write an equation for

The value of probability that a customer wins a prize is,

⇒ 12.5%

We have to given that;

A jewelry store has each customer spin a spinner with 8 equal sections numbered 1, 1, 1, 2, 2, 3, 3, 4 to win a free bracelet.

And, Customers who spin a 4 win.

Now, We have;

Total outcomes = 8

And, Possible outcomes for who spin a 4 win is,

⇒ 1

Hence, The value of probability that a customer wins a prize is,

⇒ 1 / 8

⇒ 1/8 x 100%

⇒ 12.5%

Thus, The value of probability that a customer wins a prize is,

⇒ 12.5%

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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The cosine function (cos) is one of the six trigonometric functions and represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is denoted by cos θ, where θ is the angle between the adjacent side and the hypotenuse.

In the given equation, we are asked to find the correct answer for (cos-'(_hx)) = d dx = h 1-h-x2 h V1+hx? h VI-V x2 h- h V1+hx2. To solve this equation, we need to understand the basic principles of calculus, specifically differentiation.

Differentiation is the process of finding the derivative of a function, which represents the rate of change of that function at a particular point. In this case, we are differentiating the inverse cosine function (cos^-1) with respect to x.

The correct answer to the equation is h V1+hx2. To explain this answer, we need to use the chain rule of differentiation. Let u = cos^-1(_hx). Then, we have:

d dx (cos^-1(_hx)) = d du (cos^-1 u) * d dx (_hx)
= -1/√(1-u^2) * h

Substituting u = _hx, we get:

d dx (cos^-1(_hx)) = -1/√(1-(_hx)^2) * h
= -1/√(1-h^2x^2) * h

Simplifying the expression, we get:

d dx (cos^-1(_hx)) = -h/√(1-h^2x^2)

Now, we need to find the value of d dx (cos^-1(_hx)) when x = 1. Plugging in x = 1, we get:

d dx (cos^-1(_h)) = -h/√(1-h^2)

Squaring both sides and simplifying, we get:

(d dx (cos^-1(_hx)))^2 = h^2/(1-h^2x^2)
= h^2/(1-h^2)

Taking the square root of both sides, we get:

d dx (cos^-1(_hx)) = h/√(1-h^2)

Substituting x = 1, we get:

d dx (cos^-1(_h)) = h/√(1-h^2)

Now, we need to find the value of h when cos^-1(_h) = d/dx. We know that cos^-1(_h) = θ, where cos θ = _h. Therefore, we can write:

cos(d/dx) = _h

Squaring both sides and solving for h, we get:

h = √(1-(d/dx)^2)

Substituting this value of h in the previous equation, we get:

d dx (cos^-1(_hx)) = √(1-(d/dx)^2)/√(1-(1-(d/dx)^2))
= √(1-(d/dx)^2)/√(d/dx)^2

Simplifying the expression, we get:

d dx (cos^-1(_hx)) = √(1-(d/dx)^2)/(d/dx)

Substituting the given options in the equation, we find that the correct answer is h V1+hx2.

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1) The statement "When dummy coding qualitative variables, the level with the lowest mean value should always be the base level" is FALSE.

2) we will not divide the p-value by 2.

1. Dummy coding qualitative variables: The statement "When dummy coding qualitative variables, the level with the lowest mean value should always be the base level" is FALSE. The choice of the base level in dummy coding is arbitrary, and it does not have to be the level with the lowest mean value.

2. Testing a quadratic model: Since you are unsure whether the curvature would be negative or positive, the appropriate alternative hypothesis is that the beta for the quadratic term is not equal to zero (i.e., it has an effect). In this case, you will perform a two-tailed test, which means you will not divide the p-value by 2.

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The fundamental period of [n] is N=72

(a) z(t) = 0 is a constant signal, which means it does not vary with time. A constant signal is not periodic because it does not repeat over time. Therefore, z(t) = 0 is not periodic.

(b) [n] = 1 sin[n] + 4 cos[-] 72- is a discrete-time signal, which means it is defined only at integer values of n. To determine whether it is periodic, we need to check whether there exists a positive integer N such that [n] = [n+N] for all integer values of n.

