The following inequalities are equivalent
except...
A -x>-3
B x <3
C x+1 <4
D -x<3

The value at which x does the tangent to the curve y=cosx have a slope of 1 is x =  3π/2.

To find the values of x at which the tangent to the curve y = cos(x) has a slope of 1, if x belongs to the interval [0,2π], we need to find the points where the derivative of the function is equal to 1. The derivative of y = cos(x) with respect to x is -sin(x). So, we need to solve the equation:

-sin(x) = 1

sin(x) = -1

Now, we want to find the values of x in the interval [0, 2π] that satisfy this equation. The sine function takes the value -1 at x = 3π/2. Therefore, x does the tangent to the curve y=cosx have a slope of 1 is x = 3π/2.

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The Surface Area of Cone is 124.344 cm².

We have,

Diameter= 11 cm

slant height = 7.2 cm

Radius = 5.5 cm

So, Surface Area of Cone

= πrl

= 3.14(5.5)(7.2)

= 124.344 cm²

Thus the Surface Area of Cone is 124.344 cm².

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The dimensions of the screen are:

length = L = 12 inches

width = W = (13L - 182) / 3 = (13*12 - 182) / 3 = 2 inches

a. The area of the browser window is given by:

area = length * width

24 = x(x-2)

Simplifying and solving for x, we get:

[tex]x^2[/tex] - 2x - 24 = 0

(x - 6)(x + 4) = 0

Since the length cannot be negative, we take x = 6. Therefore, the length of the browser window is x = 6 inches.

b. Let L and W be the length and width of the computer screen, respectively. We are given that:

length of browser = x inches = L - 7 inches

width of browser = x - 2 inches

The area of the browser is:

area of browser = x(x-2) = (L-7)(L-9)

We are also given that the browser covers 3/13 of the screen, so:

area of browser = (3/13) * area of screen

x(x-2) = (3/13) * L * W

Substituting x = L - 7 and simplifying, we get:

(L-7)(L-9) = (3/13) * L * W

Expanding and simplifying, we get:

13L[tex]^2[/tex] - 182L + 546 = 3LW

Since L and W are positive, we can divide both sides by 3L to get:

W = (13L - 182) / 3

We also know that the browser covers 3/13 of the screen, so:

area of browser = (3/13) * area of screen

x(x-2) = (3/13) * L * W

6(4) = (3/13) * L * ((13L-182)/3)

24 = (L/3) * (13L-182)

8 = L(13L-182)/3

24 = L(13L-182)

13L^2 - 182L - 24 = 0

(L - 12)(13L + 2) = 0

Since the length cannot be negative, we take L = 12. Therefore, the dimensions of the screen are:

length = L = 12 inches

width = W = (13L - 182) / 3 = (13*12 - 182) / 3 = 2 inches

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Step-by-step explanation:

b+g = 297 - eqn1

let the number of boys be b and the number of girls be g

(4/9)*b = (7/9)*g

cross multiply

4x9b = 7x9g

36b=63g - eqn2

from eqn 2

36b = 63g

we can divide both sides by 36 to make b the subject of formula

b=7/4 g

substitute b in eqn 1

b+g=297

7/4 g + g = 297

11/4 g = 297

11g = 297*4

11g=1188

g = 1188/11 = 108

since b+g=297

b+108=297

b=297-108=189

therefore the no of girls is 108 and of boys is 189

189-108 = 81 so there are 81 more boys than the girls

Based on the mentioned informations, at the time when the first bucket is empty, it is calculated that the second bucket will contain approximately 365.72 ounces volume of water.

The first step is to convert the 5-gallon volume to ounces. There are 128 ounces in one gallon, so 5 gallons is equal to 640 ounces.

The water is leaking out of the bucket at a rate of 7 ounces per minute. Therefore, the amount of water remaining in the bucket after t minutes can be calculated as:

Remaining water in the bucket = 640 - 7t

We want to find out when the remaining water in the bucket reaches zero, so we set the above equation equal to zero and solve for t:

640 - 7t = 0

7t = 640

t = 91.43 minutes

Therefore, it will take approximately 91.43 minutes for the water level in the bucket to reach zero.

