y and x form a proportional relationship, and = 14 and y = 35 What is y when x equals 8?

14 boxes of erasers should Mr. Franklin need to buy.

A division is a process of splitting a specific amount into equal parts.

Given that Franklin wants to buy an eraser for every fourth-grade student

There are 12 erasers in each box.

There are 7 fourth-grade classes with 24 students in each class

The total number of students=7×24

=168

We need to find the number of boxes are required to buy an eraser for each student.

Since there are 12 erasers in each box, we can divide the total number of students  by 12 to find the number of boxes Mr. Franklin needs to buy:

number of boxes = total students / erasers per box

number of boxes = 168 / 12

number of boxes = 14

Hence, 14 boxes of erasers should Mr. Franklin need to buy.

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The set B' ∩ D' represents the event "the numbers do not add to 6." It contains 20 elements.

The complement of event B is B' = {2, 3, 4, 5, 7, 8, 9, 10, 11, 12}. The complement of event D is D' = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12}. Taking the intersection of their complements, we get B' ∩ D' = {2, 3, 4, 5, 7, 8, 10, 11}. This set represents the event "the numbers do not add to 6." It contains 8 elements out of the 36 possible outcomes of rolling two dice, so the probability of this event is 8/36 or 2/9.

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Answer:

-2

Step-by-step explanation:

Standard equation of a proportional relationship:

y = kx

We use the given point to find k.

4 = k × (-16)

k = 4/(-16)

k = -1/4

The equation of this proportional relationship is

y = -1/4 x

Now we use x = 8 in the equation just above to find its corresponding y value.

x = 8

y = -1/4 × 8

y = -2

Answer: -2

16%  percentage of the students have grade point averages that are greater than 2.11

Using the empirical rule, we know that for a bell-shaped distribution, approximately:

68% of the data falls within one standard deviation of the mean

95% of the data falls within two standard deviations of the mean

99.7% of the data falls within three standard deviations of the mean

To find the percentage of students with a grade point average greater than 2.11

We first need to calculate how many standard deviations away from the mean 2.11 is:

z = (2.11 - 2.56) / 0.45

= -1

This tells us that 2.11 is 1 standard deviation below the mean.

Since the distribution is symmetric.

The percentage of students with a GPA greater than 2.11 is the same as the percentage of students with a GPA less than 2.56 + 1*0.45, which is:

68% + 95% = 163%

So, approximately

100% - 163%

= 16% of the students have a GPA greater than 2.11.

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The required time hen will the rock be 9 meters from ground level is 6.40 seconds.

We can use the kinematic equations of motion to solve this problem. The equation we need is:

h = vi(t) + (1/2)at²

where h is the height of the rock above the ground at time t, vi is the initial velocity (positive when upward), a is the acceleration due to gravity (negative), and t is the time elapsed.

At the top of the cliff, the initial height h₀ = 45 meters and the initial velocity vi = 22 meters per second. When the rock is 9 meters above ground level, its height h = 9 meters. We want to find the time t at which this occurs.

First, we can use the equation of motion to find the time it takes for the rock to reach its maximum height. At the highest point, the velocity of the rock is zero, so vi = 22 m/s, a = -9.8 m/s^2, and h = h₀ + 0. We can solve for the time t1:

[tex]h=v_{i}t_{1}+\frac{1}{2}at_{1} ^{2}[/tex]

[tex]0=22t_{1}-4.9t_{1}^{2}[/tex]

[tex]t_{1} = 4.49 seconds[/tex]

Next, we can use the same equation of motion to find the time it takes for the rock to reach a height of 9 meters on the way back down. This time, the initial height is h0 = 45 - (maximum height) = 45 - 22.45 = 22.55 meters, and the initial velocity is -vi = -22 m/s. We can solve for the time t2:

[tex]h=v_{i}t_{2}+\frac{1}{2}at_{2} ^{2}[/tex]

[tex]9 = 22t_{2} + 4.9*t_{2}^{2}[/tex]

[tex]t_{2} = 1.91 seconds[/tex]

The total time for the rock to reach a height of 9 meters on the way back down is t = [tex]t_{1}+t_{2}[/tex] = 6.40 seconds (rounded to two decimal places).