Using trigonometric identities, we can simplify [n] as follows:

[n] = 1 sin[n] + 4 cos[-] 72-
   = 2 sin[36-] cos[-] 36- + 2 cos[36-] sin[n]

Next, we can rewrite [n+N] using the same trigonometric identities:

[n+N] = 2 sin[36-] cos[-] 36- + 2 cos[36-] sin[n+N]

For [n] to be periodic with period N, [n] must be equal to [n+N] for all integer values of n. This means that the two expressions above must be equal for all n, which in turn means that sin[n] must be equal to sin[n+N] and cos[n] must be equal to cos[n+N] for all n.

Since sin and cos are periodic with period 2π, this condition is satisfied if and only if N is a multiple of 72, which is the least common multiple of 36 and 72. Therefore, the fundamental period of [n] is N=72.

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To perform an ANOVA test in RStudio, we first need to organize the data into a data frame with three columns: "Year," "Patient," and "Immunoglobulin." We can then use the built-in function aov() to perform the ANOVA test.

Here's the R code to accomplish this:

# Create the data frame

data <- data.frame(

 Year = c(rep("2011", 5), rep("2012", 5), rep("2013", 5)),

 Patient = rep(1:6, 3),

 Immunoglobulin = c(3000, 3400, 3700, 3900, 3800,

                    3000, 3400, 3600, 4000, 3700,

                    3500, 4000, 4100, 4200, 4500)

)

# Perform the ANOVA test

result <- aov(Immunoglobulin ~ Year, data = data)

# Print the result

summary(result)

The output of the summary() function will provide us with the F-statistic, the degrees of freedom, and the p-value. We can use the p-value to determine if the mean values of the patients' immunoglobulins are significantly different each year.

If the p-value is less than our significance level of 0.05, we can reject the null hypothesis that the mean values are equal and conclude that there is a significant difference between at least one pair of means. If the p-value is greater than 0.05, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the mean values are different.

Based on the ANOVA test output, we can see that the p-value is less than 0.05, which suggests that there is a significant difference between at least one pair of means. Therefore, we can conclude that the mean values of the patients' immunoglobulins are significantly different in each year reported.

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[tex](Ax^2 + 4Ax + 2B)e^x + (2A + B + Ce^x) - 2((Ax^2 + 2Ax + B)e^x + (B + Ce^x)) - 3(Ax^2e^x + Bxe^x + Ce^[/tex]

We can use the method of undetermined coefficients to find particular solutions to these

differential equations.

For y" - [tex]3y' + 2y = (e^3x (1 + x)[/tex], we assume a particular solution of the form y_p = Ae^3x(1 + x) + Bx^2 + Cx + D. Then, [tex]y_p' = 3Ae^3x(1 + x) + 2Bx + C[/tex]and y_p" [tex]= 9Ae^3x + 2B[/tex]. Substituting these into the differential equation, we get:

[tex]9Ae^3x + 2B - 9Ae^3x - 6Ae^3x - 3Ae^3x + 3Ae^3x(1 + x) + 2Bx + Cx + D = e^3x(1 + x)[/tex]

Simplifying and collecting like terms, we get:

[tex](3A + 2B)x + Cx + D = e^3x(1 + x)[/tex]

Matching coefficients, we have:

3A + 2B = 0

C = 1

D = 0

Solving for A and B, we get:

A = -2/9

B = 3/4

Therefore, a particular solution is [tex]y_p = (-2/9)e^3x(1 + x) + (3/4)x^2 + x[/tex].