At the same time, the empty bucket is being filled with water from the leaky faucet at a rate of 4 ounces per minute. Therefore, the amount of water in the empty bucket after t minutes can be calculated as:

Water in the empty bucket = 4t

We want to find out how much water will be in the empty bucket at the time when the first bucket is empty, so we substitute t = 91.43 into the above equation:

Water in the empty bucket = 4 x 91.43

Water in the empty bucket = 365.72 ounces

Therefore, at the time when the first bucket is empty, the second bucket will contain approximately 365.72 ounces of water.

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The complete question is :

If the bucket that is filled with water initially contained 5 gallons of water and the leak in that bucket started at time zero, how long will it take for the water level in the bucket to reach zero, and how much water will be in the empty bucket at that time assuming that both leaks continue at the same rate of 7 ounces per minute and 4 ounces per minute, respectively?

The discount percentage that has been offered to Sydney if after the discount she has to pay $20.79 for an item marked at $21 is 1%

Discount refers to a reduction in price offered on articles and items in order to increase sales. It is a marketing technique.

Listed price =$21

After discount price = $20.79

Discount offered = 21 - 20.79

= $0.21

Discount percentage = [tex]\frac{LP - DP}{LP}*100[/tex]

where LP is Listed Price

DP is Discount Price

Discount percent = [tex]\frac{21-20.79}{21}*100[/tex]

= [tex]\frac{0.21}{21}*100[/tex]

= 0.01 * 100

= 1%

Therefore, the discount she gets is equal to 1% after she uses her loyalty card at her local hardware store.

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The circumference of a circle in terms of π with radius 31½ yards is 63π yards ( optionD)

The circumference is the perimeter of a circle or ellipse. This means that the circumference would be the arc length of the circle, as if it were opened up and straightened out to a line segment.

The circumference of a circle is expressed as;

C = 2πr

where r is the radius

r = 31 ½ = 63/2 yards

therefore C = 2 × 63/2 × π

C = 63π yards

Therefore the circumference of the circle in term of π with radius 31 ½ is 63π yards

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Since S is the average of the random sample, the expected value (E(S)) and variance (V(S)) are: E(S) = µ and V(S) = σ² / n

For W with v degrees of freedom, we know that E(W) = v and V(W) = 2v.

Now, let's consider the random sample X = (X1, X2, ..., Xn) from a normal distribution with mean y and variance o. The sample variance S^2 is defined as:

S^2 = (1/n-1) * sum(i=1 to n) (Xi - y)^2

where y is the sample mean.

Using properties of the normal distribution, we can derive that E(S^2) = o * (n-1)/n and V(S^2) = (2o^2 * (n-1)^2)/(n(n-2)).

To find E(S) and V(S), we take the square root of E(S^2) and V(S^2) respectively. Thus:

E(S) = sqrt(o * (n-1)/n)

V(S) = sqrt((2o^2 * (n-1)^2)/(n(n-2)))


First, let's discuss the chi-square distribution. If W has a chi-square distribution with v degrees of freedom, the expected value (E(W)) and variance (V(W)) are as follows:

E(W) = v
V(W) = 2v

Now let's consider the random sample from a normal distribution. If X, i = 1, 2,...,n is a random sample from a normal distribution with mean y (µ) and variance σ², we can find the sample mean (S) as:

S = (ΣX_i) / n

Since S is the average of the random sample, the expected value (E(S)) and variance (V(S)) are:
E(S) = µ and V(S) = σ² / n

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Sarah can make 14 gallons of lemonade with 21 scoops of lemonade mix.

Given information:

Sarah was mixing up some lemonade to sell for her lemonade stand.

The recipe calls for 3 scoops of lemonade mix to make 2 gallons of lemonade.

If 3 scoops of lemonade mix are needed to make 2 gallons of lemonade, then one scoop of lemonade mix can make 2/3 of a gallon of lemonade.

As per the multiplication,

So, with 21 scoops of lemonade mix, Sarah can make:

21 scoops x (2/3 gallon/scoop) = 14 gallons

Therefore, Sarah can make 14 gallons of lemonade with 21 scoops of lemonade mix.

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True.

the solution set of the linear system whose augmented matrix [a1 a2 a3 | b] is the same as the solution set of the equation x1a1 + x2a2 + a3x3 = b.

