√₂º · Va This can be transformed into a basic integral by letting Consider the indefinite integral U= x' +9 ✓ and du = 7x6 ✓dx · √x + 9 dx: Performing the substitution yields the integral​

Answer:

[tex]\frac{15}{49}[/tex]

Step-by-step explanation:

[tex]\frac{3}{14} /\frac{7}{10} =\frac{3}{14} *\frac{10}{7} = \frac{3}{7} *\frac{5}{7} =\frac{15}{49}[/tex]

Answer:

f(g(x)) = f(2x - 3) = 3(2x - 3) + 5 = 6x - 9 + 5 = 6x - 4

f(g(2)) = f(2*2 - 3) = f(4 - 3) = f(1) = 3*1 + 5 = 3 + 5 = 8

The composite function values:

f(g(x)) = 6x- 4

f(g(2)) = 8.

Any function that is created by combining two or more other functions is referred to as a composite function. To put it another way, given two functions, f, and g, the composite function of f and g, denoted as f(g(x)), is a new function that applies g to the input x before applying f to the output of g. (x).

To find the value of fg(x), we need to compute g(x) first and then substitute it into f(x).

g(x) = 2x - 3

Substitute g(x) into f(x):

f(g(x)) = 3(2x - 3) + 5

f(g(x)) = 6x - 4

Therefore, fg(x) = 6x - 4.

To find the value of fg(2), substitute 2 into fg(x):

fg(2) = 6(2) - 4

fg(2) = 8

Therefore, fg(2) = 8.

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D

Step-by-step explanation:

The correct answer is option D, which is 105, 90, 75, 60, 45.

The factors of polynomial [tex]3x^{2} +25x-18[/tex] is (3x-2)(x+9) that's why the correct option is (3x-2) which is option c.

Given a polynomial [tex]3x^{2} +25x-18[/tex].

We are required to find the factors of polynomial given as [tex]3x^{2} +25x-18[/tex].

Polynomial is a combination of algebraic terms which is formed by using algebraic operations.

Factors are those numbers which when divided gives the number whose factors they are.

[tex]3x^{2} +25x-18[/tex]

To find the factors we need to break the middle term of the polynomial so that the broken parts when multiplied gives the product of both the first term and last term.

[tex]3x^{2}[/tex]+27x-2x-18

3x(x+9)-2(x+9)

(3x-2)(x+9)

Hence the factors of polynomial [tex]3x^{2} +25x-18[/tex] is (3x-2)(x+9) that's why the correct option is (3x-2) which is option c.

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The probability of pulling a red or green card, written as a reduced fraction is 1/4

From the question, we have the following probabilities:

P(White) = 1/10

P(Pink) = 1/5

P(Green) = 1/20

P(Red) = 1/5

The probability of pulling a red or green card, written as a reduced fraction is the calculated as:

P(Red or Green card) = P(Red card) + P(Green card)

Substitute the known values in the above equation

P(Red or Green card) = 1/5 + 1/20

Express 1/5 as 4/20

P(Red or Green card) = 4/20 + 1/20

Take the LCM

P(Red or Green card) = (4+1)/20

Evaluate the sum

P(Red or Green card) = 5/20

Simplify the fraction

P(Red or Green card) = 1/4

Hence, the probability of pulling a red or green card, written as a reduced fraction is 1/4

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Answer:

x=70, y=55

Step-by-step explanation:

Since the angle "y" and 2x-15 form a straight line, that means the sum of the angles, must be 180 degrees.

So using this we can derive the equation: [tex]y+2x-15=180[/tex]

The next thing you need to know is that the sum of interior angles of a triangle is 180 degrees, so if we add all the angles, we should get 180.

So using these we can derive the equation: [tex]x+2y=180[/tex]

So, in this case we simply have a systems of equations. We can solve this by solving for x in the second equation (sum of interior angles), and plug that into the first equation.