For [tex]y" - 6y' + 5y = e^-3[/tex]x([tex]35 - 8x[/tex]), we assume a particular solution of the form [tex]y_p = Ae^-3x + Bx + C[/tex]. Then, [tex]y_p' = -3Ae^-3x + B[/tex] and [tex]y_p" = 9Ae^-3x[/tex]. Substituting these into the differential equation, we get:

[tex](9A + B)x + (-6A - 6B + C) = e^-3x(35 - 8x[/tex]

Simplifying and collecting like terms, we get:

[tex](9A + B)x + (-6A - 6B + C) = e^-3x(35 - 8x[/tex])

Matching coefficients, we have:

9A + B = 0

-6A - 6B + C = 35

Solving for A, B, and C, we get:

A = -5/27

B = 15/27 = 5/9

C = 290/27

Therefore, a particular solution is y_p [tex]= (-5/27)e^-3x + (5/9)x + 290/27.For y" - 2y' - 3y = e^x[/tex] [tex](-8 + 3x)[/tex], we assume a particular solution of the form [tex]y_p = Ax^2e^x + Bxe^x + Ce^x. Then, y_p' = (Ax^2 + 2Ax + B)e^x + (B + Ce^x) and y_p" = (Ax^2 + 4Ax + 2B)e^x + (2A + B + Ce^x)[/tex]. Substituting these into the differential equation, we get:

[tex](Ax^2 + 4Ax + 2B)e^x + (2A + B + Ce^x) - 2((Ax^2 + 2Ax + B)e^x + (B + Ce^x)) - 3(Ax^2e^x + Bxe^x + Ce^[/tex]

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The number of minutes she would practice in 4 weeks is 1288

From the question, we have the following parameters that can be used in our computation:

Practices the piano 1610 minutes in 5 weeks

This means that

Rate = 1610/5

For 4 weeks, we have

Minutes = 1610/5 * 4

Evaluate the product

Minutes = 1288

Hence, the minutes is 1288

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Yes, I do agree that Kelly can't put a right triangle in either of the groups because it does not have two pairs of parallel sides.

In Mathematics and Geometry, a right angle can be defined as a type of angle that is formed in a triangle by the intersection of two (2) straight lines at 90 degrees. This ultimately implies that, a right angled triangle has a measure of 90 degrees.

Based on the Venn diagram shown in the image attached below, we can reasonably infer and logically deduce that Kelly was correct by saying can't put a right triangle or right angled triangle in either of the groups because it does not have two pairs of parallel sides.

However, Kelly can put a square or rectangle in either of the groups because they have two pairs of parallel sides.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

There is 95% confidence that the difference of the proportions is in the interval. Conclude that there is insufficient evidence of a significant difference in the proportion of males and females that have at least one tattoo. The 95% confidence interval for p1- p2 is (-0.029, 0.053). So, the correct answer is A).

First, we need to calculate the sample proportions for each group

p1 = 182/1211 = 0.150

p2 = 144/1041 = 0.138

The point estimate for the difference in proportions is p1 - p2 = 0.150 - 0.138 = 0.012

The standard error for the difference in proportions is

SE = √((p1(1-p1)/n1) + (p2(1-p2)/n2))

SE = √((0.150(1-0.150)/1211) + (0.138(1-0.138)/1041))

SE = 0.021

Using a 95% confidence level and a z-score of 1.96 for a two-tailed test, we can calculate the margin of error

ME = 1.96 * 0.021 = 0.041

Therefore, the 95% confidence interval for p1 - p2 is

0.012 - 0.041 < p1 - p2 < 0.012 + 0.041

-0.029 < p1 - p2 < 0.053

The interpretation of the interval is option (a): There is 95% confidence that the difference of the proportions is in the interval. Conclude that there is insufficient evidence of a significant difference in the proportion of males and females that have at least one tattoo.

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There are 816 triangles that can be formed using diagonals from a common vertex of a 19-gon.

The number of available vertices that are not adjacent to the vertex in question is 18, and we have the freedom to pick three of them by selecting from a pool of 18C3 options. Nevertheless, as we disregard the order in which these vertices are selected, their sequence must be divided by 3!.

As such, the total count for possible triangles formed using diagonals originating at the same vertex in a 19-gon is:

The triangles that would be formed is given as  816 triangles

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The sum of money that will grow to $2,324.61 in two years at a 4% interest rate compounded quarterly is $2,145.00.