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Driving without an app would have been a much different experience. Instead of having turn-by-turn directions, one would need to rely on maps and other forms of navigation, such as written directions from friends or family, or using landmarks and street signs.

Besides converting distances, there are a number of other math concepts that one might need to use when driving without an app. For example, one might need to use basic arithmetic to calculate the estimated time of arrival based on average speed and distance. One might also need to use spatial reasoning to understand the layout of a city and the relative location of different streets and landmarks. Additionally, one might need to use problem-solving skills to make decisions about which route to take and how to handle unexpected road closures or detours.

The measurement closest to the area of the paper surface of the Do-Not-Disturb sign once the circle has been cut out is 120.79 [tex]cm^2[/tex].

What is area of rectangle ?

In geometry, a rectangle is a four-sided flat shape with four right angles (90-degree angles) and opposite sides that are parallel and equal in length. The two pairs of opposite sides in a rectangle are congruent (i.e., have the same length), and the perimeter of a rectangle is the sum of the lengths of its four sides.

The area of a rectangle is a measure of the amount of space enclosed by the rectangle and is always expressed in square units. The formula for the area of a rectangle is A = length x width, where A is the area, length is the distance between the two longer sides of the rectangle (also called the length of the rectangle), and width is the distance between the two shorter sides (also called the width of the rectangle). The unit of measurement for the length and width of a rectangle must be the same for the area calculation to be correct.

The area of a rectangle is a useful measure when calculating the amount of material needed to cover a flat surface, such as the floor area of a room or the area of a piece of paper. It is also useful in solving geometry problems, such as finding the area of a composite shape made up of several rectangles or finding the dimensions of a rectangle given its area and one of its side lengths.

According to given information :

To find the area of the Do-Not-Disturb sign once the circle has been cut out, we need to subtract the area of the circle from the area of the rectangle.

The rectangle has dimensions of 17 cm by 8 cm, so its area is:

Area of rectangle = length x width = 17 cm x 8 cm = 136 [tex]cm^2[/tex]

The circle has a radius of 2.2 cm, so its area is:

Area of circle = π[tex]r^2[/tex] = π(2.2 cm)[tex]^2[/tex] ≈ 15.21 [tex]cm^2[/tex]

To find the area of the Do-Not-Disturb sign once the circle has been cut out, we need to subtract the area of the circle from the area of the rectangle:

Area of Do-Not-Disturb sign = Area of rectangle - Area of circle

= 136 [tex]cm^2[/tex] - 15.21 [tex]cm^2[/tex]

≈ 120.79 [tex]cm^2[/tex]

Therefore, the measurement closest to the area of the paper surface of the Do-Not-Disturb sign once the circle has been cut out is 120.79 [tex]cm^2[/tex].

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An expression that represents the amount of canned food collected so far by the three friends is 15x^2 + 4xy - 1.

An expression that represents the number of cans the friends still need to collect to meet their goal is 9x^2 - 10xy - 1.

From the information provided above, we can logically deduce the following number of canned food collection expressions for each friend:

Jessa; 9x^2

Theresa: 6x^2 - 4

Ben: 4xy + 3

Canned food collection goal: 24x^2 - 6xy - 2

Therefore, an expression that represents the total amount of canned food collected so far by the three friends can be calculated and written as follows;

Total amount collected = 9x^2 + 6x^2 - 4 + 4xy + 3

Total amount collected = 15x^2 + 4xy - 1

In order to meet the goal, they need to collect this amount:

Remaining canned foods = 24x^2 - 6xy - 2 - (15x^2 + 4xy - 1)

Remaining canned foods = 24x^2 - 6xy - 2 - 15x^2 - 4xy + 1

Remaining canned foods = 9x^2 - 10xy - 1.

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The slope is 3 which represents the cost per mile and the y-intercept is 5 which represents the fixed cost of the taxi.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

The linear equation is given as,

y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

The equation is given below.

C = 3m + 5

Where 'C' represents the cost and 'm' represents the number of miles.

The slope is 3 which represents the cost per mile and the y-intercept is 5 which represents the fixed cost of the taxi.

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The complete question is given below.

The equation c=3m+5 represents a line of the best fit for a scarer plot where c represents the total cost of a taxi in dollars and m represents the number of the trip. What do the numbers 5 and 3 represent?

The equation of the line through point (6,2) and perpendicular to 4x+5y+7=0 is 5x-4y+14=0.

