Which steps should be taken to calculate the volume of the prism?Check all that apply

Answer:

Minimum Unit Cost = $14,362

Step-by-step explanation:

The standard form of a quadratic is given by:

ax^2 + bx + c

So for our function, we can say,

a = 0.6

b = -108

c = 19,222

We can find the vertex (x-coordinate where minimum value occurs) by the formula -b/2a

So,

-(-108)/2(0.6) = 108/1.2 = 90

Plugging this value into original function would give us the minimum (unit cost):

[tex]c(x)=0.6x^2-108x+19,222\\c(90)=0.6(90)^2-108(90)+19,222\\=14,362[/tex]

Answer:

The minimum unit cost is 14,362

Step-by-step explanation:

The minimum unit cost is given by a quadratic equation. Therefore the minimum value is at its vertex

For a quadratic function of the form

[tex]ax ^ 2 + bx + c[/tex]

the x coordinate of the vertex is

[tex]x=-\frac{b}{2a}[/tex]

In this case the equation is: [tex]c(x) = 0.6x^2-108x+19,222[/tex]

Then

[tex]a= 0.6\\b=-108\\c=19,222[/tex]

Therefore the x coordinate of the vertex is:

[tex]x=-\frac{(-108)}{2(0.6)}[/tex]

[tex]x=90[/tex]

Finally the minimum unit cost is:

[tex]c(90)=0.6(90)^2-108(90)+19,222\\\\c(90)=14,362[/tex]

Answer:

4.382 cubiti

Step-by-step explanation:

That's a simple exercise of cross-multiplication:

[tex]\frac{x}{1.95}  = \frac{1}{0.445}[/tex]

x being the measure in cubitus we're looking for. We can isolate it:

x = (1.95 m * 1 ) / (0.445 m/cubitus) = 4.382 cubiti

1.95 m = 4.382 cubiti

Which totally makes sense... since a cubitus is roughly half a meter long... and the basketball is 2 meters high... so there are roughly 4 cubiti in 2 meters.

The height of the basketball forward in cubiti is approximately 4.38.

To convert the height from meters to cubiti, we use the conversion factor provided in the question:

1 cubitus = 0.445 m

Given the height of the basketball forward is 1.95 m, we divide this value by the conversion factor to find the height in cubiti:

Height in cubiti = Height in meters / Conversion factor

Height in cubiti = 1.95 m / 0.445 m/cubitus

Now, we perform the division:

Height in cubiti ≈ 4.38 cubiti

Since the value of 1 cubitus is considered to be exact, the number of significant figures in the answer is determined by the height in meters, which is 1.95 m (three significant figures). Therefore, the answer is rounded to three significant figures as well.

The answer is:

The correct option is:

A. 10 in.

To calculate the length of the rectangle using its perimeter and one of its sides (width), we need to remember the formula to calculate the perimeter of a rectangle.

[tex]Perimeter_{rectangle}=2width+2length[/tex]

Now, we are given the following information:

[tex]Perimeter=34in\\Width=7in[/tex]

Then, substituting and calculating, we have:

[tex]Perimeter_{rectangle}=2width+2length[/tex]

[tex]34in=2*7in+2length[/tex]

[tex]34in-14in=2length\\\\2length=20in\\\\length=\frac{20in}{2}=10in[/tex]

Hence, we have that the length of the rectangle is equal to 10 inches.

So, the correct option is:

A. 10 in.

Have a nice day!

Answer:

The correct answer is option A.  10 in

Step-by-step explanation:

Points to remember

Perimeter of rectangle = 2(Length + width)

It is given that, Perimeter = 34 inches

Width = 7 inches

To find the length of rectangle

Perimeter = 2(Length + width)

34 = 2(Length + 7)

17 = Length + 7

Length = 17 - 7 = 10 inches

Therefore the length of rectangle = 10 inches

The correct answer is option A.  10 in

The probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room or both is .54

Probability is the likeliness of the occurrence of an event.

