Help please!
What is the magnitude of the position vector whose terminal point is (-2, 4)?

Answer: The contrapositive statement :"If you do not have a gerbil, then you are not a pet owner.”

It is false.

Counter example : If you have a dog, then you are a pet owner.

Step-by-step explanation:

We know that the contrapositive of a statement of the form " If a then b" is "If not a then not b.

Given: The conditional statement is "If you have a gerbil, then you are a pet owner.”

Then the contrapositive statement will be "If you do not have a gerbil, then you are not a pet owner.”

The contrapositive statement is false.

Counter example : If you have a dog, then you are a pet owner.

Means if you area pet owner then it can be any pet not just gerbil.

Answer:

The contrapositive is false.

Step-by-step explanation:

Given conditional statement is :

If you have a gerbil, then you are a pet owner (conditional statement is in the form of if p, then q)

The contrapositive is represented as (if not q, then not p)

So, the contrapositive sentence will be :

If you are not a pet owner, then you do not have a gebril.

This statement is false as if you are not a pet owner then you will not have any animal with you and not only gebril.

Answer:

  233 cartons of food; 467 cartons of clothing

Step-by-step explanation:

This linear programming problem can be formulated as two inequalities (in addition to the usual constraints that the variables be non-negative). One of these expresses the constraint on weight. Let f and c represent numbers of food and clothing containers, respectively.

  40f +25c ≤ 21000

The other expresses the limit on volume.

  20f + 5c ≤ 7000

_____

Feasible Region vertex

We can subtract the boundary line equation of the first inequality from that of 5 times the second to find f:

  5(20f +5c) -(40f +25c) = 5(7000) -21000

  60f = 14000

  f = 233 1/3

The second boundary line equation can be rearranged to find c:

  c = 1400 -4f = 466 2/3

The nearest integer numbers to these values are ...

  (f, c) = (233, 467)

The other vertices of the feasible region are associated with one or the other variable being zero: (f, c) = (0, 840) or (350, 0).

Check of Integer Solution

Trying these in the constraint inequalities gives ...

Selection of the Answer

The answer to the question will be the feasible region vertex that maximizes the number of people helped. That is, we want to maximize ...

  p = 13f + 6c

The values of p at the vertices are ...

  p = 13·233 + 6·467 = 5831

  p = 13·0 + 6·840 = 5040

  p = 13·350 + 6·0 = 2100

The most people are helped when the plane is filled with 233 food cartons and 467 clothing cartons.

This is a linear programming problem. We create two constraints based on weight and volume, and create an equation to maximize the number of people helped. The exact solution depends on solving this using a suitable tool.

This problem can be approached as a linear programming problem where the goal is to maximize the number of people helped. Total weight cannot exceed 21000 pounds, and total volume must not exceed 7000 cubic feet.

Let X be the number of cartons of food, and Y be the number of cartons of clothing.

So, we have two constraints: 40X + 25Y ≤ 21000 (weight constraint) and 20X + 5Y ≤ 7000 (volume constraint). We want to maximize the number of people helped (Z), represented by 13X + 6Y (since each food carton helps 13 people and each clothing carton helps 6 people).

The exact number of food and clothing cartons will depend on how we solve this linear programming problem. This is typically done using tools like a graphing calculator or software, which can give us the number of X and Y which maximizes Z.

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Answer:

The radius is: 1

Step-by-step explanation:

The equation of a circle in center-radius form is:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

Where the center is at the point (h, k) and the radius is "r".

Given the equation of the circle:

[tex]x^2+y^2=1[/tex]

You can identify that:

[tex]r^2=1[/tex]

Then, solving for "r", you get that the radius of the circle is:

[tex]r=\sqrt{1}\\\\r=1[/tex]

Answer:

Radius = 1

Step-by-step explanation:

It is given that equation of a circle is x² + y² = 1

Points to remember

Equation of a circle with center(h, k) and radius r is given by,

(x - h)²  + (y - k)² = r²

To find the radius of circle

Compare the two equations

x² + y² = 1  and (x - h)²  + (y - k)² = r²

we get the center is (0, 0)

therefore we can write

r² = 1

r = 1

Therefore radius r = 1

Answer:

c. the time it takes to run the 100-meter race

Step-by-step explanation:

An independent variable is the variable which is not being controlled and it does not depend on the other variables. This is the variable which is being studied/measured in the experiment.