To find the equation of a line perpendicular to another line, we first need to determine the slope of the original line. We can rewrite the original line as 5y = -4x - 7, which is in the slope-intercept form y = (-4/5)x - 7/5. The slope of this line is -4/5.

Since the line we want is perpendicular to this line, it must have a slope that is the negative reciprocal of -4/5. The negative reciprocal is 5/4. We can use the point-slope form of the equation of a line to find the equation of the new line. Plugging in the point (6,2) and the slope 5/4, we get:

y - 2 = (5/4)(x - 6)

Simplifying and rearranging, we get:

5x - 4y + 14 = 0

So, the equation of the line through the point (6,2) and perpendicular to 4x+5y+7=0 is 5x-4y+14=0.

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Answer:

Step-by-step explanation:

Mean value for the given data is 8.129, median = 7.7, Standard deviation is 1.699, Probability for MPa value greater than 10 is 0.148 and Coefficient of variation = 20.9

a) To estimate the mean value, we have to calculate all the observations and divide it by number of observations.

  x = ∑ xi / n = 219.5/27 = 8.129

So here the estimator used is x

b) To estimate the strength value or to separate weakest 50% from strongest 50%, we have to calculate the median. For that we have to arrange in ascending order and find the middle most term.

Since  n is odd, middle term is (n+1)/2 th term = (27+1)/2 = 28/2 = 14th term. Ascending order is given as image. the 14th term is 7.7.

c) Next we have to calculate standard deviation

 σ =  √(∑xi² - (∑x)²/n)/(n-1) = [1859.53 - (219.5²/27)] /(27-1)

                                          = (1859.53- 1784.454) / 26

                                         = 1.699

d) We have to calculate the probability flexural strength greater than 10. Here there are 4 values above 10.

So, p = 4/27 = 0.148

e) Coefficient of variation = (σ/x) ×100

                                           = [tex]\frac{1.699}{8.129} * 100[/tex]

                                            = 20.9

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6 cos(π/2 t) is the best model for the position of the weight.

The motion of the weight on the spring can be modeled by a sine or cosine function because it oscillates back and forth around its equilibrium position.

We know that, the weight is initially pulled down 6 inches from its equilibrium position, so the function should have an amplitude of 6.

The time it takes for the spring to complete one oscillation is 4 seconds, so the period of the function is 4 seconds.

The general form of a sine or cosine function with amplitude A and period T is:

f(t) = A sin(2πt/T) or f(t)

= A cos(2πt/T)

Substituting the given values,

we get:

f(t) = 6 sin(2πt/4) or f(t)

= 6 cos(2πt/4)

Simplifying, we get:

f(t) = 6 sin(π/2 t) or f(t)

= 6 cos(π/2 t)

Therefore,

the function that best models the position of the weight is

s(t) = 6 cos(π/2 t).

So, the answer is option (D).

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(a) Average for a student with 99 attendance is 81.1% and (b) The average for a student with 21 attendance is 78.8%.

Professor Cramer determines a final grade based on attendance, two papers, three major tests, and a final exam, the activities carry different weights.

Attendance is worth 3%,

each paper is worth 5%,

each test is worth 13%,

and the final is worth 48%.

(a) The average for a student with 99 on attendance, 75 on the first paper, 61 on the second paper, 94 on test 1, 86 on test 2, 77 on test 3, and 79 on the final exam

= [(99×3%) + (75×5%) + (61×5%) + (94×13%) + (86×13%) + (77×13%) + (79×48%)] ÷ [0.03 + 2(0.05) + 3(0.13) + 0.48]

= [2.97 + 3.75 + 3.05 + 12.22 + 11.18 + 10.01 + 37.92] ÷ 1

= 81.1 %

(b) The average for a student with the above scores on the papers, tests, and final exam, but with a score of only 21 on attendance

= [(21×3%) + (75×5%) + (61×5%) + (94×13%) + (86×13%) + (77×13%) + (79×48%)] ÷ [0.03 + 2(0.05) + 3(0.13) + 0.48]

= [0.63 + 3.75 + 3.05 + 12.22 + 11.18 + 10.01 + 37.92] ÷ 1

= 78.8 %

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Answer:

The area of the triangle formed by the tangent to the graph of the function f(x) = (x-6)/(x-2) at the point x = 3 with the axes of the coordinate system is 28.125 square units.