The augmented matrix [a1 a2 a3 | b] corresponds to the system of linear equations:

a1x1 + a2x2 + a3x3 = b

The solution set of this system is the same as the solution set of the vector equation:

[x1, x2, x3] * [a1, a2, a3] = b

which can be written as:

x1a1 + x2a2 + x3a3 = b

Therefore, the solution set of the linear system whose augmented matrix [a1 a2 a3 | b] is the same as the solution set of the equation x1a1 + x2a2 + a3x3 = b.

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The angle which is supplemental to the 6 is the angle 8, which makes option a correct.

When a transversal line intersects a pair of parallel lines, several angles are formed which includes: Corresponding angles, vertical angles, alternate angles, complementary and supplementary angles.

Supplementary angles are a pair of angles that add up to 180 degrees. The angles 6 and 8 lie on a straight line which implies they are supplementary as the sum of angles on a straight line is also 180°

Therefore, the angle which is supplemental to the 6 is the angle 8, which makes option a correct.

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To construct a confidence interval for the ratio of the variances of the two populations sampled, we can use the F-distribution. The formula for the F-statistic is: F = (s1^2 / s2^2) / (n1 - 1) / (n2 - 1) Where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Using the given data, we have:

s1 = 19.4
s2 = 18.8
n1 = 61
n2 = 41

The F-statistic is then:

F = (19.4^2 / 18.8^2) / (61 - 1) / (41 - 1) = 1.399

To find the confidence interval, we need to look up the critical values of the F-distribution with degrees of freedom (df) of (n1 - 1) and (n2 - 1) at the 1% level of significance.

Using a table or calculator, we find that the critical values are 0.414 and 2.518.

Thus, the confidence interval for the ratio of the variances is:

1 / (2.518 / sqrt(F)) < σ1^2 / σ2^2 < 1 / (0.414 / sqrt(F))

1 / (2.518 / sqrt(1.399)) < σ1^2 / σ2^2 < 1 / (0.414 / sqrt(1.399))

0.266 < σ1^2 / σ2^2 < 2.083

Therefore, we can be 98% confident that the ratio of the variances of the two populations sampled lies between 0.266 and 2.083.
To construct a 98% confidence interval for the ratio of the variances of the two populations sampled, we will use the F-distribution and the following formula:

CI = (s1^2 / s2^2) * (1 / Fα/2, df1, df2, F1-α/2, df1, df2)

Here, s1 and s2 are the standard deviations of the first and second kinds of equipment, and df1 and df2 are the degrees of freedom for each sample. Fα/2 and F1-α/2 are the F-distribution critical values at the α/2 and 1-α/2 levels, respectively.

Step 1: Calculate the variances (s1^2 and s2^2).
Variance1 = (19.4)^2 = 376.36
Variance2 = (18.8)^2 = 353.44

Step 2: Calculate the degrees of freedom (df1 and df2).
df1 = n1 - 1 = 61 - 1 = 60
df2 = n2 - 1 = 41 - 1 = 40

Step 3: Find the F-distribution critical values (Fα/2, df1, df2, F1-α/2, df1, df2) for a 98% confidence interval (α = 0.02).
F0.01, 60, 40 = 0.4611
F0.99, 60, 40 = 2.1080

Step 4: Calculate the confidence interval using the formula.
CI = (376.36 / 353.44) * (1 / 0.4611, 2.1080)
Lower limit = (376.36 / 353.44) * 0.4611 = 0.5925
Upper limit = (376.36 / 353.44) * 2.1080 = 2.2444

The 98% confidence interval for the ratio of the variances of the two populations sampled is (0.5925, 2.2444).

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The distance from the wall, that the ladder slipped during the day, would be 21 cm.

First, use the Pythagorean theorem to find the original height of the ladder from the wall:

50² + b² = 300²

2, 500 + b ² = 90, 000

b ² = 90, 000 - 2, 500

b = √ 87, 500

b = 295. 8 cm

We can then find the new distance from the wall to be :
= 50 + 70

= 120 cm

The new height of the ladder to show the slip would be:

120² + b² = 300²

14,400 + b ² = 90, 000

b ² = 90, 000 - 14, 400

b = √ 75, 600

b = 275 cm

The distance slipped is:

= 295. 8 - 275

= 21 cm

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A: Liam is correct. The graph shows a positive correlation between ice cream sales and temperature. As the temperature increases, ice cream sales also increase.