Option a. The previous running experience is not being considered and is not a variable under study in this case. So this is not the answer.

Option b. The teenagers who are being studied constitute the sample. These are not the variables.

Option c. Time is the independent variable, as it is being measured during the experiment and the conclusion is being drawn based on it.

Option d. Amount of caffeine is being decided by Cosella and is therefore not the independent variable.

Therefore, the correct answer is option c

Answer:

21°

Step-by-step explanation:

the angles of triangle abc has to = 180 so subtract the known angles to get 21

angles bca and daf have to be the same so its 21

Answer:

[tex]x=21[/tex]

Step-by-step explanation:

We have been given an image of two parallel lines cut by a transversal. We are asked to find the measure of x.

We know that corresponding angles of two parallel lines are equal. We can see that angle BCA and angle DAF are corresponding angles.

Let us find measure of angle BCA using angle sum property.

[tex]54^{\circ}+105^{\circ}+m\angle BCA=180^{\circ}[/tex]

[tex]159^{\circ}+m\angle BCA=180^{\circ}[/tex]

[tex]159^{\circ}-159^{\circ}+m\angle BCA=180^{\circ}-159^{\circ}[/tex]

[tex]m\angle BCA=21^{\circ}[/tex]

Therefore, the value of x is 21 degrees.

Answer:

  63

Step-by-step explanation:

The two tangents to a circle from the same point are the same length.

  SD = SG

  5x +18 = 8x -9

  27 = 3x . . . . . . . . add 9-5x

  9 = x . . . . . . . . . . divide by 3

SG = 8x -9 = 8·9 -9

SG = 63

Answer:

  y = -x +5

Step-by-step explanation:

The 2-point form of the equation of a line is useful for this.

  y = (y2 -y1)/(x2 -x1)(x -x1) + y1 . . . 2-point form of equation for a line

  y = (-2 -6)/(7 -(-1))/(x -(-1)) +6 . . . . substitute the give points

  y = -8/8(x +1) +6 . . . . . . . . . . . . . . simplify a bit; next, simplify more

  y = -x +5

The equation of the line passing through the point s (-1, 6) and (7, -2) is y = -x + 5

The formula for calculating the equation of a line is expressed as y = mx + b where;

m is the slope of the line

b is the y-intercept

Given the coordinate points  (-1,6) and (7, -2)

Get the slope:

Slope = -2-6/7-(-1)

Slope = -8/8

Slope = - 1

Get the y-intercept:

-2 = -1(7) + b

-2 = -7 + b

b = -2 + 7

b = 5

Get the required equation:

Recall that y = mx + b

Substituting m = -1 and b = 5 into the equation:

y = -x + 5

Hence the equation of the line passing through the point s (-1, 6) and (7, -2) is y = -x + 5

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Answer:

  4 m

Step-by-step explanation:

Use the formula for the area of a triangle. Fill in the given numbers and solve for the unknown.

  A = (1/2)bh

  48 m² = (1/2)(24 m)h . . . . . put in the given numbers

  (48 m²)/(12 m) = h = 4 m . . . . divide by the coefficient of h

The height of the triangle is 4 meters.

[tex]\huge{\boxed{\frac{5}{2}}}[/tex]

Slope-intercept form is [tex]f(x)=mx+b[/tex], where [tex]m[/tex] is the slope and [tex]b[/tex] is the y-intercept.

[tex]\frac{5x}{2}=\frac{5}{2}x[/tex], so change the equation to represent this. [tex]f(x)=\frac{5}{2}x+3[/tex]

Great! Now the function is in slope-intercept form, so we can just see that [tex]m[/tex], or the slope, is equal to [tex]\boxed{\frac{5}{2}}[/tex]

Answer:

Option A (Exterior)

Step-by-step explanation:

To understand this question, it is important to understand the concept of the exterior angle. An exterior angle is an angle which is made by two intersecting lines outside of the shape. Basically, one of the two lines is extended outside the shape. The angle between the extended line and the other line which is not extended is the exterior angle. It is outside the shape. The interior angle is the angle which is made by the same two lines but inside the shape.