Step-by-step explanation:

Differentiation is an algebraic process that finds the gradient (slope) of a curve.  At a point, the gradient of a curve is the same as the gradient of the tangent line to the curve at that point.

Given function:

[tex]f(x)=\dfrac{x-6}{x-2}[/tex]

Differentiate the given function using the quotient rule.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Quotient Rule for Differentiation}\\\\If $y=\dfrac{u}{v}$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}$\\\end{minipage}}[/tex]

[tex]\implies f'(x)=\dfrac{(x-2)\cdot 1-(x-6)\cdot 1}{(x-2)^2}[/tex]

[tex]\implies f'(x)=\dfrac{4}{(x-2)^2}[/tex]

To find the gradient of the tangent lines at x = 3, substitute x = 3 into the differentiated function:

[tex]\implies f'(3)=\dfrac{4}{(3-2)^2}=4[/tex]

Substitute x = 3 into the function to find the y-value of the point on the curve when x = 3:

[tex]\implies f(3)=\dfrac{3-6}{3-2}=-3[/tex]

The slope-intercept form of a linear equation is y = mx + b, where m is the gradient and b is the y-intercept.

Substitute the point (3, -3) and the found gradient m = 4 into the slope-intercept formula and solve for b:

[tex]\begin{aligned}y&=mx+b\\\implies-3&=4(3)+b\\-3&=12+b\\b&=-15\end{aligned}[/tex]

Therefore, the equation of the tangent to the curve at point x = 3 is:

To calculate the point at which the tangent line intersects the x-axis, substitute y = 0 into the equation of the tangent:

[tex]\begin{aligned}\implies 0&=4x-15\\4x&=15\\x&=3.75\end{aligned}[/tex]

To calculate the point at which the tangent line intersects the y-axis, substitute x = 0 into the equation of the tangent:

[tex]\implies y=4(0)-15=-15[/tex]

Therefore, the tangent line intersects the x-axis at (3.75, 0) and the y-axis at (0, -15).  

This means the triangle formed by the tangent and the axes of the coordinate system has a height of 15 units and a base of 3.75 units.  

[tex]\begin{aligned}\textsf{Area of a triangle}&=\dfrac{1}{2}\sf \cdot base \cdot height\\\\\implies \sf Area&=\dfrac{1}{2} \cdot 3.75 \cdot 15\\\\&=\dfrac{225}{8}\\\\&=28.125\;\; \sf square\;units\end{aligned}[/tex]

Therefore, the area of the triangle is 28.125 square units.

The measure of angle B in given triangle is approximately 59°.

Triangles are fundamental three-sided polygons having three internal angles. Due to the connection between the three vertices, it is one of the basic geometric shapes.

When combined, any two triangle sides will always be longer than the third side. A triangle's surface area is determined by the product of its base and height. The angles are created by connecting the triangle's three sides end to end at a single point. The sum of the triangle's three angles is 180 degrees.

The inverse tangent function, often known as tan1, can be used in this situation. Using the following equation:

tan(B) = adjacent/opposite

where the lengths of the triangle's opposite and adjacent sides, which together make up the desired angle B, are given. In this case, we know that side b is next to angle B and that side an is perpendicular to it.

We thus have:

tan(B) = a/b = 500/300 = 5/3

To find the value of angle B, we can take the inverse tangent of both sides of the equation, using table of trigonometric functions. We get:

B = tan⁻¹(5/3) ≈ 59°

Hence the correct answer is 59°

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Answer:

he months where Jacy's hometown gets at least 10.5 centimeters of rainfall are July (x = 7) and all the months after July, which are August (x = 8), September (x = 9), October (x = 10), November (x = 11), and December (x = 12).

Step-by-step explanation:

The given function is:

f(x) = 0.2x^2 - 1.8x + 6.5

where x represents the number of months passed and f(x) represents the average monthly rainfall in centimeters.