Original Equation:

[tex]x+2y = 180[/tex]

Subtract 2y from both sides

[tex]x = 180-2y[/tex]

Now let's plug this into the first equation

[tex]y+2x-15=180[/tex]

Plug in 180-2y as x

[tex]y+2(180-2y)-15=180[/tex]

Distribute the 2

[tex]y+360-4y-15=180[/tex]

Combine like terms

[tex]-3y + 345 = 180[/tex]

Subtract 345 from both sides

[tex]-3y = -165[/tex]

Divide both sides by -3

[tex]y=55[/tex]

So we can plug this into either equation to solve for x

[tex]x+2y=180[/tex]

Substitute in 55 as y

[tex]x+2(55)=180[/tex]

[tex]x+110=180[/tex]

Subtract 110 from both sides

[tex]x=70[/tex]

Answer:

  x = 70°

  y = 55°

Step-by-step explanation:

The angle sum theorem and the definition of a linear pair can be used to write two equations in the two unknowns. Those can be solved for the angle values.

  x + y + y = 180° . . . . . . angle sum theorem

  y + (2x -15) = 180° . . . . definition of linear pair

We can use the first equation to write an expression for x that can be substituted into the second equation:

  x = 180 -2y

  y +(2(180 -2y) -15) = 180 . . . . substitute for x

  345 -3y = 180 . . . . . . . . . . . collect terms

  115 -y = 60 . . . . . . . . . . . . .divide by 3

  y = 55 . . . . . . . . . . . . . . add (y-60)

  x = 180 -2(55) = 70

The values of the variables are ...

  x = 70°

  y = 55°

  exterior angle = 125°

The contract price is $8,750,000

What is contract price?

Contract price means the amount Lao Construction Company charged the customer for total contract's execution.

We need to ascertain the percentage completion of the project first and foremost, which is the total costs incurred to date divided by the contract's total costs.

cost incurred to date=$ 1,500,000

total contract's cost=cost incurred to date+ expected future costs

total contract's cost=$1,500,000+$6,000,000

total contract's cost=$7,500,000

% completion=$1,500,000/$7,500,000

% completion=20%

gross profit recognized=(contract price*% completion)-costs incurred till date

gross profit recognized=$250,000

contract price=unknown(assume it is X)

% completion=20%

cost incurred to date=$ 1,500,000

$250,000=(20%*X)-$1,500,000

$250,000+$1,500,000=0.20X

$1,750,000=0.20X

X=$1,750,000/0.20

X=$8,750,000

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We conclude that the exponential function is:

f(x) = -1*(2)ˣ

Here we know that we have an exponential function of the form:

f(x) = a*b^x

And we know two points on the function, that are:

f(1) = -2 = a*b^1 = a*b

f(2) = -4 = a*b^2

Then we have a system of equations to solve, which is:

-2 = a*b

-4 = a*b^2

From the first equation we can solve:

-2/a = b

Replacing that in the other equation we can get:

-4 = a*(-2/a)^2 = 4/a

a = 4/-4 = -1

Now that we know the value of a, we can get the value of b:

-2/a = b

-2/-1 = 2 = b

In this way, we conclude that the exponential function is:

[tex]f(x) = -1*(2)^x[/tex]

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Since the Dilating polygon WXYZ will alter the side lengths of the polygon. The correct option is option  (b) the slope of WX is 3/4, and the length of W'X' is 15.

The coordinates in the question were:

(3,2), and  (7,5).

Then the slope of WX is:

[tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex]

So m   = [tex]\frac{{5} - 2 }{7-3} }[/tex]

    =3/4

The length of WX is calculated by:

[tex]WX = \sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1}) ^ 2}[/tex]

[tex]\sqrt{(7-3)^2 + (5-2)^2} \\\\= \sqrt{25} \\\\= 5[/tex]

Since, The scale factor is  =  3.

Hence, WX = 3 x 5 = 15

Therefore, Since the Dilating polygon WXYZ will alter the side lengths of the polygon. The correct option is option  (b) the slope of WX is 3/4, and the length of W'X' is 15.