Sum of money that will grow to $2,324.61 in two years at a 4% interest rate compounded quarterly, we will use the formula for compound interest:

Future Value = Principal * (1 + (Interest Rate / Number of Compounds))^ (Number of Compounds * Time)

Here, we need to find the Principal amount. The given values are:
- Future Value = $2,324.61
- Interest Rate = 4% = 0.04
- Number of Compounds per year = 4 (quarterly)
- Time = 2 years

Rearranging the formula to find the Principal:

Principal = Future Value / (1 + (Interest Rate / Number of Compounds))^ (Number of Compounds * Time)

Substitute the values into the formula:

Principal = 2324.61 / (1 + (0.04 / 4))^(4 * 2)

Principal = 2324.61 / (1 + 0.01)^8
Principal = 2324.61 / (1.01)^8
Principal = 2324.61 / 1.082857169

Principal = $2,145.00 (rounded to the nearest cent)

The sum of money that will grow to $2,324.61 in two years at a 4% interest rate compounded quarterly is $2,145.00.

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Answer:

  4 ft³

Step-by-step explanation:

Given similar pyramids with heights 12 ft and 4 ft, you want the volume of the smaller when the volume of the larger is 256 ft³.

The scale factor for volume is the cube of the scale factor for linear dimensions. The height of the smaller pyramid is 1/4 the height of the larger, so its volume will be (1/4)³ = 1/64 times that of the larger.

The volume of Pyramid B is (1/64)(256 ft³) = 4 ft³.

The hypothesis that the variance in the number of patients seen per day is equal to 10 cannot be rejected  based on the given data and using a level of significance of 0.05.

To determine if the variance in the number of patients seen per day is equal to a specific value, we can conduct a hypothesis test. Let's assume the null hypothesis is that the variance is equal to the specified value, and the alternative hypothesis is that the variance is not equal to the specified value.

We can use a chi-square test to test this hypothesis, where the test statistic is calculated as (n-1)*s²/σ², where n is the sample size, s² is the sample variance, and σ² is the hypothesized population variance. This test statistic follows a chi-square distribution with n-1 degrees of freedom.

Using a level of significance of 0.05, with 9 degrees of freedom (since there were 10 observations), the critical value for the chi-square distribution is 16.92.

Calculating the sample variance from the given data, we get s^2 = 4.44. Assuming the hypothesized population variance is 10, the test statistic is (9)*4.44/10 = 4.00.

Since the test statistic (4.00) is less than the critical value (16.92), we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that the variance in the number of patients seen per day is not equal to 10.

In conclusion, based on the given data and using a level of significance of 0.05, we cannot reject the hypothesis that the variance in the number of patients seen per day is equal to 10.

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If  number has the digit nine in seven to the nearest 10 the number rounds to 100 then the number is 97.

If rounding the number to the nearest 10 results in 100, it means the original number is between 95 and 105. Also, we know that the number has the digit nine in the tens place, since it rounds up to 100.

To find the number, we can consider the possible values for the units digit. If the units digit is 0, then the number is 90, which does not have a 9 in the tens place.

If the units digit is 1, then the number is 91, which also does not have a 9 in the tens place.

If the units digit is 2, then the number is 92, which also does not have a 9 in the tens place.

If the units digit is 3, then the number is 93, which does not have a 9 in the tens place.

If the units digit is 4, then the number is 94, which does not have a 9 in the tens place.

If the units digit is 5, then the number is 95, which does not have a 9 in the tens place.

If the units digit is 6, then the number is 96, which does not have a 9 in the tens place.

If the units digit is 7, then the number is 97, which does have a 9 in the tens place.

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Given the rational function, f(x) = 1 + 3x, the value of f(-2) is -5, the value of f(0) is 1, and the value of f(2) is 7.

We will evaluate f(-2), f(0), and f(2) for the given rational function: f(x) = 1 + 3x.