To find the months where the average rainfall is at least 10.5 centimeters, we need to set the function f(x) greater than or equal to 10.5 and solve for x:

0.2x^2 - 1.8x + 6.5 ≥ 10.5

0.2x^2 - 1.8x - 4 ≥ 0

Multiplying both sides by 5, we get:

x^2 - 9x - 20 ≥ 0

We can factor the left-hand side of the inequality as:

(x - 5)(x - 4) ≥ 0

The solution to this inequality is the set of values of x that make the inequality true. This includes the intervals where:

(x - 5) ≥ 0 and (x - 4) ≥ 0, which gives x ≥ 5

OR

(x - 5) ≤ 0 and (x - 4) ≤ 0, which gives x ≤ 4

Thus, the months where Jacy's hometown gets at least 10.5 centimeters of rainfall are July (x = 7) and all the months after July, which are August (x = 8), September (x = 9), October (x = 10), November (x = 11), and December (x = 12).

Jacy's hometown gets at least 10.5 centimeters of rainfall in the months of September, September of the following year, September of the year after that, and so on.

An inequality is a relation between two numbers or expressions that are not equal.

It can show which of them is greater or smaller by using symbols like < or >. It can also be a statement of fact about the order relationship of quantities.

According to the question given,

We need to solve the inequality:

A(t) ≥ 10.5

Substituting the given function, we get:

2.3sin(π/6)t ≥ 10.5

sin(π/6)t ≥ 4.57.

Since the sine function has a maximum value of 1, the inequality is only satisfied when:

sin(π/6)t = 1

Solving for t, we get:

π/6t = π/2 + 2πk or π/6t = 3π/2 + 2πk

where k is any integer.

Simplifying each equation, we get:

t = 3 + 12k or t = 9 + 12k, where k is any integer.

Therefore, the only solution is given by:

t = 9 + 12k

where k is any integer.

Hence, Jacy's hometown gets at least 10.5 centimeters of rainfall in the months of September, September of the following year, September of the year after that, and so on.

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The correct constraint that ensures no more than 2 projects will be selected is: (B). X1 + X2 + X3 + X4 ≤ 2

This restriction specifies that the corporation can choose no more than two projects since the total of the choice factors X1, X2, X3, and X4 is less than or equal to 2. The other restrictions do not specifically place a cap on the number of chosen projects.

Binary constraint C guarantees that X1 is more than or equal to X2, but it does not place a cap on the number of projects that can be chosen.

The three-way constraint D does not guarantee that the corporation chooses no more than two projects, but it does limit the sum of X1, X2, and X3 to be less than or equal to X4.

Therefore, the correct option is (B). X1 + X2 + X3 + X4 ≤ 2

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(a) Expression for  P(3x−4<5) is ∫[from -∞ to 3] fx(x) dx

(b) Expression for P(X>Y)(c) P(X+Y=5) is ∬[over (x,y) satisfying x > y] fx(x) fY(y) dxdy

(c) Expression for E(cov(XY)) is ∬xyfx(x) fY(y) dxdy - E(X)E(Y)

(d) Expression for Mx+y(t) (the mgf of X +Y) is Mx(t) My(t)

(a) P(3x-4<5) can be written as:

P(3x < 9)

P(x < 3)

The expression involving fx(x) for this probability would be:

∫[from -∞ to 3] fx(x) dx

(b) P(X>Y) can be written as:

P(X-Y > 0)

The expression involving fx(x) and fY(y) for this probability would be:

∬[over (x,y) satisfying x > y] fx(x) fY(y) dxdy

(c) P(X+Y=5) can be written as:

P(Y = 5 - X)

The expression involving fx(x) and fY(y) for this probability would be:

∬[over (x,y) satisfying x+y=5] fx(x) fY(y) dxdy

(d) The covariance of X and Y is defined as:

cov(X,Y) = E(XY) - E(X)E(Y)

So, E(cov(X,Y)) can be written as:

∬xyfx(x) fY(y) dxdy - E(X)E(Y)

(e) The MGF of X+Y can be written as:

Mx+y(t) = E(e^(t(X+Y)))

Since X and Y are independent, we can write this as:

Mx+y(t) = E(e^(tX) e^(tY))

Using the fact that X and Y have their own MGFs,

We can write this as:

Mx+y(t) = Mx(t) My(t)

Where Mx(t) and My(t) are the MGFs of X and Y, respectively.

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Answer:

25

Step-by-step explanation: just like that

The ratio of bananas to strawberries is 11:3.