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Answer:    slope 3/4 and length 15

Step-by-step explanation: plato

The value of c such that the function f is a probability density function is 2

The density function is given as:

f(x) = cxe^(−x^2) if x ≥ 0

f(x) = 0 if x < 0.

We start by integrating the function f(x)

∫f(x) = 1

This gives

∫ cxe^(−x^2) = 1

Next, we integrate the function using a graphing calculator.

From the graphing calculator, we have:

c/2 * (0 + 1) = 1

Evaluate the sum

c/2 * 1 = 1

Evaluate the product

c/2 = 1

Multiply both sides of the equation by 2

c = 2

Hence, the value of c such that the function f is a probability density function is 2

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Fraction is a topic that deals with expressing the relationship between two numbers or terms in the form of a ratio. So that the required symbolic language required in the question is: 17 - [tex]\frac{x}{17}[/tex] = [tex]\frac{17}{3}[/tex] + 6

Thus the value of x is 90[tex]\frac{2}{3}[/tex].

Fraction is a topic that deals with expressing the relationship between two numbers or terms in the form of a ratio. Some types of fractions are mixed fractions, proper fractions, and improper fractions.

Thus to express the given question in a symbolic language, let the fraction of 17 taken away be represented by x.

So that;

i. a fraction of 17 is taken away from 17 can be expressed as 17 - [tex]\frac{x}{17}[/tex].

ii. remains exceeds one-third of seventeen by six can be expressed as  [tex]\frac{17}{3}[/tex] + 6

Therefore the required symbolic language to the question is:

                  17 - [tex]\frac{x}{17}[/tex]  =  [tex]\frac{17}{3}[/tex] + 6

So that,

[tex]\frac{289 - x}{17}[/tex] = [tex]\frac{17 + 18}{3}[/tex]

cross multiply to have

3(289 - x) = 17(17 + 180)

867 - 3x = 595

3x = 867 - 595

    =272

x = [tex]\frac{272}{3}[/tex]

  = 90[tex]\frac{2}{3}[/tex]

x = 90[tex]\frac{2}{3}[/tex]

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a. Central angle: Angle BAC

b. A major arc is: Arc BEC

c. A minor arc is: Arc BC

d. Measure of arc BEC in circle A = 260°

e. Measure of arc BC = 100°

According to the central angle theorem the measure of central angle (i.e. angle BAC in circle A) is the same as the measure of the intercepted arc (i.e. arc BC in circle A).

Referring to the image given, a central angle (i.e. angle BAC) is formed by two radii of a circle (i.e. AB and AC in circle A), where the vertex of the angle (i.e. vertex A in circle A) is at the center of the circle.

An arc that is bigger than a semicircle (half a circle) or with a measure greater than 180 degrees is called a major arc of a circle.  

An arc that is smaller than a semicircle (half a circle) or with a measure less than 180 degrees is called a minor arc of a circle.  

a. Central angle in circle A is: ∠BAC

b. Major arc in circle A is: Arc BEC

c. Minor arc in circle A is: Arc BC.

d. Based on the central angle theorem, we have:

Measure of arc BEC in circle A = 360 - 100

Measure of arc BEC in circle A = 260°

e. m∠BAC = 100° [given]

Based on the central angle theorem, we have:

m(arc BC) = m∠BAC

Measure of arc BC = 100°

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The irrational number on set B is given by: [tex]\sqrt{5}[/tex]

Irrational numbers are numbers that cannot be represented by fractions. The two most common examples are:

In the set B given in this problem, the only number that is irrational is [tex]\sqrt{5}[/tex], as it is a non-exact root.

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The percentage of people who preferred cheesecake if 50% prefferd fruit cake 1/5 preferred sponge is 40%.

