and probability:

E = -10,000·1.00 + 80,000·0.10 = -10,000 +8,000 = -2,000

The detective can be expected to lose money on the job, so should not take it.

Answer:

B

Step-by-step explanation:

You are given the inequality

[tex]8a-15>73[/tex]

Add 15 to both sides:

[tex]8a-15+15>73+15\\ \\8a>88[/tex]

Now divide both sides by 8:

[tex]a>11[/tex]

You should choose option B, because this number line shows all values of x which are greater than 11.

Hold on I’m figuring it out right now

Answer:

[tex]\frac{6x-14}{x^{4} +4x}[/tex]

Step-by-step explanation:

I have to [tex]\frac{\frac{3x-7}{x^{2} } }{\frac{x^{2} }{2}+\frac{2}{x}}[/tex]

Let's start by joining the macro denominator with a common denominator. So, by applying a minimum common multiple [tex]\frac{x^{2} }{2} +\frac{2}{x}=\frac{x^{3}+ 4 }{2x}[/tex]

Now I can write the expression as

[tex]\frac{\frac{3x-7}{x^{2}}}{\frac{x^{3}+ 4 }{2x}}[/tex]

Now to convert both fractions into one, I multiply the numerator of the one above by the denominator of the one below, and the denominator of the one above with the numerator below, remaining that way.

[tex]\frac{\frac{3x-7}{x^{2}}}{\frac{x^{3}+4}{2x}}=\frac{(3x-7)(2x)}{(x^{2})(x^{3}+ 4)}[/tex]

Having the fraction in this way, I could simplify the x of the "2x" of the numerator with an x^2 (x^2=x*x) of the denominator

[tex]\frac{(3x-7)(2x)}{(x^{2})(x^{3}+4)}=\frac{2(3x-7)}{x(x^{3}+ 4)}[/tex]

finally, applying distributive property, I have to

[tex]\frac{(6x-14)}{(x^{4}+ 4x)}[/tex]

Done

Combine fractions with common denominator [tex]\(2x^2\),[/tex] then simplify numerator and denominator separately: [tex]\(\frac{12x^2 - 28x^3}{x^4 + 8x^3}\).[/tex]

let's simplify the complex fraction step by step:

1. Start by finding a common denominator for all the fractions involved. In this case, the least common denominator is [tex]\(2x^2\).[/tex]

2. Rewrite each fraction with the common denominator:

  - [tex]\(\frac{3x - 7}{x^2}\)[/tex]  becomes [tex]\(\frac{3x - 7}{x^2} \cdot \frac{2}{2}\)[/tex] to match the denominator [tex]\(2x^2\).[/tex] So, it becomes[tex]\(\frac{6 - 14x}{2x^2}\).[/tex]

  -[tex]\(\frac{x^2}{2}\)[/tex] remains the same.

  - [tex]\(\frac{2}{x}\)[/tex]  becomes [tex]\(\frac{2}{x} \cdot \frac{2x}{2x}\)[/tex] to match the denominator [tex]\(2x^2\).[/tex] So, it becomes [tex]\(\frac{4x}{2x^2}\).[/tex]

3. Now, combine all the fractions:

[tex]\[ \frac{\frac{6 - 14x}{2x^2}}{\frac{x^2}{2} + \frac{4x}{2x^2}} \][/tex]

4. Combine the terms in the numerator and denominator:

  - The numerator remains the same: [tex]\(6 - 14x\).[/tex]

  - The denominator becomes[tex]\(\frac{x^4 + 8x^3}{2x^2}\).[/tex]

5. Divide the numerator by the denominator:

[tex]\[ \frac{6 - 14x}{\frac{x^4 + 8x^3}{2x^2}} \][/tex]

6. Multiplying the numerator by the reciprocal of the denominator:

[tex]\[ (6 - 14x) \cdot \frac{2x^2}{x^4 + 8x^3} \][/tex]

7. Distribute the numerator:

 [tex]\[ \frac{12x^2 - 28x^3}{x^4 + 8x^3} \][/tex]

So, the simplified form of the complex fraction is [tex]\(\frac{12x^2 - 28x^3}{x^4 + 8x^3}\).[/tex]

Answer:

True

Step-by-step explanation:

True.

The solution to a system of equations is the point where two equations intercept. In both cases, we can see that the interception of the two systems shown occurs at (0, 1). So the statement is correct.

Answer:

It's True

Step-by-step explanation:

Answer:

y = 5 cos ((2π/3)x - 2) + 2

Step-by-step explanation:

Cosine function takes a general form of  y = A cos (Bx + C) + D

Where

A is the amplitude

2π/B is the period

C is the phase shift ( if -C, then phase shift right, if +C phase shift left)

D is the vertical displacement (+D is above and -D is below)

Given the conditions of the function to build and the general form, we can write:

** Note: period needs to be 3, so 2π/B = 3, hence B = 2π/3

Now we can write:

y = 5 cos ((2π/3)x - 2) + 2

first answer choice is right.

Final answer:

The equation for a cosine function with an amplitude of 5, a period of 3, a phase shift of 2, and a vertical displacement of 2 is y = 5 cos(2π/3 (x - 2)) + 2.

Explanation:

To write an equation for a cosine function with the given parameters, we need to understand several key features of the function: amplitude, period, phase shift, and vertical displacement.

The general form of a cosine function is y = A cos(B(x - C)) + D, where A is the amplitude, T = 2π/B (where T is the period), C is the phase shift, and D is the vertical displacement.