Let :

P(A) = Probability of feeling guilty about wasting food = .39

P(B) = Probability of feeling guilty about leaving lights on = .27

P(A∩B) = Probability of feeling guilty for both of these reasons = .12

The probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room or both is :

P(A∪B) = P(A) + P(B) - P(A∩B)

P(A∪B) = .39 + .27 - .12

Different Birthdays: https://brainly.com/question/7567074

Dependent or Independent Events: https://brainly.com/question/12029535

Grade: High School

Subject: Mathematics

Chapter: Probability

Keywords: Person, Probability, Outcomes, Random, Event, Room, Wasting, Food

Answer:

A shows an original investment of $200

Step-by-step explanation:

If you plug in x=0, you will get the value of the original investment

When you plug x=0 into A  you get

[tex]y=200(1.02)^{0}[/tex]

This simplifies to

[tex]y=200(1)[/tex]

And finally to

[tex]y=200[/tex]

ANSWER

Zero(s)

[tex]x = 0[/tex]

The function is discontinuous at

[tex]x = - 3 \:and \: x = 3[/tex]

EXPLANATION

The given rational function is

[tex] y = \frac{3x}{ {x}^{2} - 9 } [/tex]

For this function to be equal to zero, then the numerator must be zero.

Equate the numerator to zero and solve for x.

[tex]3x = 0[/tex]

This implies that

[tex]x = \frac{0}{3} = 0[/tex]

The rational function is discontinuous when the denominator is equal to zero.

[tex] {x}^{2} - 9 = 0[/tex]

Solve this quadratic equation using the square root method or otherwise.

[tex] {x}^{2} = \pm \sqrt{9} [/tex]

[tex]{x} = \pm 3[/tex]

There is discontinuity at

[tex]x = - 3 \:and \: x = 3[/tex]

Answer:

about 10.06 oz.

Step-by-step explanation:

Let x represent the number of ounces of pure gold you need to add. Then the amount of gold in the mix is ...

100%·x + 75%·5 = 91.7%·(x+5)

8.3%·x = 5·16.7% . . . . . . subtract 91.7%·x +75%·5

x = 5 · 16.7/8.3 . . . . . . . . divide by the coefficient of x

x ≈ 10.06 . . . . oz

_____

Alternate solution

The amount of non-gold in the given material is 25%·5 oz = 1.25 oz. That is allowed to be 8.3% of the final mix, so the weight of the final mix will be ...

(1.25 oz)/0.083 ≈ 15.06 oz

Since that weight will include the 5 oz you already have, the amount of pure gold added must be ...

15.06 oz - 5 oz = 10.06 oz

_____

Comment on these answers

If you work directly with carats instead of percentages, you find the amount of pure gold you need to add is 10.00 ounces, double the amount you have.

Answer:

36

Step-by-step explanation:

⇒The question is on third quartile

⇒To find the third quartile we calculate the median of the upper half of the data

Arrange the data in an increasing order

20, 21, 24, 25, 28, 29, 35, 37, 42

Locate the median, the center value

20, 21, 24, 25, 28, 29, 35, 37, 42

The values 20, 21, 24, 25 ------------lower half used in finding first quartile Q1

The value 28 is the median

The vlaues 29, 35, 37, 42...............upper half used in finding 3rd quartile Q3

Finding third quartile Q3= median of the upper half

upper half= 29,35,37,42

median =( 35+37)/2 = 36

Answer:

[tex]N= 4543[/tex] Labrador retrievers

Step-by-step explanation:

We know that the mean [tex]\mu[/tex] is:

[tex]\mu = 62.5[/tex]

and the standard deviation [tex]\sigma[/tex] is:

[tex]\sigma=2.5[/tex]

The probability that a randomly selected Labrador retriever weighs less than 65 pounds is:

[tex]P(X<65)[/tex]

We calculate the Z-score for X =65

[tex]Z = \frac{X-\mu}{\sigma}\\\\Z =\frac{65-62.5}{65}=1[/tex]

So

[tex]P(X<65) = P(Z<1)[/tex]

Looking in the table for the standard normal distribution we have to:

[tex]P(Z<1) =0.8413[/tex].