Percentage remaining = 100% - 50%

= 50%

Sponge cake = 1/5

Percentage of sponge cake = 1/5 × 50%

= 1/5 × 0.5

= 10%

Percent that likes cheesecake = Total - (fruits cake + sponge cake)

= 100% - (50% + 10%)

= 100% - (60%)

= 40%

Therefore, the percentage of people who preferred cheesecake if 50% prefferd fruit cake 1/5 preferred sponge is 40%.

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Answer: 28

Step-by-step explanation:

[tex]8+(7+1)2+4\\\\8+(8)2+4\\\\8+16+4\\\\24+4\\\\28[/tex]

Based on the residual plot given, the statement that describes if the model is good is B. The regression line is a good model because the residuals are randomly distributed.

For a model to be considered ideal or good, the residuals from the model should be randomly distributed in an equal manner around the regression line.

The residual plot shows that the residuals are randomly distributed which means that the regression line is a good model.

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Based on the calculations, the interest rate on the stock in four (4) years is equal to 7.1%.

Given the following data:

Amount borrowed (Principal) = $22,000.

Simple interest, I = $78.40.

Time = 4 year.

To determine the interest rate on the stock in four (4) years:

Mathematically, simple interest can be calculated by using this formula:

I = PRT

Where:

Making R the subject of formula, we have:

R = I/PT

Substituting the given parameters into the formula, we have;

R = 6260/(22,000 × 4)

R = 6260/(88,000)

Interest rate = 0.071 = 7.1%.

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Step-by-step explanation:

5)f(g(8))

from the table g(8)=4

f(4) =4 from the table

7)g(f(5))

from the table f(5)=0

g(0)=9 from the table

9)f(f(4))

from the table f(4)=4

f(4)=4 from the table

Answer:

AB = 6

Step-by-step explanation:

from the diagram

AB + BC = AC , that is

x + 3x = 24

4x = 24 ( divide both sides by 4 )

x = 6

then

AB = x = 6

The length of AB calculated is 6 units

What is Algebraic expression ?

Algebraic expressions are the idea of expressing numbers using letters or alphabets without specifying their actual values. The basics of algebra taught us how to express an unknown value using letters such as x, y, z, etc. These letters are called here as variables. An algebraic expression can be a combination of both variables and constants. Any value that is placed before and multiplied by a variable is a coefficient.

AC = 24

Here, from the number line the data given are :

AB = x

BC = 3x

CD = 4x-13

Now calculating length AB or x by adding AB and BC and equating to 24 :

AC = AB+BC

24 = x+3x

4x = 24

x = 24/4

x = 6

Therefore, the length of AB calculated is 6 units

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Applying implicit differentiation, it is found that dy/dt when y=π/4 is of:

a-) -√2 / 2.

Implicit differentiation is when we find the derivative of a function relative to a variable that is not in the definition of the function.

In this problem, the function is:

xcos(y) = 2.

The derivative is relative to t, applying the product rule, as follows:

[tex]\cos{y}\frac{dx}{dt} - x\sin{y}\frac{dy}{dt} = 0[/tex]

[tex]\frac{dy}{dt} = \frac{\cos{y}\frac{dx}{dt}}{x\sin{y}}[/tex]

Since dx/dt=−2, we have that:

[tex]\frac{dy}{dt} = -2\frac{\cos{y}}{x\sin{y}}[/tex]

When y = π/4, x is given by:

xcos(y) = 2.

[tex]x = \frac{2}{\cos{\frac{\pi}{4}}} = \frac{2}{\frac{\sqrt{2}}{2}} = \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = 2\sqrt{2}[/tex]

Hence:

[tex]\frac{dy}{dt} = -2\frac{\cos{y}}{x\sin{y}}[/tex]

[tex]\frac{dy}{dt} = -\frac{1}{\sqrt{2}}\cot{y}[/tex]

Since cot(pi/4) = 1, we have that:

[tex]\frac{dy}{dt} = -\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{\sqrt{2}}{2}[/tex]

Which means that option a is correct.