The sum of the interior angle and the exterior angle is 180 degrees. It is also interesting to note that the sum of the angles in the triangle is 180 degrees.

Suppose that the angles in the triangles are A, B, and C, and the associated exterior angle with the angle A is angle D. By the argument, A+B+C=180 degrees and A+D=180 degrees. Since 180 degrees = 180 degrees, therefore A+B+C = A+D. Angle A cancels on both sides and reduces to B+C=D. This proves that the exterior angle of a triangle is equal to the sum of the two remote interior angles!!!

Answer:

Step-by-step explanation:

We know that the function is measured in terms of time. And the constant distance traveled in 50 miles. The inverse of the function is supposing say t(d) which measures the function in terms of distance traveled. The inverse of the function is obtained by dividing the distance by 50 as in the original function then the function is multiplied by 50. So the inverse of function obtained is: t(d) = d/50

Steps: d(t) = 50t

          t = d/50

          t(d) = d/50

I hope this helps!

Answer:

16. the expression below.

17. about 470,933 MD's and about 18,743 DO's

The expression for the total number of doctors is [tex]\frac{-1374t^2+43192.22t+67805.35}{(65-2t)(85-t) }[/tex]

a. MD + DO

[tex]\frac{28390+693t}{85-t} +\frac{776-12t}{65-2t}[/tex]

This can be simplified as

[tex]\frac{(28390+693t)(65-2t)+(776-12t)(85-t) }{(85-t)(65-2t) }[/tex]

[tex]= \frac{-1374t^2+43192.22t+67805.35}{(65-2t)(85-t) }[/tex]

b. If t = 0 in 1980, in 2010, t = 30

The number of MD would be

[tex]\frac{28390+693*30}{85-30} \\= 379[/tex]

The number of DO would be

[tex]\frac{776-12*30}{65-2*30}\\\\= 83[/tex]

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We have the following expression:

[tex]tan(\theta)cot(\theta)-sin^{2}(\theta)=cos^2(\theta)[/tex]

We know that:

[tex]cot(\theta)=\frac{1}{cot(\theta)}[/tex]

Therefore, by substituting in the original expression:

[tex]tan(\theta)\left(\frac{1}{tan(\theta)}\right)-sin^{2}(\theta)=cos^2(\theta) \\ \\ \\ Simplifying: \\ \\ 1-sin^2(\theta)=cos^2(\theta)[/tex]

We know that the basic relationship between the sine and the cosine determined by the Pythagorean identity, so:

[tex]sin^2(\theta)+cos^2(\theta)=1[/tex]

By subtracting [tex]sin^2(\theta)[/tex] from both sides, we get:

[tex]\boxed{cos^2(\theta)=1-sin^2(\theta)} \ Proved![/tex]

We have the following expression:

[tex]\frac{cos(\alpha)}{cos(\alpha)-sin(\alpha)}=\frac{1}{1-tan(\alpha)}[/tex]

Here, let's multiply each side by [tex]cos(\alpha)-sin(\alpha)[/tex]:

[tex](cos(\alpha)-sin(\alpha))\left(\frac{cos(\alpha)}{cos(\alpha)-sin(\alpha)}\right)=(cos(\alpha)-sin(\alpha))\left(\frac{1}{1-tan(\alpha)}\right) \\ \\ Then: \\ \\ cos(\alpha)=\frac{cos(\alpha)-sin(\alpha)}{1-tan(\alpha)}[/tex]

We also know that:

[tex]tan(\alpha)=\frac{sin(\alpha)}{cos(\alpha)}[/tex]

Then:

[tex]cos(\alpha)=\frac{cos(\alpha)-sin(\alpha)}{1-\frac{sin(\alpha)}{cos(\alpha)}} \\ \\ \\ Simplifying: \\ \\ cos(\alpha)=\frac{cos(\alpha)-sin(\alpha)}{\frac{cos(\alpha)-sin(\alpha)}{cos(\alpha)}} \\ \\ Or: \\ \\ cos(\alpha)=\frac{\frac{cos(\alpha)-sin(\alpha)}{1}}{\frac{cos(\alpha)-sin(\alpha)}{cos(\alpha)}} \\ \\ Then: \\ \\ cos(\alpha)=cos(\alpha).\frac{cos(\alpha)-sin(\alpha)}{cos(\alpha)-sin(\alpha)} \\ \\ \boxed{cos(\alpha)=cos(\alpha)} \ Proved![/tex]