Finally the amount N of Labrador retrievers that weigh less than 65 pounds is:

[tex]N = P(X<65) *5400[/tex]

[tex]N = 0.8413*5400[/tex]

[tex]N= 4543[/tex] Labrador retrievers

Taking the cubic root of a number is the same as raising that number to the power of 1/3.

Moreover, we have

[tex]64 = 2^6[/tex]

So, we have

[tex]\sqrt[3]{64} = \sqrt[3]{2^6} = (2^6)^{\frac{1}{3}} = 2^{6\cdot\frac{1}{3}} = 2^2 = 4 [/tex]

Answer:

4

Step-by-step explanation:

Since we see a cube root, we will attempt to rewrite 64 as a number with an exponent of 3.

[tex]\sqrt[3]{64}[/tex]

[tex]= \sqrt[3]{4^3}[/tex]

[tex]= 4 [/tex]

Answer: First option.

Step-by-step explanation:

You know that the meaning of the symbol of the inequality "<" is: Less than.

So, you can check each option to find the number that could be a length of this piece of wood.

Given [tex]w<5[/tex], you can substitute each number given in the options into this inequality. Then:

[tex]1)\ w<5\\\\4.5<5\ (This\ is\ true)[/tex]

[tex]2)\ w<5\\\\6<5 (This\ is\ not\ true)[/tex]

[tex]3)\ w<5\\\\11.3<5\ (This\ is\ not\ true)[/tex]

[tex]4)\ w<5\\\\13<5\ (This\ is\ not\ true)[/tex]

Therefore, a lenght of the piece of wood could be 4.5

Answer:

  (n -13)/(n -7)

Step-by-step explanation:

Simplify the fraction on the left, then add the two fractions.

[tex]\displaystyle\frac{n^2-10n+24}{n^2-13n+42}-\frac{9}{n-7}=\frac{(n-6)(n-4)}{(n-6)(n-7)}-\frac{9}{n-7}\\\\=\frac{n-4}{n-7}-\frac{9}{n-7}\\\\=\frac{n-4-9}{n-7}\\\\=\frac{n-13}{n-7}[/tex]

_____

Comment on the graph

The vertical asymptote tells you the simplified form has one zero in the denominator at x=7. That is, the denominator is x-7.

The x-intercept at 13 tells you that x-13 is a factor of the numerator.

The horizontal asymptote at y=1 tells you there is no vertical scaling, so the simplest form is ...

  (n -13)/(n -7)

The hole at x=6 is a result of the factor (x-6) that is cancelled from the first fraction in the original expression. At that value of x, the fraction is undefined. So, the above solution should come with the restriction x ≠ 6.

The answer is:

The difference between the circle and the square is:

[tex]Difference=4\pi -8[/tex]

To solve the problem, we need to find the area of the circle and the area of the square, and then, subtract them.

For the square we have:

[tex]side=2\sqrt{2}[/tex]

We can calculate the diagonal of a square using the following formula:

[tex]diagonal=side*\sqrt{2}[/tex]

So,

[tex]diagonal=2\sqrt{2}*\sqrt{2}=2*(\sqrt{2})^{2}=2*2=4units[/tex]

The area will be:

[tex]Area_{square}=side^{2}= (2\sqrt{2})^{2} =4*2=8units^{2}[/tex]

For the circle we have:

[tex]radius=\frac{4units}{2}=2units[/tex]

The area will be:

[tex]Area_{Circle}=\pi *radius^{2}=\pi *2^{2}=\pi *4=4\pi units^{2}[/tex]

[tex]Area_{Circle}=4\pi units^{2}[/tex]

Then, the difference will be:

[tex]Difference=Area_{Circle}-Area{Square}=4\pi -8[/tex]

Have a nice day!

ANSWER

[tex]4\pi - 8[/tex]

EXPLANATION

The diagonal of the square can be found

using Pythagoras Theorem.

[tex] {d}^{2} = {(2 \sqrt{2} )}^{2} + {(2 \sqrt{2} )}^{2} [/tex]

[tex]{d}^{2} = 4 \times 2+ 4 \times 2[/tex]

[tex]{d}^{2} = 8+ 8[/tex]

[tex]{d}^{2} = 16[/tex]

Take positive square root

[tex]d = \sqrt{16} = 4[/tex]

The radius is half the diagonal because the diagonal formed the diameter of the circle.

