An open-topped box is constructed from a rectangular piece of cardboard that is twice as long as it is wide by removing a square of size 3 inches from each corner and turning up the edges.
If the box is to hold 1,620 in3, how big should the originial piece of cardboard be?

The parent functions that have an infinite number of zeroes are sine and cosine.  A zero of a function is a value of x for which the function equals zero. For example, the function f(x) = x has a zero at x = 0, because f(0) = 0.

The parent functions sine and cosine have an infinite number of zeroes because they are periodic functions, meaning they repeat their values at regular intervals.

The sine function has zeroes at every multiple of π, or 0, π, 2π, 3π, etc. The cosine function has zeroes at every odd multiple of π/2, or π/2, 3π/2, 5π/2, etc.

In contrast, the other parent functions listed have a finite number of zeroes. The linear function has one zero, the quadratic function has at most two zeroes, the exponential function has at most one zero, the reciprocal function has no zeroes, the absolute value function has at most one zero, and the square root function has at most one zero.

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The constraints should also include the fact that the total donated StandardSoap must equal 500L

Q1: You should remove the constraints related to the purchasing limits from your LP, such as the limit of 4000L of lotion, 5000L of soap, and 750L of SpecialX.

Q2: In Excel, your objective function should be to maximize the monthly profit, which can be determined by subtracting the total costs of producing Moisturizing+ and StandardSoap from the total revenue of the hand sanitizer sold on the market and donated to charity. Your decision variables should be the amount of Moisturizing+ and StandardSoap you produce. Your constraints should include the relationships between the amounts of lotion, soap, and SpecialX used to make Moisturizing+ and StandardSoap. In addition, your constraints should also include the fact that the total donated StandardSoap must equal 500L.

Solver's response will suggest an optimal production mix for maximizing the monthly profit. It makes sense to follow Solver's suggested production mix if it meets your company's requirements and goals.

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For Part 1 of 2 (a): The exact distance between the points (2,73) and (-22,3) is 74.
For Part 2 of 2 (b): The midpoint of the line segment whose endpoints are the given points is (−10,38).

(a) The exact distance between the points is found using the distance formula:

d = √[(x2 - x1)² + (y2 - y1)²]

Where (x1, y1) and (x2, y2) are the coordinates of the given points. Plugging in the values from the given points:

d = √[(-22 - 2)² + (3 - 73)²]

Simplifying:

d = √[(-24)² + (-70)²]

d = √[576 + 4900]

d = √[5476]

d = 74

Therefore, the exact distance between the points is 74.


(b) The midpoint of the line segment whose endpoints are the given points is found using the midpoint formula:

M = [(x1 + x2)/2, (y1 + y2)/2]

Where (x1, y1) and (x2, y2) are the coordinates of the given points. Plugging in the values from the given points:

M = [(2 + (-22))/2, (73 + 3)/2]

Simplifying:

M = [(-20)/2, (76)/2]

M = [-10, 38]

Therefore, the midpoint is (-10, 38).

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Answer:

  10.81 ft

Step-by-step explanation:

You want the length of the shortest ladder that will reach over a 2 ft high fence to reach a building 6 ft from the fence.

Relevant trig relations are ...

  Sin = Opposite/Hypotenuse

  Cos = Adjacent/Hypotenuse

In the attached diagram, the ladder is show as line segment CD, intersecting the top of the fence at point B. The length of the ladder is the sum of segment lengths BD and CB.

Using the above trig relations, we can write expressions that let us find these lengths in terms of the angle at D:

  sin(D) = AB/BD   ⇒   BD = AB/sin(D)

  cos(D) = BG/CB   ⇒   CB = BG/cos(D)

Then the ladder length is ...

  CD = BC +CB = AB/sin(D) +BG/cos(D)

  CD = 2/sin(D) +6/cos(D)

The minimum can be found by differentiating the length with respect to the angle. This lets us find the angle that gives the minimum length.

  CD' = -2cos(D)/sin²(D) +6sin(D)/cos²(D)

  CD' = 0 = (6sin³(D) -2cos³(D))/(sin²(D)cos²(D)) . . . common denominator

  0 = 3sin³(D) -cos³(D) . . . . the numerator must be zero

Factoring the difference of cubes, we have ...