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Step-by-step explanation:

Divide, the same as ÷

divide 6 by 3

by is the same as the symbol

6 ÷ 3 = 2

Difference, This is another way of saying take away or minus

the difference of 6 and 3 is

6 - 3 = 3

Product, This is another word for multiply or times

the product of 2 and 3 is

2 × 3 = 6

divide (÷)  the difference of 20 and 6 (20 - 6) by the product of 7 and 2       (7 × 2)

so

(20 - 6) ÷ (7 × 2)

can you do the rest???

The solutions for the given equations are:

Writing a number or an equation as a product of its factors is said to be the factorization.

A linear equation has only one factor, a quadratic equation has 2 factors and a cubic equation has 3 factors.

1. Solving x² - 49 = 0; (quadratic equation)

⇒ x² - 7² = 0

This is in the form of a² - b². So, a² - b² = (a + b)(a - b)

⇒ (x + 7)(x - 7) =0

By the zero-product rule,

x = -7 and 7.

2. Solving 3x³ - 12x = 0

⇒ 3x(x² - 4) = 0

⇒ 3x(x² - 2²) = 0

⇒ 3x(x + 2)(x - 2) = 0

So, by the zero product rule, x = -2, 0, 2

3. Solving 12x² + 14x + 12 = 18; (quadratic equation)

⇒ 12x² + 14x + 12 - 18 = 0

⇒ 12x² + 14x - 6 = 0

⇒ 2(6x² + 7x - 3) = 0

⇒ 6x² + 9x - 2x - 3 = 0

⇒ 3x(2x + 3) - (2x + 3) = 0

⇒ (3x - 1)(2x + 3) = 0

∴ x = 1/3, -3/2

4. Solving -x³ + 22x² - 121x = 0

⇒ -x³ + 22x² - 121x = 0

⇒ -x(x² - 22x + 121) = 0

⇒ -x(x² - 11x - 11x + 121) = 0

⇒ -x(x(x - 11) - 11(x - 11)) = 0

⇒ -x(x - 11)² = 0

∴ x = 0, 11, 11

5. Solving x² - 4x = 5; (quadratic equation)

⇒ x² - 4x - 5 = 0

⇒ x² -5x + x - 5 = 0

⇒ x(x - 5) + (x - 5) = 0

⇒ (x + 1)(x - 5) =0

∴ x = -1, 5

Hence all the given equations are solved.

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Answer:

Step-by-step explanation:

To find the value of t, isolate it on one side of the equation.

[tex]\sf{\dfrac{3}{2}+t=\dfrac{1}{2}}[/tex]

First, you subtract by 3/2 from both sides.

[tex]\Longrightarrow: \sf{\dfrac{3}{2}+t-\dfrac{3}{2}=\dfrac{1}{2}-\dfrac{3}{2}}[/tex]

Solve.

1/2-3/2

1-3/2

1-3=-2

-2/2

Divide.

-2/2=-1

[tex]\Longrightarrow: \boxed{\sf{t=-1}}[/tex]

Therefore, the solution is t=-1, which is our answer.

I hope this helps, let me know if you have any questions.

STEP BY STEP EXPLANATION;

1.To solve the equation,the least common multiple of the denominators must be found.

LCM=2

Therefore,

3/2 +t =1/2

2.Each term must be multiplied by the LCM.

i.e

2(3/2)+2(t)=2(1/2)

3+2t=1

2t=1-3  ( subtracting 3 from each side of the equation)

2t=1-3

2t/2=-2/2 (dividing both sides of the equation by the co-efficient of t)

t=-1

Applying the Pythagorean theorem, the length of the radius is: 35.3 units.

According to the Pythagorean theorem, the sum of the squares of the shorter sides of a right triangle equals the square of the longest side. For example, if a and b are two smaller legs of a right triangle, and c is the longest leg (hypotenuse) of the right triangle, then the Pythagorean theorem states that:

a² + b² = c².