We have the following expression:

[tex]\frac{cos(x+y)}{cosxsiny}=coty-tanx[/tex]

From Angle Sum Property, we know that:

[tex]cos(x+y)=cos(x)cos(y)-sin(x)sin(y)[/tex]

Substituting this in our original expression, we have:

[tex]\frac{cos(x)cos(y)-sin(x)sin(y)}{cosxsiny}=coty-tanx[/tex]

But we can also write this as follows:

[tex]\\ \frac{cosxcosy}{cosxsiny}-\frac{sinxsiny}{cosxsiny}=coty-tanx \\ \\ Simplifying: \\ \\ \frac{cosy}{siny}-\frac{sinx}{cosx} =coty-tanx \\ \\ But: \\ \\ \frac{cosy}{siny}=coty \\ \\ \frac{sinx}{cosx}=tanx \\ \\ Hence: \\ \\ \boxed{coty-tanx=coty-tanx} \ Proved![/tex]

We have the following expression:

[tex]\ln\left|1+cos \theta\right|+\ln\left|1-cos \theta\right|=2\ln\left|sin \theta\right|[/tex]

By Logarithm product rule, we know:

[tex]log_{b}(x.y) = log_{b}(x) + log_{b}(y)[/tex]

So:

[tex]\ln\left|1+cos \theta\right|+\ln\left|1-cos \theta\right|=\ln\left|(1+cos \theta)(1-cos \theta)\right|[/tex]

The Difference of Squares states that:

[tex]a^2-b^2=(a+b)(a-b) \\ \\ So: \\ \\ (1+cos \theta)(1-cos \theta)=1-cos^2 \theta[/tex]

Then:

[tex]\ln\left|(1+cos \theta)(1-cos \theta)\right|=\ln\left|1-cos^{2} \theta\right|[/tex]

By the Pythagorean identity:

[tex]sin^2(\theta)+cos^2(\theta)=1 \\ \\ So: \\ \\ sin^2 \theta = 1-cos^2 \theta[/tex]

Then:

[tex]\ln\left|1-cos^{2} \theta\right|=\ln\left|sin^2 \theta|[/tex]

By Logarithm power rule, we know:

[tex]log_{b}(x.y) = ylog_{b}(x)[/tex]

Then:

[tex]\ln\left|sin^2 \theta|=2\ln\left|sin \theta|[/tex]

In conclusion:

[tex]\boxed{\ln\left|1+cos \theta\right|+\ln\left|1-cos \theta\right|=2\ln\left|sin \theta\right|} \ Proved![/tex]

Answer:

51.496 miles

Step-by-step explanation:

circumference of circle= 2pir

here we will tale radius as 8.2 because 16.4 is diameter of circle.

2*3.14*8.2=

6.28*8.2=51.496 miles

Based on the properties of a parallelogram, the missing angle measures are:

The opposite angles of a parallelogram are defined as congruent angles, while the adjacent angles in a parallelogram are supplementary.

Thus:

m∠U = m∠S = 70°

m∠R = 180 - 70 = 110°

m∠T = m∠R =110°

Therefore, based on the properties of a parallelogram, the missing angle measures are:

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The missing angle measures are:

A parallelogram's opposing angles are known as congruent angles, whilst its adjacent angles are known as supplementary angles.

So, by the property of parallelogram

m∠U = m∠S = 70°

m∠R = 180 - 70

m∠R = 110°

and, m∠T = m∠R =110°

Thus, by the properties of a parallelogram, the angle are:

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Answer:

[tex]z=0.923[/tex]

Step-by-step explanation:

we know that

A relationship between two variables, x, and z, represent an inverse variation if it can be expressed in the form [tex]z*x=k[/tex] or [tex]z=k/x[/tex]

step 1

Find the value of k

For x=6, z=2

[tex]z*x=k[/tex]

substitute

[tex]2*6=k[/tex]