  0 = (∛3·sin(D) -cos(D))·(∛9·sin²(D) +∛3·sin(D)cos(D) +cos²(D))

The second factor is always positive, so the value of D can be found from

  ∛3·sin(D) = cos(D)

  D = arctan(1/∛3) . . . . . . . divide by ∛3·cos(D), take inverse tangent

  D ≈ 37.736°

  CD = 2/sin(37.736°) +6/cos(37.736°) = 3.51 +7.30 = 10.81 . . . feet

The shortest ladder that reaches over the fence to the building is 10.81 feet.

__

Additional comments

The second attachment shows a graphing calculator solution to finding the minimum of the length versus angle in degrees.

The ladder length can also be found in terms of the distance AD.

  L = BD(1 +BG/AD) = (1 +6/AD)√(4+AD²)

The minimum L is found when AD=∛(BG·AD²) = ∛24 ≈ 2.884.

Answer:

x=10

Step-by-step explanation:

solve for x: 1/2x+4=9

First, you need to get the variable x to be on a side of the equation, by itself. That is your goal throughout the problem. The easiest way to start this is by removing 4 from the left side of the equation. We know that before the number 4 is a plus sign. In order to remove 4 from the left side of the equation, we must do the opposite of the plus sign. The plus sign represents addition, and the opposite of addition is subtraction. This means we need to subtract 4 from the left side. Remember, that whatever we do to one side, we must do to the other side! This means we must subtract 4 on BOTH sides of the equation.

On the left side of the equation, we have 4 and we subtract it by 4.

4-4=0

Since the answer is 0 we can forget about the number, because it does not hold any value.

Next, on the right side of the equation we have 9, and we subtract it by 4, as well because whatever we do to one side, we must do to the other side.

9-4=5

Our new equation should be: 1/2x=5

Our last step is to continue our goal from the beginning of the problem; to get x alone on one side of the equation. In order to complete that goal, we must remove 1/2 from the left side of the equation. Remember when we subtracted 4 one each side, we took the opposite of the sign. In this equation, we don't necessarily see a sign like before. However, whenever x is directly beside another number, that means that it is being multiplied by that number. Just like how the opposite of addition is subtraction, the opposite of multiplication is division. We need to divide 1/2 on both sides.

On the left side, we divide 1/2x by 1/2.

1/2 divided by 1/2 is 10

One isn't necessary to keep in the equation ONLY if it is next to x, like in this case.

Lastly, whatever we do to one side, we must do to the other. We finish the problem by dividing 5 by 1/2.

5 divided by .5 is 10

We are left with x=10.

The answer is 10.

Answer:

We can use the standard normal distribution to solve this problem, by standardizing the salary value of $53,000 using the given mean and standard deviation:

z = (X - μ) / σ

where X is the salary value of $53,000, μ is the mean of $40,756, and σ is the standard deviation of $7500.

z = (53,000 - 40,756) / 7500 = 1.63547

Using a standard normal distribution table or calculator, we can find the proportion of employees who earn at most $53,000 by finding the area to the left of the standardized value of 1.63547:

P(Z ≤ 1.63547) = 0.9514

Therefore, approximately 95.14% of employees in entry-level positions at the company earn at most $53,000.

Answer:

Below

Step-by-step explanation:

p^2 + 7mp + 4 - (-2p^2 - mp +1) =

(p^2 + 2p^2)  + ( 7mp  +   mp)  + ( 4   -  1)     <==== two errors made here

3p^2            + 8mp     + 3  

Answer:

The correct answer is 4th degree polynomial.

According to the model, the expected salary for a CEO of a firm with $5,000,000 in sales and $20,000,000 in market value is $2,178,357 or between $1,139,522 and $4,056,537.  Therefore, asking for $500,000 as an annual salary would be significantly lower than what the model predicts.