Hence r=2 units.

Area of circle is

[tex]\pi {r}^{2} =\pi \times {2}^{2} = 4\pi[/tex]

The area of the square is

[tex] {l}^{2} = {(2 \sqrt{2)} }^{2} = 4 \times 2 = 8[/tex]

The difference in area is

[tex]4\pi - 8[/tex]

Answer:

If you are looking for the dimensions of the playground, they are that the width is 4 yards and the length is 16 yards

Step-by-step explanation:

We need to know 2 things here:  first, the area of a rectangle which is A = l×w,

and then we need to know how to express one dimension in terms of the other, since we have way too many unknowns right now to solve for anything!

We are told that the length is 4 times the width, so if the width is "w", then the length is "4w".  We know the area is 64, so let's sub in those values where they belong in the area formula:

64 = 4w(w).  Multiplying to simplify we get

[tex]64=4w^2[/tex]

The easiest way to do this is to divide both sides by 4 to get

[tex]16=w^2[/tex]

and when you take the square root of 16 you get 4 and -4.  However, the two things in math that will never ever be negative are distance measurements and time.  So the -4 won't do.  That means that w = 4.  If that be the case, and the length is 4 times the width, then the length is 16.  And there you go!

Answer:

2947 lb

Step-by-step explanation:

Find the volume of the sphere

v=4/3 ×pi×r³

r=2ft and pi=3.14

v=4/3 × 3.14×2³

v=33.49 ft³

Given that;

Density ⇒ 88 lb/ft³

Volume⇒33.49 ft³

Mass=?-------------------------------------find the mass

But we know density=mass/volume -----so mass=density × volume

Mass= 88×33.49 =2947.41 pounds

                           ⇒2947 lb

Explanation:

Multiply it out.

n^2 -n -(n^2 +5n+6)

= -6n -6

= -6(n +1)

For any integer value of n, this is divisible by 6. (The quotient is -(n+1).)

Part A

Yes, triangle ABC and triangle APQ are similar because of Angle-Angle similarity.

Angle BAC is congruent to Angle PAQ because of reflexive property (they share the same angle).

It is given that Segment BC is parallel to Segment PQ, so Angle ABC is congruent to Angle APQ because the corresponding angles postulate.

Part B

Segment PQ corresponds to Segment BC because they are parallel to each other.

Part C

Angle APQ corresponds to Angle B because of the corresponding angles postulate.

The volume of the solid generated by revolving the plane region about the y-axis is approximately 35,929.77 cubic units.

Here,

To use the shell method to find the volume of the solid generated by revolving the plane region bounded by the curves [tex]y = x^{(5/2)}, y = 32[/tex],

and x = 0 about the y-axis, we need to integrate the circumference of cylindrical shells along the y-axis.

The volume V can be expressed as the integral of the circumference of the cylindrical shells from y = 0 to y = 32:

V = ∫[0 to 32] 2π * x * h(y) dy

where h(y) represents the height (or thickness) of each shell, and x is the distance from the y-axis to the curve [tex]y = x^{(5/2)[/tex].

To find h(y), we need to express x in terms of y by rearranging the equation [tex]y = x^{(5/2)[/tex]:

[tex]x = y^{(2/5)[/tex]

Now, we can express the volume integral:

V = ∫[0 to 32] 2π * [tex]y^{(2/5)[/tex] * (32 - y) dy

Now, we'll evaluate the integral:

V = 2π ∫[0 to 32] ([tex]32y^{(2/5)} - y^{(7/5)[/tex]) dy

Integrate each term separately:

[tex]V = 2\pi [(32 * (5/7) * y^{(7/5)}) - (5/12) * y^{(12/5)}] | [0 to 32]\\V = 2\pi [(32 * (5/7) * (32)^{(7/5)}) - (5/12) * (32)^{(12/5)}] - [0][/tex]

Now, evaluate the expression:

[tex]V = 2\pi [(32 * (5/7) * 2^7) - (5/12) * 2^{12}][/tex]

V = 2π [(32 * 1280/7) - (5/12) * 4096]

V = 2π [81920/7 - 341.33]

V ≈ 2π * 81920/7 - 2π * 341.33

V ≈ 36608π - 678.13

The volume of the solid generated by revolving the plane region about the y-axis is approximately 35,929.77 cubic units.