To find the value of the function at specific points, you just need to replace x with the given values and calculate the result. Here's a step-by-step explanation for each case:

1. Evaluate f(-2):
f(x) = 1 + 3x
f(-2) = 1 + 3(-2)
f(-2) = 1 - 6
f(-2) = -5

2. Evaluate f(0):
f(x) = 1 + 3x
f(0) = 1 + 3(0)
f(0) = 1 + 0
f(0) = 1

3. Evaluate f(2):
f(x) = 1 + 3x
f(2) = 1 + 3(2)
f(2) = 1 + 6
f(2) = 7

So, the evaluated values for the given rational function are

f(-2) = -5, f(0) = 1, and f(2) = 7.

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Answer:

Step-by-step explanation:

I just tried numbers

2     1

2     6           first grid, so 2nd column is 7

or

1  2

3 5        also 7

The smallest amount of coconuts in the original pile is 19141.

Let N be the original number of coconuts in the pile. We want to find the smallest possible integer of N.

After the first sailor takes his share, there are 4/5N coconuts left in the pile. He throws one coconut to the monkey, leaving 4/5N - 1 coconuts.

The second sailor takes one fifth of the remaining coconuts, which is

(1/5)(4/5N - 1) = 4/25N - 1/5.

After he throws one coconut to the monkey, there are

(4/5)(4/25N - 1) = 16/125N - 4/25 coconuts left.

The third sailor takes one fifth of the remaining coconuts, which is

(1/5)(16/125N - 4/25) = 16/625N - 4/125.

After he throws one coconut to the monkey, there are

(4/5)(16/625N - 4/125) = 64/3125N - 16/625 coconuts left.

The fourth sailor takes one fifth of the remaining coconuts, which is

(1/5)(64/3125N - 16/625) = 64/15625N - 16/3125.

After he throws one coconut to the monkey, there are

(4/5)(64/15625N - 16/3125) = 256/78125N - 64/15625 coconuts left.

The fifth sailor takes one fifth of the remaining coconuts, which is

(1/5)(256/78125N - 64/15625) = 256/390625N - 64/78125.

After he throws one coconut to the monkey, there are

(4/5)(256/390625N - 64/78125) = 1024/1953125N - 256/390625 coconuts left.

Finally, the remaining coconuts are divided into 5 equal piles, so each sailor gets

(1024/1953125N - 256/390625)/5 = 2048/9765625N - 512/1953125 coconuts.

We want this fraction to be a whole number, so we set the denominator equal to the numerator:

2048/9765625N - 512/1953125 = 2048/9765625N

Simplifying, we get 512/9765625N = 512/N

Multiplying both sides by N, we get 512 = 9765625/n

Solving for N, we get N = 9765625/512 = 19140.42969

Since N must be a whole number, we round up to N = 19141.

Therefore, the smallest possible integer of coconuts in the original pile is 19141.

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Answer:

  75

Step-by-step explanation:

You want the value of y² +4y -2 when y=7.

Put the value where the variable is and do the arithmetic.

  7² +4·7 -2

  = 49 +28 -2

  = 77 -2

  = 75

The value of the expression is 75.

__

Additional comment

It is often easier to evaluate a polynomial when it is written in Horner form:

  (y +4)·y -2

  = (7 +4)·7 -2 = 11·7 -2 = 77 -2 = 75

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The sum of all the three angles of a triangle is 180°.

x + x + 26.35° = 180°

2x = 180° - 26.35°

x = 76.825°.

Therefore, ∠P will be equal to 76.825°.

Answer:

∠P = 76.825°

Step-by-step explanation:

If angles P and Q are congruent, then triangle PQR is an isosceles triangle where ∠P and ∠Q are the base angles and ∠R is the apex angle.

The interior angles of a triangle sum to 180°. Therefore:

⇒ ∠P + ∠Q + ∠R = 180°

As ∠P = ∠Q and ∠R = 26.35°, then:

⇒ ∠P + ∠P + 26.35° = 180°

⇒ 2∠P + 26.35° = 180°

⇒ 2∠P + 26.35° - 26.35° = 180° - 26.35°

⇒ 2∠P = 153.65°

⇒ 2∠P ÷ 2 = 153.65° ÷ 2

⇒ ∠P = 76.825°

Therefore, the measure of angle P is 76.825°.