According to the model from part (a), the equation for the logarithm of annual salary is:

log(salary) = 4.62 + 0.16 sales + 0.11 log(mktval) + u

where u is the error term. This model has a multiple R-squared of 0.30, which means that it explains 30% of the variation in salaries based on sales and market value.

log(salary) = 4.62 + 0.16 x log(500000) + 0.11 x log(20000000)

log(salary) = 4.62 + 0.16 x 13.122 + 0.11 x 16.811

log(salary) = 7.625

salary = exp(7.625)

salary = $2,178,357

We can also use the coefficients from the model to calculate the expected salary directly, without taking logarithms.

salary = exp(β0) x sales^β1 x mktval^β2 x e^u

salary = exp(4.62) x 5,000,000^0.16 x 20,000,000^0.11 x e^u

salary = $2,178,357 x e^u

Using this assumption, we can calculate a 95% confidence interval for the expected salary:

log(salary) = 7.625

standard error = 535.9 x sqrt(0.16^2 + 0.11^2) = 136.6

95% confidence interval = exp(log(salary) ± 1.96 x standard error)

95% confidence interval = $1,139,522 to $4,056,537

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The solution of the given quadratic system above would be = 6 , -10 for X and y respectively. That is option B.

To calculate the value of x and y substitution method should be used.

3x + y= 8 ---> equation 1

y=-x² + 3x + 8 ---> equation 2

Make y the subject of formula in equation 1;

y = 8 - 3x

Substitute y = 8 - 3x into equation 2;

8 - 3x = -x² + 3x + 8

x² = 3x +3x +8 -8

x² = 6x

X = 6

Substitute X = 6 into equation 1;

3(6) + y = 8

Make y the subject of formula;

y = 8-18

y = -10

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Answer and Explanation:

The expression [tex](3y + 5) + \frac{1}{2}(3y + 5)[/tex] is equivalent to the expression [tex]1.5(3y + 5)[/tex] because of the distributive property and the equivalence of [tex]1 \frac{1}{2}[/tex] and [tex]1.5[/tex].

In other words:

We know that

[tex]1 + \frac{1}{2} = 1 \frac{1}{2} = 1.5[/tex],

and that

[tex](B + C)(A) = BA + CA[/tex].

Therefore,

[tex]1A + \frac{1}{2}A = 1 \frac{1}{2}A = 1.5A[/tex]

and

[tex](3y + 5) + \frac{1}{2}(3y + 5) = 1.5(3y + 5)[/tex].

To simplify [tex]1.5(3y + 5)[/tex], we can distribute the 1.5.

[tex]1.5(3y + 5) = (1.5) (3y) + (1.5)(5)[/tex]

[tex]= \boxed{4.5y + 7.5}[/tex]

Answer:$168 per hour

Step-by-step explanation: so in a half hour they charge $84 .

a half hour is 30 minutes and you need another half hour so double $84

$84x2=168

Mateo takes 20 minutes to run around the track once.

The rate at which an object goes from one location to another is known as its speed. It is described as the distance that an object covers in a certain amount of time. The distance traveled in one unit of time is typically used to express speed, such as meters per second (m/s), kilometers per hour (km/h), or miles per hour (mph) (mph).

[tex]Speed = \frac{distance}{time }[/tex]

Here we assume that the distance around the track is 'd', Adrian's speed is 'a' (in distance per minute) and Mateo's speed is 'm' (also in distance per minute).

Since Adrian takes 15 minutes to run around the track once, we have:

a = d/15

Let's use the formula: time = distance/speed

to write two equations for the total distance traveled by each boy.

For Adrian: 2d = a × 30

For Mateo: 2d = m × t

where 't' is the time it takes Mateo to run around the track once.

Since we know that Mateo runs faster than Adrian, we have:

m > a

Substituting the expression we found for 'a' into the equation for Adrian, we get:

2d = (d/15) × 30

d = 225

Now, we can solve for Mateo's speed using the equation for the total distance traveled by Mateo:

2d = m × t

2(225) = m × t

m = 450/t

Substituting this expression for 'm' into the inequality m > a, we get:

450/t > d/15

15×450 > d×t

6750 > d×t

Substituting d = 225, we get:

6750 > 225t

t < 30

Since Mateo takes less than 30 minutes to run around the track once, and we know that Adrian takes 15 minutes, the possible value for t is 20 minutes: t = 20.

Therefore, Mateo takes 20 minutes to run around the track once.