B: Armonte is not correct. There are a few reasons why the graph represents a correlation and not causation:

Correlation does not imply causation. Just because two variables are correlated, it does not mean that one variable causes the other.

There could be other variables that are affecting both ice cream sales and temperature. For example, if the graph was plotted over a long period of time, there may be seasonal factors that affect both variables, such as summer months having higher temperatures and more ice cream sales.

The graph only shows a relationship between ice cream sales and temperature, but it does not prove causation. To establish causation, a controlled experiment would need to be conducted to show that changes in temperature cause changes in ice cream sales.

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For Question 7, the man must take out 4 socks to be sure that he has at least two socks of the same color. This is because he could potentially take out 3 brown socks and 3 black socks before getting a matching pair, but with 4 socks, there must be at least two of the same color.

For Question 8, the man must take out 3 socks to be sure that he has at least two black socks. This is because he could potentially take out all 12 brown socks before getting a black one, but with 3 socks, there must be at least one black sock and then a potential for a second.

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Answer:

1ab

Step-by-step explanation:

Hope this helps! Pls give brainliest!

Answer:

1ab or just ab

Step-by-step explanation:

All 3 parts of the equation have the variables ab at the end, meaning that the solution will also end with ab.

Let's first solve by doing the first property, subtraction:

3ab-9ab=-6ab

Now, we add that sum to 7ab:
-6ab+7ab=1ab

Depending on what they ask, you can either write 1ab or ab, they are the exact same thing.

Joe is about 10 times more likely to bowl at least three strikes out of four tries than to bowl zero strikes out of four tries.

To find the probability of Joe bowling at least three strikes out of four tries, we need to consider the different combinations of strikes and non-strikes he can get.

There are four possible outcomes:
- strike, strike, strike, non-strike
- strike, strike, non-strike, strike
- strike, non-strike, strike, strike
- non-strike, strike, strike, strike

The probability of getting a strike is 0.6, and the probability of not getting a strike (a non-strike) is 0.4. So for each outcome, we can calculate the probability as follows:

- Probability of strike, strike, strike, non-strike = 0.6 x 0.6 x 0.6 x 0.4 = 0.0864
- Probability of strike, strike, non-strike, strike = 0.6 x 0.6 x 0.4 x 0.6 = 0.0864
- Probability of strike, non-strike, strike, strike = 0.6 x 0.4 x 0.6 x 0.6 = 0.0864
- Probability of non-strike, strike, strike, strike = 0.4 x 0.6 x 0.6 x 0.6 = 0.0864

To find the probability of Joe bowling at least three strikes, we need to add up the probabilities of the last three outcomes, since they all have at least three strikes:

0.0864 + 0.0864 + 0.0864 = 0.2592

So the probability of Joe bowling at least three strikes out of four tries is 0.2592, or about 26% (rounded to the nearest whole number).

To find the probability of Joe bowling zero strikes, we can use the same approach:

- Probability of non-strike, non-strike, non-strike, non-strike = 0.4 x 0.4 x 0.4 x 0.4 = 0.0256

So the probability of Joe bowling zero strikes out of four tries is 0.0256, or about 3% (rounded to the nearest whole number).

To find how many times more likely it is for Joe to bowl at least three strikes than to bowl zero strikes, we can divide the probability of the first outcome by the probability of the second outcome:

0.2592 / 0.0256 = 10.125

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Once you have followed these steps, you should be able to help the virologist conclusively decide what quantities a and b model by analyzing the given differential equations.

It seems that the specific functions a and b, as well as the differential equations, are not provided in your question. However, I can guide you on how to approach this problem using the given terms and general concepts.

A virologist studies the dynamics of infectious disease using mathematical models. The three quantities of interest are infected people (I), susceptible people (S), and immune people (R). Functions a and b will represent two of these three quantities. To determine which quantities a and b model, we can analyze the given differential equations and follow these steps:

Step 1: Identify the variables and their relationships
Look for the variables (S, I, and R) in the differential equations and analyze how they are related to each other. Determine if there are any rate constants or parameters that link the variables.

Step 2: Analyze the equations' behavior
Study the differential equations' behavior over time, considering different initial conditions. Observe if the equations exhibit any trends, such as an increase or decrease in the quantities.