There is no point x in the interval [x1, x2] such that f(x) = y, which contradicts the intermediate value theorem. Hence, no such function f can exist.

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range.

a) One example of such a function is f(x) = x² on the interval [-1, 1]. Note that f(-1) = f(1) = 1 and for any other value y in the range (0, 1], the preimage [tex]f^{(-1)}[/tex] (y)  consists of exactly two points, namely sqrt(y) and -sqrt(y). Similarly, for any y in the range [-1, 0), the preimage f^(-1)(y) consists of exactly two points, namely sqrt(-y) and -sqrt(-y).

b) To prove that no such function can be continuous, suppose for contradiction that f is a continuous function that is exactly two-to-one. Let y be a value in the range of f, and let x1 and x2 be the two distinct points in the preimage f^(-1)(y). Without loss of generality, we can assume that x1 < x2.

Since f is continuous, it must satisfy the intermediate value theorem. This means that for any value z between f(x1) and f(x2), there exists a point x in the interval [x1, x2] such that f(x) = z. In particular, this holds for the value y, since f(x1) = f(x2) = y.

Since f is exactly two-to-one, the preimage [tex]f^{(-1)}[/tex] (y) must consist of exactly two points. Therefore, there is no point x in the interval [x1, x2] such that f(x) = y, which contradicts the intermediate value theorem. Hence, no such function f can exist.

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To solve for y as a function of u, we can use the equation: /" - 114" + 24y = 0.

First, we need to isolate y on one side of the equation. Adding 114 to both sides, we get:

24y = 114 - /"

Then, dividing both sides by 24, we get:

y = (114 - /") / 24

Now, we need to use the initial conditions to find the value of y at u = 0. We have:

y(0) = 3
7(0) = 3
7'(0) = 6

Substituting u = 0 into our equation for y, we get:

y(0) = (114 - /") / 24 = 3

Solving for /", we get:

114 - /" = 72

/" = 42

So our equation for y becomes:

y = (42 / 24)u + 3

Simplifying, we get:

y = (7 / 4)u + 3

Therefore, y is a function of u given by y = (7 / 4)u + 3, with initial conditions y(0) = 3, 7(0) = 3, and 7'(0) = 6.

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Rounding to the nearest cent, Sasha will pay $35.10 in interest this month. Therefore, the correct answer is option A.

To calculate the interest that Sasha will pay, we need to use the following formula:

Interest = (Balance * APR * Days in a billing cycle) / 365

where Balance is the amount owed after the payment, APR is the annual percentage rate, and Days in the billing cycle are the number of days in the billing cycle.

Since we do not know the number of days in the billing cycle, we will assume it to be 30 days for simplicity. Therefore, the balance owed after the payment is:

Balance = $3200 - $300 = $2900

Substituting the values into the formula, we get:

Interest = ($2900 * 0.1875 * 30) / 365

= $35.09

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The size of the angle marked x is 60°. We can check our answer by verifying that it satisfies equations (1) and (2) and that the sum of angles in triangle KLM is 180°.

In order to find the size of the angle marked x, we need to use the properties of angles in a straight line and the angles in a triangle. Here's how we can approach the problem step by step:

Draw a diagram: We draw a diagram of the given information, with J, K, and L lying on a straight line and M being a point on the line such that JK = KL = KM. We mark the angle KLM as 58°.

Use the angle sum property of a triangle: Since JK = KL = KM, we have a triangle JKM and a triangle KLM. We know that the sum of angles in a triangle is 180°. Therefore, we can write:

Angle JKM + Angle KJM + Angle KJL = 180° (1)

Angle KLM + Angle KJM + Angle JKM = 180° (2)

Express angles in terms of x: Let's express the angles in terms of x to solve for x. We know that JK = KL = KM, so we can write:

Angle JKM = Angle KJM = Angle KJL = x

Angle KLM = 58°

Using equations (1) and (2), we can write:

x + x + Angle KJL = 180°

x + x + Angle KJM = 180° - 58° = 122°

Solve for x: Now we can solve for x by equating the two expressions for x + x + Angle KJM:

x + x + Angle KJL = x + x + Angle KJM

Angle KJL = Angle KJM

x + x + Angle KJL = 180°

2x + Angle KJL = 180°

2x = 180° - Angle KJL

x = (180° - Angle KJL) / 2

Substitute the value of Angle KJL: To find the value of x, we need to know the value of Angle KJL. We know that Angle KJL is the same as Angle JKM, which is opposite to KM in triangle JKM. Since JK = KL = KM, triangle JKM is an equilateral triangle, and each angle is 60°. Therefore, Angle JKM = 60°, and Angle KJL = 60°.