A ratio is a comparison between two amounts that is calculated by dividing one amount by the other. The quotient a/b is referred to as the ratio between a and b if a and b are two values of the same kind and with the same units, such that b is not equal to 0. Ratios are represented by the colon symbol (:). As a result, the ratio a/b has no units and is represented by the notation a: b.

Given:

Number of bananas = 11 ounces,

Number of strawberries = 3 ounces

The ratio of bananas to strawberries

= 11/3

= 11:3.

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On solving the provided question, we can say that the equation will be

[tex]N = Fg * cos(15°) = 44.6 N * cos(15°) = 42.8 N[/tex]

A mathematical equation is a formula that links two assertions and signifies equivalence with the equals sign (=). In, a mathematical statement that proves the equality of two mathematical expressions is referred to as an equation. The equal sign, for instance, separates the variables [tex]3x + 5[/tex] and 14 in the equation [tex]3x + 5 = 14.[/tex]There is a mathematical formula that explains the link between the two phrases that appear on either side of a letter. Frequently, the symbol and the single variable are the same. like [tex]2x - 4 = 2[/tex], for example.

Since the weight is not moving, the sum of the three forces acting on it must be zero. This means that the force of gravity acting on the weight is balanced by the normal force and the force due to friction.

The force of gravity on the weight is given by:

[tex]Fg = m * g[/tex]

where m is the mass of the weight (which we can find by dividing the weight in pounds by the acceleration due to gravity, g), and g is the acceleration due to gravity, which is approximately [tex]9.81 m/s^2.[/tex]

[tex]m = 10.0 pounds / (2.205 pounds/kilogram) = 4.54 kg[/tex]

[tex]Fg = 4.54 kg * 9.81 m/s^2 = 44.6 N[/tex]

The normal force N acts perpendicular to the bench, so it can be found using trigonometry. The angle between the bench and the horizontal is 15 degrees, so the angle between the normal force and the horizontal is also 15 degrees.

[tex]N = Fg * cos(15°) = 44.6 N * cos(15°) = 42.8 N[/tex](rounded to one decimal place)

The force due to friction F acts parallel to the bench, in the opposite direction to the component of the weight that is parallel to the bench. This component can be found using trigonometry:

[tex]Fp = Fg * sin(15°) = 44.6 N * sin(15°) = 12.2 N[/tex]

Since the weight is not moving, the force due to friction must be equal and opposite to this component:

[tex]F = -Fp = -12.2 N[/tex] (rounded to one decimal place)

So the magnitude of the normal force is 42.8 N and the magnitude of the force due to friction is 12.2 N.

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The Corollary to the polygon is explained below and the  formula to find the measure of each interior angle is  " (n - 2)×180°/n "   .

The Corollary to the polygon Angle Sum Theorem states that : the sum of the interior angles of a regular n gon is written as :

that means , ⇒ Sum of interior angles = (n - 2) × 180° ...equation(1)

In a regular "n-gon" , all the interior angles are said to be congruent.

Let "x" be  measure of each interior angle of a regular n-gon.

So , we can write ;

⇒ Sum of interior angles = (n)×(x)  ;

Equating the above expressions with equation(1),

⇒ (n)×(x) = (n - 2) × 180°  ;

On Solving for x,

⇒ x = (n - 2)×180°/n  ;

Therefore, the measure of each interior angle of a regular "n-gon" is x = (n - 2)×180°/n  .

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Answer: a

Step-by-step explanation:

Answer:

Tyler is Correct

Step-by-step explanation:

Total Hike = 2.25 (0.75 + 1.5)

2.25/0.75 = 3

Tyler is correct, since the total hike is 3 times as long as the distance travelled from the parking lot

The constant of proportionality is 2. (True)

The equation n = 2m represents this relationship. (True)

Frederick needs 5 cups of water for 10 tablespoons of the mic. (False)

Point (2, 4) means 2 tablespoons of ms is needed for 4 cups of water. (False)

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction

If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,

then the equation of a line is

y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)

To find the slope;

m =  (y₂- y₁) / (x₂ - x₁)

Given:

Frederick is making hot chocolate from a mix.

The graph models, the relationship between tablespoons of mixing cups of water.

Let m be the variable that represents the number of cups of water and n is the number of tablespoons in the mix.

From the graph;

The line passes through (0, 0) and (1, 2),

So, the equation of the line is,

n - 0 = (0 - 2)/(0 - 1) m

n = 2m

Therefore, the constant of proportionality is 2.