Given the following parameters of the right triangle:

Triangle ABC is a right triangle with angle ACB as the right angle according to the tangent theorem since segment BC is tangent to Circle A

Segment AB = 58 (hypotenuse of the right triangle)

Segment BC = 46 (small leg of the right triangle)

Radius = AC (small leg of the right triangle)

Using the Pythagorean theorem, we have:

AC = √(AB² - BC²)

AC = √(58² - 46²)

AC = 35.3 units (to the nearest tenth).

Therefore, by using the Pythagorean theorem, the length of the radius of the circle, which is segment AC, to the nearest tenth is: 35.5 units.

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Step-by-step explanation:

we know from the laws of motion that in the equation

h = 20t - 1.86t²

the gravitational acceleration is in the factor of the t² term.

h = v0t + 1/2 × g × t²

v0 being the initial velocity (20 m/s).

and we can therefore see, that the gravitational acceleration "a" on Mars is 2×1.86 = 3.72 m/s²

(a)

the velocity v after 2 seconds is (first law of motion)

v = v0 + at = 20 - 3.72×2 = 12.56 m/s

gravity pulling down, so negative acceleration.

(b)

first we need the time when the rock is at 25 m.

25 = 20t - 1.86t²

0 = -1.86t² + 20t - 25

the solution to a quadratic equation is always

x = (-b ± sqrt(b² - 4ac))/(2a)

in our case

x = t

a = -1.86

b = 20

c = -25

t = (-20 ± sqrt(400 - 4×-1.86×-25))/(2×-1.86) =

= (-20 ± sqrt(214))/-3.72

t1 = (-20 + 14.62873884...)/-3.72 =

= 1.443887409... s ≈ 1.44 s

t2 = (-20 - 14.62873884...)/-3.72 =

= 9.308800763... s ≈ 9.31 s

that means, on its way up the rock reached 25m after

1.44 s.

on its way down the rock reached 25 m after

9.31 s.

velocity 0 (the rock came to a stop before falling back down) was after

0 = 20 - 3.72t

3.72t = 20

t = 20/3.72 = 5.376344086... s

that means the rock was falling after this point back to the ground and was gaining speed again.

that accelerating phase until the rock was again at 25 m was

9.308800763... - 5.376344086... = 3.932456677... s

long

the velocity at these points was

v-up = 20 - 3.72×1.443887409... = 14.62873884... m/s ≈

≈ 14.63 m/s

v-down = 0 + 3.72×3.932456677... = 14.62873884... m/s ≈

≈ 14.63 m/s

as expected : the rock did a perfectly symmetric flight path between the two 25 m marks.

so time and speed on both sides had to be identical.

but we have proven it.

Answer:

a)  12.56 m/s

b) up:  14.63 m/s (2 d.p.)

   down:  -14.63 m/s (2 d.p.)

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{9 cm}\underline{The Constant Acceleration Equations (SUVAT)}\\\\s = displacement in m (meters)\\u = initial velocity in m s$^{-1}$ (meters per second)\\v = final velocity in m s$^{-1}$ (meters per second)\\a = acceleration in m s$^{-2}$ (meters per second per second)\\t = time in s (seconds)\\\\When using SUVAT, assume the object is modeled\\ as a particle and that acceleration is constant.\end{minipage}}[/tex]

The average gravitational acceleration on Mars is 3.721 m/s² (about 38% of that of Earth).  The fact that the rock is thrown from the surface of Mars rather than the surface of Earth is of no consequence when using the equations of constant acceleration (SUVAT equations) as long as acceleration is not used in the calculations.

Given:

[tex]s=20t-1.86t^2, \quad u=20, \quad t=2[/tex]

[tex]\begin{aligned}\textsf{Using }\: s&=\dfrac{1}{2}(u+v)t\\\\\implies 20(2)-1.86(2)^2 & = \dfrac{1}{2}(20+v)(2)\\\\32.56 & = 20+v\\\\v & = 32.56-20\\\\v & = 12.56\:\: \sf m/s\end{aligned}[/tex]

Therefore, the velocity of the rock after 2 s is 12.56 m/s.