Step 3: Compare the equations with known epidemic models
Compare the given differential equations with known epidemic models, such as the SIR model or the SEIR model. These models have well-defined equations that describe the rates of change for susceptible, infected, and immune individuals.

Step 4: Rule out alternatives
Based on your analysis, eliminate the alternatives that don't match the behavior exhibited by the differential equations. Ensure that your conclusion is supported by a logical argument.

Once you have followed these steps, you should be able to help the virologist conclusively decide what quantities a and b model by analyzing the given differential equations.

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Euler's method is useful for providing an approximate solution to ODEs that may not have an easily obtainable closed-form solution for differential equations.

Euler's method is a numerical technique for solving first-order ordinary differential equations (ODEs) with an initial value. Here's a step-by-step explanation:

1. Start with a given initial value problem (IVP): dy/dx = f(x, y), with an initial condition y(x0) = y0.

2. Choose a step size, h, which represents the interval between consecutive points in your solution.

3. Apply Euler's method formula to find the approximate value of y at the next point: y1 = y0 + h * f(x0, y0).

4. Continue this process, using the previously calculated value as the new initial condition, to find the approximate solution at other points. For the nth point, use the formula: yn = yn-1 + h * f(xn-1, yn-1).

5. Compare your approximate solution to the exact solution, if available, to evaluate the accuracy of Euler's method.

Keep in mind that the accuracy of Euler's method depends on the step size, h. Smaller values of h will generally yield more accurate results, but require more calculations.

Euler's method is useful for providing an approximate solution to ODEs that may not have an easily obtainable closed-form solution. It's a simple and easy-to-understand method, which is why it's often covered in introductory differential equations courses, such as the one you mentioned.

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keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above

[tex]y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{2}{3}}x+4\qquad \impliedby \qquad \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{2}{3}} ~\hfill \stackrel{reciprocal}{\cfrac{3}{2}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{3}{2} }}[/tex]

so we're really looking for the equation of a line whose slope is -3/2 and it passes through (2 , -6)

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{-6})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{3}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-6)}=\stackrel{m}{- \cfrac{3}{2}}(x-\stackrel{x_1}{2}) \implies y +6 = - \cfrac{3}{2} ( x -2) \\\\\\ y+6=- \cfrac{3}{2}x+3\implies {\Large \begin{array}{llll} y=- \cfrac{3}{2}x-3 \end{array}}[/tex]

The base angles of the isosceles triangle is A = 50°

Given data ,

Let the isosceles triangle be represented as ΔABC

Now , the measure of ∠ABC = 80°

So , the measure of ∠ACB = ∠BAC = A°

And , In an isosceles triangle, the two equal sides are called legs, and the remaining side is called the base. The angle opposite the base is called the vertex angle

So , A° + A° + 80° = 180°

Subtracting 80 on both sides , we get

2A° = 100°

Divide by 2 on both sides , we get

A = 50°

Hence , each base angle measures 50°

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Yes, the statement is true. This is a restatement of the definition of the limit of a function of two variables.

Formally, we say that the limit of f(x,y) as (x,y) approaches (a,b) is L if and only if for every number ε > 0, there exists a number δ > 0 such that if the distance between (x,y) and (a,b) is less than δ, then the distance between f(x,y) and L is less than ε. In symbols:

For every ε > 0, there exists a δ > 0 such that if 0 < sqrt((x-a)^2 + (y-b)^2) < δ, then |f(x,y) - L| < ε.

The condition that f(x,y) approaches L along every straight line through (a,b) is equivalent to saying that the limit of f(x,y) as (x,y) approaches (a,b) along any path is also L. This is a stronger condition than the usual definition of the limit, and implies that the limit exists and is equal to L.

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The surface area of the Rubik's cube is 1176 cm².

Surface area is a measurement of the overall space occupied by an object's surface. Often, it is expressed in terms of square measurements like square meters (m²) or square feet (ft²).

A Rubik's cube has six faces, each of which is a square with sides equal to the length of a side of the cube. Therefore, the surface area of a Rubik's cube is six times the area of one of its faces.

The area of one face of the cube with a side of 14 cm is:

14 cm x 14 cm = 196 cm²

So, the total surface area of the Rubik's cube is:

6 x 196 cm²= 1176 cm²

Hence, the area will bed 1176 cm².