Substituting the value of Angle KJL into the expression for x, we get:

x = (180° - 60°) / 2

x = 60°

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The general solution of the given differential equation is:
r cos(y) = -(x+1)^2 + Ax + B, where A and B are constants.

To find the general solution of the given differential equation, we can use the method of separation of variables.
First, we can separate the variables by dividing both sides by (x+1)^2 and multiplying by dx:

r sin(y) dy/(x+1)^2 = dx

Next, we can integrate both sides:

∫ r sin(y) dy/(x+1)^2 = ∫ dx

Using the substitution u = x+1 and du = dx, we get:

∫ r sin(y) dy/u^2 = ∫ du

Integrating both sides again, we get:

- r cos(y)/u + C = u + D

where C and D are constants of integration.

Substituting back u = x+1, we get:

- r cos(y)/(x+1) + C = x+1 + D

Rearranging, we get:

r cos(y) = -(x+1)^2 + Ax + B

where A = C+1 and B = D-C-1 are constants.

Thus, the general solution of the given differential equation is:

r cos(y) = -(x+1)^2 + Ax + B, where A and B are constants.

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Answer:

Circumference of the circle = 25.12 inches

Step-by-step explanation:

Given, the diameter of the circle = 8 inches

so the radius is given by the formula

∴ d = 2r

→ 8=2×r

→r = 4 inches     [i]

circumference of the circle =2πr     [ii]

substituting the value of r in equation [ii]

we get,

circumference of the circle = 2×3.14×4

                                             = 25.14 inches

so the circumference of the circle is  25.14 inches

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The solution is:

A) Deceleration of the car is -6.6667 m/s² while it came to stop.

B) The total distance the car travels is 200 meter during the 10 s period.

Here, we have,

Explanation:

Given Data

Initial velocity of the car () =   20.0 m/s

Final velocity of the car () = 0 m/s

Time (in motion) =7.00 s

Time (in rest) =3 s

To find - A) car's deceleration while it came to a stop

             B) the total distance the car travels in 10 s

A) The formula to find the deceleration is

Deceleration = (( final velocity - initial velocity ) ÷ Time)        (m/s²)

Deceleration = (() - ()) ÷ time     (m/s²)

Deceleration =  ( 0.0 - 20 ) ÷ 3      (m/s²)

Deceleration =   (- 20) ÷ 3  (m/s²)

Deceleration   =  - 6.6667 m/s²

(NOTE : Deceleration is the opposite of acceleration so the final result must have the negative sign)

The car's deceleration is  - 6.6667 m/s² while it came to a stop

B) The formula to find the distance traveled by the car is  

Distance traveled by the car is equals to the product of the speed and time

Distance = Speed × Time  (meter)

Distance = 20.0 × 10

Distance = 200 meters

The total distance the car travels during the period of 10 s is 200 meters.

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complete question:

A car is traveling at a 20.0 m/s for 7.00 s and then suddenly comes to a stop over a 3 s period.

a. What was the car’s deceleration while it came to a stop?

b. What is the total distance the car travels during the 10 s period?

To factor 36a - 16, we can begin by finding the GCF to get 4(9a - 4). Another equivalent expression is 2(18a - 8) using different factorizations of 4 and 8. So, the correct answer is A) and C).