Find the time when the height of the rock is 25 m:

[tex]\begin{aligned}20t-1.86t^2 & = s\\\implies 20t-1.86t^2 & = 25\\1.86t^2-20t+25 & = 0\end{aligned}[/tex]

Quadratic Formula

[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac} }{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]

Therefore:

[tex]a=1.86, \quad b=-20, \quad c=25[/tex]

[tex]\implies t=\dfrac{-(-20) \pm \sqrt{(-20)^2-4(1.86)(25)} }{2(1.86)}[/tex]

[tex]\implies t=\dfrac{20 \pm \sqrt{214}}{3.72}[/tex]

[tex]\implies t=9.308800763..., 1.443887409...[/tex]

Therefore, the height of the rock is 25 m when:

To find the velocity (v) of the rock at these times, substitute the found values of t into the equation, along with s = 25 and u 20 m/s:

[tex]\begin{aligned}\textsf{Using }\: s&=\dfrac{1}{2}(u+v)t\\\\\implies v & = \dfrac{2s}{t}-u\\\\v & = \dfrac{2(25)}{t}-20\\\\v & = \dfrac{50}{t}-20\\\\\end{aligned}[/tex]

When t = 1.443887409...s

[tex]\implies v=\dfrac{50}{1.443887409...}-20=14.63\:\: \sf m/s \: \:(2\:d.p.)[/tex]

When t = 9.308800763...s

[tex]\implies v=\dfrac{50}{9.308800763...}-20=-14.63\:\: \sf m/s \: \:(2\:d.p.)[/tex]

Therefore, the velocity of the rock when its height is 25 m is:

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Using the triangle midsegment theorem, the length of AC in the given triangle is: 40 units.

The midsegment of a triangle can be defined as the line segment that intersects two sides of a triangle at their midpoints. This means that, the sides they intersect is bisected forming two equal halves.

In a typical triangle, there are three midsegments in the triangle. For example, in the image given in the attachment below, the midsegments of the triangle are: DF, FB, and BD. All midsegments are parallel to the third sides of a triangle.

What is the Triangle Midsegment Theorem?

According to the triangle midsegment theorem, the length of the midsegment (i,e. DF) is parallel to the third side (i.e. AC) and also half the length of the third side (AC).

We are given the following:

EC = 30

DF = 20

Applying the triangle midsegment theorem, we have:

DF = 1/2(AC)

Substitute

20 = 1/2(AC)

2(20) = AC

40 = AC

AC = 40 units.

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The number of calories per ounce of soda is 10

The given parameters are:

Calories = 50

Ounces = 5

Let the number of calories be y and the ounces be x.

So, we have:

y = kx

Substitute y = 50 and x = 5

50 = 5k

Divide by 5

k = 10

Substitute k = 10 in y = kx

y = 10x

See attachment for the graph of the relationship between the number of calories and the number of ounces

In (a), we have:

k = 10

This means that the number of calories per ounce of soda is 10

The unit rate and the slope represent the same and they have the same value

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Benjamin is correct about the diameter being perpendicular to each other and the points connected around the circle.

The steps involved in inscribing a square in a circle include;

Alicia deductions were;

Draws two diameters and connects the points where the diameters intersect the circle, in order, around the circle

Benjamin's deductions;

The diameters must be perpendicular to each other. Then connect the points, in order, around the circle

Caleb's deduction;

No need to draw the second diameter. A triangle when inscribed in a semicircle is a right triangle, forms semicircles, one in each semicircle. Together the two triangles will make a square.

It can be concluded from their different postulations that Benjamin is correct because the diameter must be perpendicular to each other and the points connected around the circle to form a square.

Thus, Benjamin is correct about the diameter being perpendicular to each other and the points connected around the circle.

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