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We can rewrite y=[tex]5e^{-0.7t}[/tex] using the properties of exponents as follows:

y=[tex]5e^{-0.7t} =5(e^{-0.7)t} =5(\frac{1}{e^{0.7} })t[/tex]

We can recognize [tex]\frac{1}{e^{0.7} }[/tex] as a base for exponential function that can be written in the form 1 ±r We know that is an increasing function, so [tex]e^{0.7}[/tex]≥1

therefore [tex]\frac{1}{e^{0.7\\} }[/tex]≥1 which means we must use the form y=a[tex](1-r^){t}[/tex]

To find the percent rate of change, we can use the formula:

percent rate of change=|r|×100

So, the percent rate of change is:

=|0.503|×100

Rounding to the nearest tenth of a percent, we get a percent rate of change of approximately 50.3%.

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The living room approximately 16 feet long based on the information provided.

A closed, such a double object known as a square has four sides that are equal and four vertices. On either side, it has parallel sides. A rectangle with equal width and length is also comparable to a square.

You may determine a square's area by multiplying one side by the square itself. As a result, we can use the scale factor of the living room's size to determine its length:

√256 ft² = 16 ft

The living room becomes 16 feet long as a result.

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Var(θ3) = (a-1)(a-2) / λ^2 - θ^2.

First, let's find the method of moments estimate of θ based on Y1,...,Yn. We know that E(Y) = E(log(X/x0)) = E(log(X)) - log(x0) = θ^-1 - log(x0). Therefore, θ^-1 = E(Y) + log(x0) and θ3 = 1 / (E(Y) + log(x0)).

Next, let's find the distribution of Y. Since Y = log(X/x0), we can use the change of variables formula to find the density function of Y. Let g(y) = x0 * exp(y), then the Jacobian is |g'(y)| = x0 * exp(y). The density function of Y is fY(y) = fX(g(y)) * |g'(y)| = θ * (x0 * exp(y))^θ * (x0 * exp(y))^(-θ-1) * x0 * exp(y) = θ * x0^θ * exp(-θy).

Finally, let's find the mean and variance of θ3. We know that E(θ3) = E(1 / (E(Y) + log(x0))) = 1 / (E(Y) + log(x0)) = θ. To find the variance, we can use the fact that Var(θ3) = E(θ3^2) - E(θ3)^2. We can use the fact that if U follows a gamma distribution, then E(U^r) = Γ(a+r) / [λ^r * Γ(a)] to find E(θ3^2). Let U = E(Y) + log(x0), then E(θ3^2) = E(U^-2) = Γ(a-2) / [λ^2 * Γ(a)] = (a-1)(a-2) / λ^2. Therefore, Var(θ3) = (a-1)(a-2) / λ^2 - θ^2.

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Answer:

Step-by-step explanation:

OK so 54318+21298=75616.

So i divided 75600 by 2 and got 37800 as an answer. So that means that 37800+37800=75600.

The quotient (2x^3+ 6x²- 6x + 2)divided by (2x-3) is: [tex]x^2 + 4x + 3[/tex].

Quotients are often used in algebra to solve equations and expressions involving division. They are used in everyday life such as when calculating the average speed of a journey.

[tex]x^2 + 4x + 3[/tex]

[tex]---------[/tex]

[tex]2x - 3 | 2x^3 + 6x^2 - 6x + 2 \\ - (2x^3 - 3x^2)[/tex]

    [tex]----------[/tex]

         [tex]9x^2 - 6x \\- (9x^2 - 13x)[/tex]

             [tex]-----------[/tex]

                   [tex]7x + 2[/tex]

The quotient is:

[tex]x^2 + 4x + 3[/tex]

Therefore, (2x^3 + 6x² - 6x + 2) divided by (2x-3) is equal to (x^2 + 4x + 3) with a remainder of (7x + 2).

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The nearest 5 integers of 562817​ are:

562815​, 562816​, 562817​, 562818​, 5628179

An integer comprises a whole number which can be positive, negative, and zero and is not a fraction. Integer examples include: -5, 1, 25, 18, 97, as well as 3,043. 1.43, 21 3/4, 3.84, and other numbers that don't constitute integers are some examples.

Thus,

The nearest 5 integers of 562817​ are obtained by subtracting and adding the smallest integer that is 1 from the given number.