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Final answer:

The shell method is used to evaluate the volume of a solid created by revolving a region around the y-axis using a vertical shell element, integrating from x=0 to the x value corresponding to y=32.

Explanation:

To use the shell method to find the volume of the solid generated by revolving the given plane region about the y-axis, we consider a vertical element or 'shell' at a certain x-value with thickness dx. Given the equations [tex]x^{5/2}[/tex], y = 32, and x = 0, these will be the bounds for our region.

The volume of each infinitesimal shell with radius x and height [tex](32 - x^{5/2})[/tex], when revolved around the y-axis, is [tex]2πx(32 -x^{5/2})dx[/tex]. To find the total volume, we integrate this expression from x=0 to where y is 32, which corresponds to the x value where [tex]x^{5/2} = 32[/tex].

Using the substitution [tex]x^{5/2}[/tex] to solve for dx, we get the integral in terms of y, which simplifies the computation. Finally, we evaluate the definite integral to find the volume of the solid of revolution.

Answer:

  64

Step-by-step explanation:

The value of a+b+c is the value of the expression when x=1:

  (3+5)^2 = 8^2 = 64

Answer:

it is 27 because of my cacculations

Step-by-step explanation:

you would fist subtract your numbers and thne get rid of the 5

Answer:

27

Hope It Helps

Answer:

x = 1.5

Step-by-step explanation:

The left side of the equation is graphed as a straight line with a slope of -5/3 and a y-intercept of +7. The right side of the equation is graphed as a horizontal line at y = 4.5. The point of intersection of these lines has the x-coordinate of the solution: x = 1.5.

The answer is 5 hope this helps

Answer:

5

Step-by-step explanation:

abs(a + b) + abs(c)

abs(-3 + 7) + abs(1)

abs(4) + abs(1)

4 + 1

5

Answer:

Find the attached

Step-by-step explanation:

We have been given the following expression;

(x-2)=5(x+1)

We are required to determine the graph of the solution set. To do this we formulate the following set of equations;

y = x - 2

y = 5(x+1)

We then graph these two equations on the same cartesian plane. The solution will be the point where these two graphs intersect.

Find the attachment below;

Answer:

{-1.75}

Step-by-step explanation:

The given equation is

[tex]x-2=5(x+1)[/tex]

Let as assume f(x) be left hand side and g(x) be the right hand side.

[tex]f(x)=x-2[/tex]

[tex]g(x)=5(x+1)[/tex]

The solution set of given equation is the intersection point of f(x) and g(x).

Table of values are:

For f(x)                    For g(x)

x      f(x)                  x          g(x)

0      -2                   0           5

2       0                    -1          0

Plot these corresponding ordered pairs on a coordinate plan and connect them by straight lines

From the below graph it is clear that the intersection point of f(x) and g(x) is (-1.75,-3.75).

Therefore, the solution set of given equation is {-1.75}.

Answer:

The numbers are 14 and 25

Step-by-step explanation:

Let one of the numbers be x. The second number we are told is 3 less than twice x. The second number will thus be;

2x - 3

The sum of the numbers is thus;

x + (2x - 3) = 3x - 3

But the sum of the numbers is said to be 39, therefore;

3x - 3 = 39

3x = 42

x = 14

The second number is thus;

2(14) - 3 = 28 - 3 = 25

The set of equations representing the word problem is ( x + y = 39 ) and ( x = 2y - 3 )

The correct answer is option

a).( x + y = 39 ) and ( x = 2y - 3 )

To solve this problem, let's first understand the given information.

Let's denote x as the number of one type of fruit (let's say apples) and y as the number of another type of fruit (let's say oranges). The problem states two conditions:

1. The total number of fruits is 39.

2. The number of apples (x) is either three less than twice the number of oranges (y) or it's the same as three more than twice the number of oranges.