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The perfect square trinomial for this problem would be 5(x - (7/5))^2.

The given quadratic equation is:

5x^2 - 14x + 8

To complete the square and find the perfect square trinomial, follow these steps:

1. Make sure the coefficient of x^2 is 1. Divide the entire equation by the coefficient of x^2 (in this case, 5):

  x^2 - (14/5)x + 8/5

2. Find the term to complete the square. Take half of the coefficient of the x term, square it, and add it to both sides:

  x^2 - (14/5)x + (14/10)^2 = -8/5 + (14/10)^2

3. Rewrite the left side of the equation as the perfect square trinomial:

  (x - 7/5)^2 = -8/5 + 49/25

Now, the perfect square trinomial for the given problem is (x - 7/5)^2.

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The value of x is -4

When an integer by itself , the product obtained is a perfect square. Examples of perfect squares are 25 from 5×5, 36 from 6×6

The square root of the perfect square will give the original number.

Solving 4(x+7)^2-49=-13

collecting like terms

4( x+7)^2 = -13+49

4(x+7)² = 36

divide both sides by 4

(x+7)² = 9

Square both sides

x+7 = √9

x+7 = 3

x = 3-7

x = -4

therefore the value of x is -4

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The measures of angles are m∠A = 80°, m∠B = 50°, and m∠C = 50°, concluding that the triangle is an isosceles triangle.

Based on the given information, we can start by using the fact that the angles in a triangle sum to 180 degrees. Let's label the unknown angles as m∠A, m∠B, and m∠C:

m∠A + m∠B + m∠C = 180

We also know that m∠1 = m∠2, which means that triangle ABD is similar to triangle CBD by the Angle-Angle (AA) similarity theorem. This implies that the ratios of corresponding sides in these triangles are equal:

AB/BD = BD/DC

Since BD = DC, we have AB = DC.

Therefore, triangle ABC is isosceles with AB = AC. This means that m∠B = m∠C.

Now let's use the given information that BD = DC and m∠BDC = 100° to find the measure of m∠B. We can draw the perpendicular bisector of BC to point D, which will bisect angle BDC into two equal angles of measure x degrees.

Since triangle BDC is isosceles, we know that:

m∠DBC = m∠DCB

= (180 - m∠BDC)/2

= 40°.

Therefore, m∠B = m∠DBC + m∠DCB = 40° + x.

Now we can use the fact that the angles in triangle ABC sum to 180 degrees to solve for m∠A:

m∠A + m∠B + m∠C = 180

m∠A + 2m∠B = 180 (since m∠B = m∠C)

m∠A + 2(40° + x) = 180 (since m∠B = 40° + x)

m∠A + 80° + 2x = 180

m∠A = 100° - 2x

We also know that m∠BDC = 100°, so m∠DBC = m∠DCB = (180 - m∠BDC)/2 = 40°. Therefore, we can write:

m∠A + m∠B + m∠C = 180

m∠A + 2m∠B = 180 (since m∠B = m∠C)

m∠A + 2(40° + x) = 180 (since m∠B = 40° + x)

m∠A + 80° + 2x = 180

m∠A = 100° - 2x

Now we can solve for x by using the fact that m∠A, m∠B, and m∠C must be positive:

m∠A > 0, m∠B > 0, m∠C > 0

100° - 2x > 0

x < 50°

Therefore, the possible values of x are 10°, 20°, 30°, 40°, and 49°. We can use these values to find the measures of m∠A, m∠B, and m∠C:

If x = 10°, then m∠A = 80°, m∠B = 50°, and m∠C = 50°.

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The exact number of fractions equivalent to the given fraction is 8/10, 12/15 and 16/20, under the condition that the given fraction is 4/5.

In order to determine the equivalent fractions of 4/5, we are have to multiply the numerator and denominator by same numbers.

Then the exact numbers equivalent to the given fraction is

(4/5) × (2/2) = (4 × 2) / (5 × 2) = 8/10

(4/5) × (3/3) = (4 × 3) / (5 × 3) = 12/15

(4/5) × (4/4) = (4 × 4) / (5 × 4) = 16/20

Then, 8/10, 12/15, and 16/20 are equal to 4/5 when simplified, which means they are equivalent in nature.

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