562815​, 562816​, 562817​, 562818​, 5628179

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The Highest Common Factor is: [tex]{6x^2yz^3}[/tex].

What is HCF ?

HCF stands for "Highest Common Factor", which is the largest number that divides two or more integers without leaving a remainder. It is also known as GCD (Greatest Common Divisor).

The given expressions are:

[tex]$$24x^3yz^4, \quad 30x^2y^2z^3, \quad 36x^3y^2z^3$$[/tex]

To find the HCF, we need to find the common factors of the given expressions.

We can write each expression as a product of prime factors:

[tex]$$24x^3yz^4 = 2^3 \cdot 3 \cdot x^3 \cdot y \cdot z^4$$[/tex]

[tex]$$30x^2y^2z^3 = 2 \cdot 3 \cdot 5 \cdot x^2 \cdot y^2 \cdot z^3$$[/tex]

[tex]$$36x^3y^2z^3 = 2^2 \cdot 3^2 \cdot x^3 \cdot y^2 \cdot z^3$$[/tex]

The common factors among these expressions are [tex]2, 3, $x^2$, $y$, $z^3$.[/tex]

Therefore, the Highest Common Factor is:

[tex]{6x^2yz^3}[/tex].

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Answer:

x = 60

Step-by-step explanation: The sum of the exterior angles of any polygon is always equal to 360 degrees, including for a parallelogram. This means that if you add up all the exterior angles of a parallelogram, the total will always be 360 degrees.

hence :

2x + 2x + x + x = 360, for the given figure

=> 6x = 360

dividing by 6 both sides

x = 60

Answer:

Let's first simplify the left-hand side and right-hand side of the inequality:

2 - 5(x + 1) = 2 - 5x - 5 = -5x - 3 3(x - 1) - 24 = 3x - 27

So the inequality becomes:

-5x - 3 ≥ 3x - 27

Now we can solve for x:

-5x - 3 ≥ 3x - 27 -8x ≥ -24 x ≤ 3

The solution set is all x-values less than or equal to 3. We can express this in interval notation as:

(-∞, 3]

The solution set in interval notation is (-∞, 4].

To solve the inequality 2 − 5(x + 1) ≥ 3(x − 1) − 24 and write the solution set in interval notation, we need to follow these steps:

Distribute the -5 and 3 on the left and right sides of the inequality, respectively:
2 - 5x - 5 ≥ 3x - 3 - 24

Simplify both sides of the inequality by combining like terms:
-3x - 3 ≥ 3x - 27

Add 3x to both sides of the inequality to isolate the variable on one side:
-3 ≥ 6x - 27

Add 27 to both sides of the inequality:
24 ≥ 6x

Divide both sides of the inequality by 6 to solve for x:
4 ≥ x

Write the solution set in interval notation. Since the inequality is "greater than or equal to," we use a closed bracket for the lower bound and an open bracket for the upper bound:
(-∞, 4]

Therefore, the solution set in interval notation is (-∞, 4].

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Answer:

Let's call the number of packs of skimmed milk "S" and the number of packs of full cream milk "F".

We know that the total number of packs of milk is 1350:

F + L + S = 1350

We also know that there are 150 more packs of skimmed milk than low fat milk:

S = L + 150

And we know that there are 465 packs of low fat milk:

L = 465

We can substitute L=465 into the equation S=L+150 to get:

S = 465 + 150 = 615

Now we can use the first equation to solve for F:

F + L + S = 1350

F + 465 + 615 = 1350

F = 270

Therefore, there are 270 packs of full cream milk.

Percentage error is 17.30%, it can be said that the answer given by student is correct.

The percent error is the distinction between the estimated value and the actual value in relation to the actual value.

Estimated number of balls in each bag = 86

Real number of balls in the bag = 104

Percentage error = | ( estimated value - real value / real value ) | * 100

Percentage error = | ( 86 - 104 / 104 ) | * 100

Percentage error = | ( -18 / 104 ) | * 100

Percentage error = | - 17.30 | * 100

Percentage error = 17.30%

Thus, it can be said that the answer given by student is correct.

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As a financial analyst, I should opt for an Initial cost of $125000 and after all the making and labor charges each bike is made under $225.