Given an amplitude (A) of 5, a period (T) of 3, a phase shift (C) of 2, and a vertical displacement (D) of 2, we can substitute these values into the general form:

Putting it all together, the equation of the cosine function is: y = 5 cos(2π/3 (x - 2)) + 2

Answer:

$80

Step-by-step explanation:

If $100 is 25% more than Dwayne's original amount (a), you have ...

a + 0.25a = 100

1.25a = 100

a = 100/1.25 = 80

Dwayne originally had $80.

Answer:

The volume is equal to [tex]320\pi\ cm^{3}[/tex]

Step-by-step explanation:

we know that

The volume of the cone is equal to

[tex]V=\frac{1}{3}\pi r^{2} h[/tex]

we have

[tex]r=8\ cm[/tex]

[tex]h=15\ cm[/tex]

substitute the values

[tex]V=\frac{1}{3}\pi (8)^{2}(15)[/tex]

[tex]V=320\pi\ cm^{3}[/tex]

Final answer:

volume of 320π cm³.

Explanation:

To calculate the volume of the oblique cone with the given dimensions, use the formula for the volume of a cone,

The volume of the oblique cone can be calculated using the formula V = 1/3 * π * r^2 * h.

Answer:

x=9

Step-by-step explanation:

9+5+5=19

Answer:

[tex]l=\frac{A-2wh}{2w+2h}[/tex]

Step-by-step explanation:

We were given that; the formula for the surface area, A, of a prism is given by

[tex]A=2lw +2lh +2wh[/tex]

where is the length of the prism, w is the width, and h is the height.

We want to solve this formula for l,

Group the l terms;

[tex]A-2wh=2lw +2lh [/tex]

Factor l on the right;

[tex]A-2wh=(2w +2h)l [/tex]

Divide both sides by 2w +2h

[tex]\frac{A-2wh}{2w+2h}=l[/tex]

Therefore:

[tex]l=\frac{A-2wh}{2w+2h}[/tex]

Answer:

What he said.

Step-by-step explanation:

Answer:

4 cups

Step-by-step explanation:

Answer:

$817.08

Step-by-step explanation:

Answer:

FALSE

Step-by-step explanation:

What we have here is experimental probability:   12 heads out of 20 tosses.

The fraction 12/20 reduces to  6/10, or 0.60, which corresponds to 60%.

The answer to this question is FALSE, because this is not theoretical probability.

Answer:

false

Step-by-step explanation:

Answer:

option d.

Step-by-step explanation:

We have the following set of data:

x     0   1    2   3  4

f (x)   18 14  10 6  2

Let's assume the function is linear, then, the equation of the line would ne:

(y - y0) = m(x-x0)

where m= (y1-y0) / (x1-x0)

And (x1, y1) = (1, 14)

(x0, y0) = (0, 18)

You can choose any of the points given in the set of data.

Then,

m = (14-18)/(1-0) = -4.

Then the equation of the line is:

(y - 18) = -4x

y = -4x + 18.

If the function is linear, then all the points given in the set of data will satisfy the function. Let's try:

(2, 10):

10 = -4(2) + 18.

10 = 10

SATISFIES THE EQUATION

(3, 6):

6 = -4(3) + 18.

6 = 6

SATISFIES THE EQUATION

(4, 2)

2 = -4(4) + 18.

6 = 6

SATISFIES THE EQUATION

So, the function is linear. And the correct option is option d.

Answer:

D. Line.ar; -4x + 18

Step-by-step explanation:

Answer:

y = tan(1/2 x + π/2) ⇒ answer c

Step-by-step explanation:

* Lets revise some fact of y = tanx

- The domain of tanx is all x(≠ π/2) + nπ, where n is the number of cycle

- The range is all real numbers

- The period of tanx is π ÷ coefficient of x

* Lets revise some transformation

- A horizontal stretching is the stretching of the graph away from

 the y-axis

• if 0 < k < 1 (a fraction), the graph is f (x) horizontally stretched by

 dividing each of its x-coordinates by k (x × 1/k)

- A horizontal compression is the squeezing of the graph toward

 the y-axis.

• if k > 1, the graph f (x) horizontally compressed by dividing each

 of its x-coordinates by k. (x × 1/k)

* Look to the graph of y = tanx ⇒ red graph

- the graph of tanx intersect x-axis at the origin

- The period of tanx is π

* Look to the blue graph (the problem graph)

∵ The graph intersect x-axis at points (-π , 0)

- That means the graph of tanx moved to the left by π units

∴ y = tan(x + π)

- The period of the graph is 2π

∵ The period = π/coefficient of x

∴ 2π = π/coefficient of x ⇒ using cross multiplication

∴ Coefficient of x = π/2π = 1/2

- That means the graph stretched horizontally

∴ y = tan1/2(x + π)

* y = tan(1/2 x + π/2)

Answer:

The answer is: yes; k = -3 and y = -3x ⇒ the 3rd answer

Step-by-step explanation:

* Lets revise how to know the relation is direct proportion

- If all the ratios of x/y are proportion (equal ratios), then they

 are varies directly

* Now lets check the relation between x and y

∵ x = 1 and y = -3

∴ y/x = -3/1 = -3

∵ x = 3 and y = -9

∴ y/x = -9/3 = -3

∵ x = 5 and y = -15

∴ y/x = -15/5 = -3

∵ All ratios are equal -3 (they are proportion)

∴ y varies directly with x

* y ∝ x

∴ y = k x

∴ The constant of variation k is -3

∴ y = -3x

* The answer is: yes; k = -3 and y = -3x