[tex]k=12[/tex]

therefore

The equation of the inverse variation is equal to

[tex]z*x=12[/tex]

step 2

What is the value of z when x=13

substitute the value of x in the equation and solve for z

[tex]z*(13)=12[/tex]

[tex]z=12/13[/tex]

[tex]z=0.923[/tex]

Answer:

Surface area of the right prism =  156 square units

Step-by-step explanation:

Surface area of prism = area of 2 triangle + area of three rectangles

To find the area of triangles

Here base b = 4 units and height h = 3 units

Area of triangle = bh/2

 = (4 * 3)/2 = 6  square units

Area of 2 rectangles = 2 * 6 = 12 units

To find the area of rectangles

length of rectangle = 12 units,

Here 3 rectangles with 3 different width

width1 = √(4² + 3²) = 5 units

width2 = 4 units and width3 = 3 units

Area1 = Length * width1

 = 12 * 5 = 60 square units

Area1 = Length * width2

 = 12 * 4 = 48square units

Area1 = Length * width3

 = 12 * 3 = 36 square units

Total area of three rectangles = 60 + 48 + 36 = 144

To find the surface area of prism

Surface area = Area of triangles + area of rectangles

 = 12 + 144 = 156 square units

Answer: We can expect about 40.13% of bottles to have a volume less than 32 oz.

Step-by-step explanation:

Given : The volumes of soda in quart soda bottles can be described by a Normal model with

[tex]\mu=\text{32.3 oz}\\\\\sigma=\text{1.2 oz}[/tex]

Let X be the random variable that represents the volume of a randomly selected bottle.

z-score :[tex]\dfrac{x-\mu}{\sigma}[/tex]

For x = 32 oz

[tex]z=\dfrac{32-32.3}{1.2}=-0.25[/tex]

The probability of bottles have a volume less than 32 oz is given by :-

[tex]P(X<32)=P(z<-0.25)=0.4012937[/tex]           [Using standard normal table]

In percent, [tex]0.4012937\times100=40.12937\%\approx40.13\%[/tex]

Hence, we can expect about 40.13% of bottles to have a volume less than 32 oz.

Answer:

4 coconuts

Step-by-step explanation:

52-x/4+3+5

52-x/12

13-x/3

4

Answer:

5 coconuts

Step-by-step explanation:

If there were no loss , and 52 coconuts were distributed among 4 people, the number of coconuts each one would have obtained would be 13. Now as few coconuts are spoiled, the each one of those 4 people can get

1) 12 coconuts , and as per condition given there will be 3 more coconuts left. Hence total number of coconuts would be 12*4+3=48+3=51

Hence 1 coconuts would have been spoiled. Now let us see if this satisfies the second condition too.

If 51 coconuts are distributed equally among 3 people , each one gets 17 coconuts and none is left. But the condition says that 2 coconuts are left. Hence

Our assumption of loss of coconut was wrong.

2) 11 coconuts, as per condition given there will be 3 more coconuts left. Hence total number of coconuts would be 11*4+3=44+3=47

Hence 52-47=5 coconuts would have been spoiled. Now let us see if this satisfies the second condition too.

If 47 coconuts are distributed equally among 3 people , each one gets 15 coconuts and 2 coconuts are left. Hence it satisfies the second condition also. Let us see, if it satisfies third condition too.

If 47 coconuts are distributed among 5 people , each one gets, 9 cocnuts and coconuts left will be 2. Hence it satisfies the third condition also.

Hence , there were 5 coconuts spoiled.

Answer:

  D)  Amplitude: 2; period: π; midline: y = 1

Step-by-step explanation:

The question is much more easily answered from the graph than from the description of the graph.

The amplitude is the extent of the peak above the midline (2), or half the peak-to-peak value (4/2=2). The midline is the line halfway between the peaks (1). The period is the horizontal distance between peaks of the same polarity (π).

Answer:

Please provide the polynomial to answer this question

Step-by-step explanation:

An expression will be said a polynomial if it contains variables like x, p , q, t etc etc.