When a linear function is exposed to various constraints, it is maximized or reduced using the mathematical modeling technique known as linear programming. In corporate planning, industrial engineering, and other fields, this technique has proven helpful for guiding quantitative judgments.

Given, The Bici bicycle company is making a low-price ultra-light bicycle.

They have two plans,

I. Initial cost of $125000 and after all the making and labor charges

each bike is made under $225.

II. Initial cost of $100000 and after all the making and labor charges

each bike is made for under $275.

As they want bicycles to cost less we should opt for the first plan even if the initial cost is more, As the market demand is for a cycle that is low in price so it would sell more and an extra investment of $25000 won't be wasted.

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In the following question, Jordan needs 2200 square centimetres of carpet to cover the surface of the post, including the bottom.

To find the surface area of the scratching post, we need to add up the surface areas of all the sides.

The scratching post has a rectangular prism shape with dimensions of 10 cm x 10 cm x 90 cm. The bottom is also a 10 cm x 10 cm square.

So the surface area of the post, including the bottom, is:

2(10 cm x 10 cm) + 2(10 cm x 90 cm) + 2(10 cm x 10 cm) = 200 cm^2 + 1800 cm^2 + 200 cm^2 = 2200 cm^2

Therefore, Jordan needs 2200 square centimetres of carpet to cover the surface of the post, including the bottom.

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The r value is close to 0, it indicates that there is no correlation between the two variables.

The Pearson correlation coefficient (r value) is a measure of the strength of the linear relationship between two variables. It is calculated using the formula:

r = (sum of the products of the paired scores) / (square root of the sum of the squares of the first scores) * (square root of the sum of the squares of the second scores)

In this case, the two variables are traitanx and comanx2. To find the Pearson correlation between these two variables, we would need to calculate the sum of the products of the paired scores, the sum of the squares of the first scores, and the sum of the squares of the second scores. Then, we would plug these values into the formula and calculate the r value.

Without the actual data set, it is not possible to calculate the Pearson correlation between traitanx and comanx2. However, if the r value is close to 1, it indicates a strong positive correlation between the two variables. If the r value is close to -1, it indicates a strong negative correlation between the two variables. If the r value is close to 0, it indicates that there is no correlation between the two variables.

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Using multiplicative property of equations, the solution to the equation is x = 32.

The multiplicative property of equations states that if you multiply both sides of an equation by the same number, the equation will still be true.

In this case, we can use the multiplicative property to solve for x by multiplying both sides of the equation by (4/3) to cancel out the fraction on the right side of the equation.

24 = (3/4)x

(4/3)(24) = (4/3)(3/4)x

32 = x

So the solution to this equation is x = 32.

No need to further simplify your answer since the solution is already in its simplest form.

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The value of RU, considering the Triangle Midsegment Theorem, is given as follows:

RU = 35.

The value of RU is obtained applying the proportions in the context of the problem.

A proportion is applied as the Triangle Midsegment Theorem states that the midsegment of a triangle is parallel to it's base, and has the length half as long.

The length of the base is of ST = 70, hence the length of the midsegement is given as follows:

RU = 0.5ST

RU = 0.5 x 70

RU = 35.

(the length of the midsegment is half the length of the base, hence we apply the proportion multiplying by 0.5).

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The other vertices appears to be located in the second and fourth quadrants.

What is a polygon?

A polygon is a two-dimensional geometric shape that is made up of straight line segments connected end-to-end to form a closed shape.

Based on the image, it appears that we are dealing with a polygon JKLMNO on a coordinate grid.

First, we can see that the polygon is a hexagon (six-sided figure) with vertices at points J, K, L, M, N, and O. We can also see that the polygon is not a regular hexagon, since its sides are of different lengths and its angles are not all equal.

To determine the coordinates of the vertices of the polygon, we would need to know the scale and orientation of the coordinate grid. However, based on the image, we can make some general observations about the location of the vertices. For example, we can see that vertex J appears to be in the third quadrant (negative x and y values), while vertex N appears to be in the first quadrant (positive x and y values).

Therefore,  The other vertices appear to be located in the second and fourth quadrants.

To learn more about Polygon from given link.

https://brainly.com/question/23846997

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