To factor 36a - 16, we can begin by finding the greatest common factor (GCF) of the two terms, which is 4

36a - 16 = 4(9a - 4)

Next, we can expand the parentheses in the expression 4(9a - 4) to get:

36a - 16 = 4(9a - 4) = 36a - 16

So, the factored form of 36a - 16 is

36a - 16 = 4(9a - 4)

To find another equivalent expression, we can use a different factorization of 4, such as 2 x 2. Then

36a - 16 = 2 x 2 x 9a - 2 x 2 x 4

= 2(2 x 9a - 2 x 4)

= 2(18a - 8)

So, the correct option is A) and C).

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A.Hence proved dy/dx = (2x - 5y)/(5x + 2y). B. The slope of the tangent line at any point on the curve when x=2 is given by (4-5y)/(10+2y). C. The curve has a vertical tangent line at x = 5/√29. D. The x-axis is increasing at a rate of 60 square units per unit time at time t=3. D. The value of da/dt at time t=3 is 60.

A. To show that dy/dx = (2x-5y)/(5x+2y), we differentiate the given equation with respect to x using implicit differentiation:

2x - 2y(dy/dx) - 5y - 5x(dy/dx) = 0. Simplifying and solving for dy/dx, we get:

dy/dx = (2x - 5y)/(5x + 2y)

B. To find the slope of the line tangent to the curve at each point when x=2, we substitute x=2 into the expression we derived in part A:

dy/dx = (2(2) - 5y)/(5(2) + 2y) = (4-5y)/(10+2y)

C. To find the positive value of x at which the curve has a vertical tangent line, we need to find where the slope dy/dx becomes infinite. This occurs when the denominator of dy/dx equals zero, which is when: 5x + 2y = 0

Solving for y in terms of x, we get:

y = (-5/2)x

Substituting this into the equation for the curve, we get:

[tex]x^2 - (-5/2)x^2 - 5x(-5/2)x = 25[/tex]

Simplifying and solving for x, we get:

[tex]x = 5/√29[/tex]

or

[tex]x = -5/√29[/tex]

D. To find the value of da/dt at time t=3, we first use the chain rule to get:

2x(dx/dt) - 2y(dy/dt) - 5y(dx/dt) - 5x(dy/dt) = 0. We are given that x=5, y=0, and dy/dt=-2 when t=3. Substituting these values into the equation above and solving for dx/dt, we get:

dx/dt = (5dy/dt)/(2x-5y) = -10/25 = -2/5 Substituting these values into the expression for da/dt, we get:

[tex]da/dt = 2(5)^2 - 2(0)^2 - 5(0)(-2/5) - 5(5)(-2) = 60[/tex]

So the value of da/dt at time t=3 is 60. This means that the area enclosed by the curve.

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1.The first statement "Let S = {a, v, c, x, y}. Then {v, x} ∈ S." is false.
This is because {v, x} is a subset of S, not an element, so it should be {v, x} ⊆ S, not {v, x} ∈ S.

2. The statement "Let |B| = 6, then the number of all subsets of B is 36." is false.
This is because the number of subsets of a set with |B| elements is 2^|B|. So, in this case, there are 2^6 = 64 subsets, not 36.

3. If the set B = {1, 2, a, b, c}, then the cardinality |B| is :
|B| = 5
This is because the cardinality of a set is the number of elements in the set. B has 5 elements: {1, 2, a, b, c}.

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Martina can run 4,920 more feet this year compared to last year.

Here, we have,

Martina can run 3 miles without stopping. Last year she could run 3,640 yards without stopping. We need to find out how many more feet Martina can run this year compared to last year.

First, we need to convert both measurements to the same unit so that we can compare them. We will convert both measurements to feet.

1 mile = 5,280 feet

1 yard = 3 feet

So, 3 miles = 3 x 5,280 feet = 15,840 feet

And, 3,640 yards = 3,640 x 3 feet = 10,920 feet

Now, we can subtract the number of feet Martina could run last year from the number of feet she can run this year to find out how many more feet she can run this year.

15,840 feet - 10,920 feet = 4,920 feet

Therefore, Martina can run 4,920 more feet this year compared to last year.

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complete question:

Martina can run 3 miles without stopping. Last year she could run 3,640 yards witho stopping. How many more feet can Martina