They can be of any degree. like [tex]x^2[/tex] , [tex]y^3[/tex] , [tex]t^4[/tex] etc

The degree of a polynomial is the highest  exponent of the variable we have in the polynomial

Example : Degree of polynomial [tex]x^3-1 = 3[/tex]

If it do not have any variable , it is not called a polynomial because it basically gives us a constant

The given function violates both conditions for a polynomial, the correct answer is B. This is not a polynomial function because the variable powers are not all non-negative integers, and the coefficients are not constants.

To determine if the given function is a polynomial, we need to check two things:

The variable powers should be non-negative integers (whole numbers).

The coefficients of each term should be constants.

Let's analyze the function f(x)=√x+1 · (x+2)

Variable powers: The variable powers in the function are 1/2 and 1. The power 1/2 (square root) is not a non-negative integer, and thus violates the first condition for a polynomial.

Coefficients: The coefficients in the function are (1/2) and 1, which are not constants, but rather they involve variables. This also violates the second condition for a polynomial.

Since the given function violates both conditions for a polynomial, the correct answer is:

B. This is not a polynomial function because the variable powers are not all non-negative integers, and the coefficients are not constants.

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The complete question is:

Determine if the function is a polynomial function. If the function is a polynomial function, state the degree and the leading coefficient. If the function is not a polynomial, state why.

f(x)=√x+1 · (x+2)

A.This is not a polynomial function because there is no leading coefficient.

B. This is not a polynomial function because the variable powers are not all non-negative integers.

C. This is not a polynomial function because the factors are not all linear.

D. This is not a polynomial function because it is not written in the form f(x) = axⁿ + bxⁿ⁻¹ + ..... + rx² + sx + t.

Answer:

Question 1: the slope is -6

Question 2: the first choice is the one you want

Step-by-step explanation:

For the first one, I can't tell what fraction is on the left side with the y, but it doesn't matter.  To me it looks like 1/2, but like I said, it won't change or affect our answer regarding the slope.  That number has nothing to do with the slope.  

In order to determine the slope of that line that is currently in point-slope form, we need to change it to slope-intercept form.  Another expression for slope-intercept form is to solve it for y.  Doing that:

[tex]y - \frac{1}{2}=-6x-42[/tex]

Now we can add 1/2 to both sides.  That gives us the slope-intercept form of the line:

[tex]y=-6x- \frac{83}{2}[/tex]

The form is y = mx + b, where the number in the "m" place is the slope.  Our slope is -6.

For the second one, we will sub in the x coordinate in a pair for x in the equation of the line and do the same for y to see if the left side equals the right side.  The answer is [tex](\frac{2}{9},-7)[/tex] and I'll show you why.  I will also show you how another point DOESN'T work in the equation.  Filling in 2/9 for x and -7 for y:

[tex]-7+7=-3( \frac{2}{9} -\frac{2}{9})[/tex] which simplifies to

0 = -3(0) so

0 = 0 and this is true.

The other point I am going to use in exactly the same process is (-3, -7) since it doesn't have fractions in it.  First I'm going to distribute the -3 into the parenthesis to get:

[tex]y+7= -3 x + \frac{6}{9}[/tex]

Subbing in -3 for x and -7 for y:

[tex]-7+7=-3( -3) +\frac{6}{9}[/tex]

As you can see, the left side equals 0 but the right side does not.  If the lft side doesn't equal the right side, then the expression is not true, so the point is not on the line.

Answer:

So a translation and a rotation and that is it.

D.

Step-by-step explanation:

There is no reflection.

If we had a negative in front of the x there would have been a reflection.

We can definitely tell the graph was translated 3 units up because of the +3.

There are other ways to look at a translation of a line but it has been translated for sure.

It has also been rotated because of the factor of 1/9 in front of x.

Let me show you a graph.

Comparing to y=mx+b where m is slope and b is y-intercept, we see the slope of f(x)=x is 1 and the y-intercept of f(x)=x is 0 while

the slope of g(x)=(1/9)x+3 is (1/9) and y-intercept of g(x)=(1/9)x+3 is 3.

I drew them here on the graph.

I also drew y=x+3 which has slope 1 and y-intercept 3.

So I just wanted to show you have we translate y=x up to y=x+3, you still don't have the same line.  But notice if you rotate y=x+3 using the (0,3) as the center of rotation you could get the green line to lay on the orange line.

Is this supposed to be multiple choice? If so it's a terrible question as the answer is all of the above. I'll sketch out the transformations, avoiding details.