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[tex]{\Large \begin{array}{llll} y=ab^x \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x=2\\ y=360 \end{cases}\implies 360=ab^2\implies 360=abb\implies \cfrac{360}{b}=ab \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x=3\\ y=216 \end{cases}\implies 216=ab^3\implies 216=abb^2\implies \stackrel{\textit{substituting from above}}{216=\left( \cfrac{360}{b} \right)b^2} \\\\\\ 216=360b\implies \cfrac{216}{360}=b\implies \boxed{\cfrac{3}{5}=b} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{since we know that}}{360=ab^2}\implies 360=a\left( \cfrac{3}{5} \right)^2\implies 360=\cfrac{9a}{25} \\\\\\ \cfrac{25}{9}\cdot 360=a\implies \boxed{1000=a}~\hfill {\Large \begin{array}{llll} y=1000\left( \frac{3}{5} \right)^x \end{array}}[/tex]

The area of the quadrilateral is 140 m^2.

For closed figure made by 4 line segments joined end to end in series is called a quadrilateral.

A parallelogram in which adjacent sides are perpendicular to each other is called a rectangle.

A rectangle is always a parallelogram and a quadrilateral but reverse statement could be not be true.

Area of the quadrilateral = 2 x area of the traingle

Thus, Area of the triangle = 1/2 x base  x height

Area of 1 triangle = 1/2 x 10 x 16

Area of 1 triangle = 80 m^2

Area of the quadrilateral = 2 x 80

Area of the quadrilateral = 140 m^2

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The complete question is;

What is the area of the quadrilateral if the dimensions of the quadrilateral  are 10cm, 16cm and 22cm

1.20 If the system has three unknowns and R has three nonzero rows, then the system has at least one solution.

This statement is true.

1.21 It is only when the rank of the coefficient matrix is less than the number of unknowns that the system can have infinitely many solutions

1.22 The system below has an infinite number of solutions:

2x + 3y + 5z + 6 - 7 - 8v = 0

3x - 4y + 7z + + 8 + 5y = 0

-7x + 9y - 2z -- 4w - 5u + 2y = 0

--5x - 5y +92 +3w + 2u + 7y = 0

-9x + 3y - 9z+5w - 3u - 4y = 0

This statement is true.

When we perform row reduction on a system of linear equations, the resulting reduced row echelon form (R) will have the same number of nonzero rows as the rank of the coefficient matrix.

In other words, if R has three nonzero rows, then the rank of the coefficient matrix is also 3.

If the rank of the coefficient matrix is equal to the number of unknowns, then the system has a unique solution.

However, if the rank of the coefficient matrix is less than the number of unknowns, then the system has either no solution or infinitely many solutions.

But in this case, since the rank is equal to the number of unknowns, the system must have at least one solution.

1.21 If the system has three unknowns and R has three nonzero rows, then the system can have an infinite number of solutions.

This statement is false. If R has three nonzero rows, then the rank of the coefficient matrix is also 3.

If the rank of the coefficient matrix is equal to the number of unknowns, then the system has a unique solution.

It is only when the rank of the coefficient matrix is less than the number of unknowns that the system can have infinitely many solutions.

1.22 The system below has an infinite number of solutions:

2x + 3y + 5z + 6 - 7 - 8v = 0

3x - 4y + 7z + 8 + 5y = 0

-7x + 9y - 2z - 4w - 5u + 2y = 0

-5x - 5y + 92 + 3w + 2u + 7y = 0

-9x + 3y - 9z + 5w - 3u - 4y = 0

This statement is true.

To check if the system has infinitely many solutions, we need to check the rank of the coefficient matrix and the rank of the augmented matrix. In this case, the rank of the coefficient matrix is 3, which is less than the number of unknowns (5).

Also, when we perform row reduction on the augmented matrix, we get the following reduced row echelon form:

1 0 -1 0 1 0

0 1 2 0 -1 0

0 0 0 1 2 0

0 0 0 0 0 1

0 0 0 0 0 0

Since the rank of the augmented matrix is less than the number of unknowns, the system has infinitely many solutions.

The variables with free parameters are z, u, and y, which can take any value.

The other variables can be expressed in terms of these free parameters.

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