Let's represent these conditions mathematically:

1. x + y = 39   (Equation 1)

2. x = 2y - 3   or   x = 2y + 3   (Equation 2)

Now, let's compare these equations with the options provided:

a) ( x + y = 39 ) and ( x = 2y - 3 )

b) ( x - y = 39 ) and ( x = 2y - 3 )

c) ( x + y = 39 ) and ( x = 3y - 2 )

d) ( x - y = 39 ) and ( x = 3y - 2 )

Comparing Equation 1 with the options, we see that options a) and c) match.

Comparing Equation 2 with the options, we see that options a) and b) match.

Therefore, the correct system of equations is option a):

(x + y = 39) and (x = 2y - 3).

1. The first equation represents the total number of fruits, which should be 39. We add the number of apples (x) and oranges (y) to get 39.

2. The second equation represents the relationship between the number of apples and oranges. The number of apples is three less than twice the number of oranges. So, we set up the equation x = 2y - 3.

Thus, the correct answer is option a)( x + y = 39 ) and ( x = 2y - 3 )

Complete question

One number is 3 less than twice another. If their sum is 39, find the numbers?

Which of the following systems of equations represents the word problem?

a) ( x + y = 39 ) and ( x = 2y - 3 )

b) ( x - y = 39 ) and ( x = 2y - 3 )

c) ( x + y = 39 ) and ( x = 3y - 2 )

d) ( x - y = 39 ) and ( x = 3y - 2 )

Answer:

2 outcomes

Step-by-step explanation:

Let's list count all the possible outcomes:

(1,3) (1,4) (1,5) (1,6)

(2,3) (2,4) (2,5) (2,6)

(3,3) (3,4) (3,5) (3,6)

As expected, there are 12 (3x4) possible outcomes.

How many outcomes do not show an even number (so showing 1 or 3) on the first spinner and show a 6 on the second spinner?

There are two cases where 6 is on the second spinner and NOT an even number on the first spinner: (1,6) and (3,6)

Answer:

The correct answer is that a possible value for n could be all numbers from 101 to 120.

Step-by-step explanation:

Ok, to solve this problem:

You have that: [tex]10 <\sqrt{n} <11[/tex]

Then, applying the properties of inequations, the power is raised by 2 on both sides of the inequation:

[tex](10)^{2} <(\sqrt{n} )^{2} <(11)^{2}[/tex]

[tex]100<n<121[/tex]

Then, a possible value for n could be all numbers from 101 to 120.

Answer:

1.6 hours

Step-by-step explanation:

I started off with T(t)=70+Ce^kt

then since the initial temp was 98.6 I did T(0)=98.6=70+C so C=28.6

Then T(1) = 80 = 28.6e^k + 70

k = ln (10/28.6)

Then plugged that into

T(t)=85=28.6e^ln(10/28.6)t + 70

and got t=.61

The answer says it is about 1.6 hours.

The time that has elapsed before the body was found is 1.5 hour

The given parameters;

Apply the Newton's method of cooling equation;

[tex]T(t) = T_{s} + (T_{o} - T_{s})e^{kt}\\\\T(t) = 70 + (98.6 - 70)e^{kt}\\\\T(t) = 70 + 28.6e^{kt}[/tex]

At the time of discovery, we have the following equation,

[tex]T_{t} = 70 + 28.6e^{kt}\\\\80 = 70 + 28.6e^{kt}\\\\10 = 28.6k^{kt}[/tex]

1 hour later, t + 1, we have the second equation;

[tex]75 = 70 + 28.6e^{kt} \\\\5 = 28.6e^{k(t+ 1)} \\\\5 = 28.6e^{kt + k} ---- (2)[/tex]

divide equation 1  by equation 2;

[tex]\frac{10}{5} = \frac{28.6e^{kt}}{28.6 e^{kt + k}} \\\\2 = e^{kt - kt - k}\\\\2 = e^{-k}\\\\-k = ln(2)\\\\k = -0.693[/tex]

The time when he dead body was discovered is calculated as;

[tex]10 = 28.6e^{kt}\\\\10= 28.6e^{-0.693t}\\\\e^{-0.693t} = \frac{10}{28.6} \\\\-0.693 t = ln(\frac{10}{28.6} )\\\\-0.693t = -1.05\\\\t = \frac{1.05}{0.693} \\\\t = 1.515 \ \\\\t \approx 1.5 \ hr[/tex]

Thus, the time that has elapsed before the body was found is 1.5 hour

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Answer:

The answer is (-5,2)

Step-by-step explanation:

So we have 2 equations and we need to solve them by substitution.