Transform y=x to y=x/9 + 3

a. using rotation and reflection

Rotate around the origin to get y=-x/9 and reflect in y=3/2 to get y=x/9 +3

b. using translation, reflection, rotation

Translate to y = x - 3, reflect in the x axis to get y = x + 3 and rotate around (0,3) to get the result.

c. using translation and reflection.

Translate to y = x + 3 and reflect along the line which bisects the angle at (0,3).

d. rotation and translation

Rotate to a slope of 1/9 and translate up 3 units.

Answer:

  10-2x

Step-by-step explanation:

If x is the length of one side of the square, the total width of the cardboard has two squares cut from it. The amount of width remaining is the width of the box. Since the original width of the cardboard is 10, the width of the box is ...

  10 -2x

Answer:

See below.

Step-by-step explanation:

Graphically: the graphs of a function and its inverse are symmetric with respect to the line y = x.

Algebraically: If functions f(x) and g(x) are inverses of each other, then the composition of f and g must equal x, and the composition of g and f must also equal x.

Two functions are inverse of each other:

An inverse function is defined as a function, which can reverse into another function.

For example,

[tex]f(x) = 3x - 2\\\\g(x) = \frac{x+2}{3}[/tex]

Checking if g(x) and f(x) are inverse of each other.

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

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

Since, fog(x) = gof(x) = x, it is algebraically verified that f(x) and g(x) are inverse of each other.

To prove that graphically, we plot the two functions.

As can be observed the two functions are symmetric to each other across the line y = x, thus, they are inverse of each other. (To check if the two functions are symmetric of each other, pick the graph of one function, for every point, interchange the x and y coordinates and plot them. The new graph will be of the inverse function.)

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Answer:

Option D. Divide 15 by 2

Step-by-step explanation:

we know that

To find the y-coordinate of the midpoint of a vertical line, adds the y-coordinates of the endpoints and divide by two

we have the endpoints

(0,0) and (0,15)

The y-coordinate of the midpoint is (0+15)/2=15/2=7.5

therefore

Divide 15 by 2

Answer:

the answer is to count by hands ( A ) and divide 15 by 2 ( D )

Step-by-step explanation:

Answer:

  see below

Step-by-step explanation:

The vertex order has been reversed, so a reflection is involved. The direction of the short side has been changed by 90°, so a rotation is potentially involved. Depending on the precise rotation and/or reflections, translation may be involved.

One potential set of transformations is ...

Another potential set of transformations (shown below) is ...

Answer:

P: 20pi A: 400-100pi

Answer:

  (1, π/3 +2kπ), (-1, 4π/3 +2kπ) . . . where k is any integer

Step-by-step explanation:

Adding any multiple of 2π to the angle results in the same point in polar coordinates.

Adding 180° (π radians) to the point effectively negates the magnitude. As above, adding any multiple of 2π to this representation is also the same point in polar coordinates.

There are an infinite number of ways the coordinates can be written.

Answer:

All the polar coordinates of point P are [tex](1,2n\pi+\frac{\pi}{3})[/tex] and [tex](-1,(2n+1)\pi+\frac{\pi}{3})[/tex], where n is an integer.

Step-by-step explanation:

The given point is

[tex]P=(1,\frac{\pi}{3})[/tex]            .... (1)

If a point is defined as

[tex]P=(r,\theta)[/tex]          .... (2)

then the polar coordinates of point P is defined as

[tex](r,\theta)=(r,2n\pi+\theta)[/tex]

[tex](r,\theta)=(-r,(2n+1)\pi+\theta)[/tex]

where, n is an integer and θ is in radian.

From (1) and (2) we get

[tex]r=1, \theta=\frac{\pi}{3}[/tex]

So, the polar coordinates of point P are

[tex](r,\theta)=(1,2n\pi+\frac{\pi}{3})[/tex]

[tex](r,\theta)=(-1,(2n+1)\pi+\frac{\pi}{3})[/tex]

Therefore all the polar coordinates of point P are [tex](1,2n\pi+\frac{\pi}{3})[/tex] and [tex](-1,(2n+1)\pi+\frac{\pi}{3})[/tex], where n is an integer.