1) y = 4x + 22

2) 4x – 6y = –32

Since we already have y isolated in equation #1, we'll use that value in equation #2:

4x - 6(4x + 22) = -32

4x - 24x - 132 = -32

-20x = 100

x = -5

Then we put that value of x in the first equation:

y = 4 (-5) + 22 = -20 + 22 = 2

The answer is then (-5,2)

Answer:

(-5, 2)

Step-by-step explanation:

We have the equations:

[tex]y=4x+22[/tex] and [tex]4x-6y=-32[/tex]

Using the substitution method, with y = 4x + 22 and replace it in the equation 4x - 6y = -32

4x - 6(4x + 22) = -32

4x -24x -132 = -32

-20x = -32 + 132

x = 100/-20= -5

Substituting the value of x in the first equations of the systems to clear x.

y = 4x + 22

y = 4(-5) + 22

y= -20 + 22 = 2

Answer:

Part 1) The larger integer is 11

Part 2) The denominator is 5

Part 3) The positive integer is 4

The graph in the attached figure

Step-by-step explanation:

Part 1)

Let

x----> the smaller positive integer

y-----> the larger positive integer

we know that

[tex]x^{2} +y^{2} =185[/tex] -----> equation A

[tex]x=y-3[/tex] -----> equation B

substitute equation B in equation A and solve for y

[tex](y-3)^{2} +y^{2} =185\\ \\y^{2} -6y+9+y^{2}=185\\ \\2y^{2}-6y-176=0[/tex]

using a graphing calculator-----> solve the quadratic equation

The solution is y=11

[tex]x=11-3=8[/tex]

Part 2)

Let

x----> the numerator of the fraction

y-----> the denominator of the fraction

we know that

[tex]x=2y+1[/tex] ----> equation A

[tex]\frac{x+4}{y+4}=\frac{5}{3}[/tex] ----> equation B

substitute equation A in equation B and solve for y

[tex]\frac{2y+1+4}{y+4}=\frac{5}{3}[/tex]

[tex]\frac{2y+5}{y+4}=\frac{5}{3}\\ \\6y+15=5y+20\\ \\6y-5y=20-15\\ \\y=5[/tex]

[tex]x=2(5)+1=11[/tex]

Part 3)

Let

x----> the positive integer

we know that

[tex]x-\frac{1}{x}=\frac{15}{4}[/tex]

solve for x

[tex]x-\frac{1}{x}=\frac{15}{4}\\ \\4x^{2}-4=15x\\ \\4x^{2}-15x-4=0[/tex]

using a graphing calculator-----> solve the quadratic equation

The solution is x=4

Check the picture below.

[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ P(\stackrel{x_1}{-10}~,~\stackrel{y_1}{-2})\qquad Q(\stackrel{x_2}{4}~,~\stackrel{y_2}{6}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{4-10}{2}~~,~~\cfrac{6-2}{2} \right)\implies \left( \cfrac{-6}{2}~,~\cfrac{4}{2} \right)\implies \stackrel{\textit{center}}{(-3~,~2)} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ \stackrel{\textit{center}}{(\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})}\qquad Q(\stackrel{x_2}{4}~,~\stackrel{y_2}{6})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ \stackrel{radius}{r}=\sqrt{[4-(-3)]^2+[6-2]^2}\implies r=\sqrt{(4+3)^2+(6-2)^2} \\\\\\ r=\sqrt{49+16}\implies r=\sqrt{65} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{-3}{ h},\stackrel{2}{ k})\qquad \qquad radius=\stackrel{\sqrt{65}}{ r} \\[2em] [x-(-3)]^2+[y-2]^2=(\sqrt{65})^2\implies (x+3)^2+(y-2)^2=65[/tex]