The raidus of a circle increases at a rate of 0.5cm find the rate of change in the area of the circle with radius 7cm [take λ = 22/7]​

Given:-

[tex] \: [/tex]

[tex] \: [/tex]

To find:-

[tex] \: [/tex]

By using formula:-

[tex] {\color{hotpink}\bigstar} {\boxed{\sf {\green{ slope : m = \: \frac{y_2 - y_1}{x_2 - x_1} }}}}[/tex]

Solution:-

[tex] \sf \: m = \frac{y_2 - y_1}{x_2 - x_1} [/tex]

[tex] \: [/tex]

where ,

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \sf \: m = \frac{( -2 ) - ( -3 ) }{1 - 2} [/tex]

[tex] \: [/tex]

[tex] \sf \: m = \frac{ - 2 + 3}{ \: 1 - 2} [/tex]

[tex] \: [/tex]

[tex] \sf \: m = \cancel \frac{1}{ - 1} [/tex]

[tex] \: [/tex]

[tex] \underline{\boxed{ \sf{ \blue{ \: m = -1 \: }}}}[/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps:)

a) The necessary sample size for a 95% confidence interval with an interval width of no more than 6% is 147.

b) The necessary sample size for a 99% confidence interval with an interval width of no more than 6% is 263.

(a) To find the necessary sample size for future surveys such that a 95% confidence interval for p has an interval width of no more than 6%, we use the formula:

n = (Zα/2/ME)² × p(1-p)

where Zα/2 is the Z-score for the desired confidence level (1.96 for 95% confidence), ME is the margin of error (0.06), and p is the estimated proportion of patients with pain relief from the pilot survey (14/20 = 0.7). Substituting these values into the formula, we get:

n = (1.96/0.06)² × 0.7(1-0.7) = 146.95

Therefore, This sample size is larger than the sample size in problem 4.6-21(a), which was 100.

(b) To find the necessary sample size for future surveys such that a 99% confidence interval for p has an interval width of no more than 6%, we use the same formula as above but with a Z-score of 2.576 for 99% confidence:

n = (2.576/0.06)² × 0.7(1-0.7) = 262.43

Therefore, This sample size is larger than the sample size in problem 4.6-21(b), which was 176.

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The answers to the questions are as follows a)50  out of the 200 vehicles will fail in their first 2 years. b)The approximate probability that 50 or more vehicles fail in their first 2 years is essentially zero. c) the approximate probability that no more than 10 of the remaining vehicles fail in their first 2 years is essentially zero.

(a) The average lifespan of the vehicle follows an exponential distribution with a rate parameter of λ = 1/8 since the average lifespan is 8 years. Let X be the number of vehicles that fail in their first 2 years. Then X follows a Poisson distribution with parameter λt, where t is the time period of interest, which is 2 years. Therefore, the expected number of vehicles that fail in their first 2 years is E(X) = λt = (1/8)*2 = 1/4. So we would expect 1/4 * 200 = 50 of the 200 vehicles to fail in their first 2 years.

(b) The number of vehicles that fail in their first 2 years follows a Poisson distribution with parameter λt = 1/4. To approximate the probability that 50 or more of them fail in their first 2 years, we can use the normal approximation to the Poisson distribution. The mean of the normal distribution is μ = λt = 1/4, and the variance is σ^2 = λt = 1/4.

Therefore, the standard deviation is σ = [tex]\sqrt{\frac{1}{4} }[/tex]= 1/2. To standardize the random variable X, we use the formula Z = (X - μ)/σ. Therefore, Z = (50 - 1/4)/(1/2) = 99.5. Using a standard normal distribution table or calculator, the probability of Z being greater than or equal to 99.5 is essentially zero, so the approximate probability that 50 or more vehicles fail in their first 2 years is essentially zero.

(c) Given that 30 vehicles have already failed in under 2 years, 170 vehicles are remaining. We want to find the probability that no more than 10 fail in their first 2 years. Since the number of vehicles that fail in their first 2 years follows a Poisson distribution with parameter λt = 1/4, the number of vehicles that do not fail in their first 2 years follows a Poisson distribution with parameter μ = λt * (number of remaining vehicles) = (1/4)*170 = 42.5. Therefore, the probability that no more than 10 of the remaining vehicles fail in their first 2 years is the same as that a Poisson distribution with parameter 42.5 is less than or equal to 10.

The resulting Z-score is Z = (10 - 42.5)/sqrt(42.5) = -6.72. Using a standard normal distribution table or calculator, the probability of Z being less than or equal to -6.72 is essentially zero, so the approximate probability that no more than 10 of the remaining vehicles fail in their first 2 years is essentially zero.

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Answer: 9.9% (rounded to the nearest tenth

Step-by-step explanation:

Answer:

Step-by-step explanation:

[tex]C=2\pi r[/tex]

[tex]1=2\pi\time r[/tex]

[tex]r=\frac{1}{2\pi}[/tex]           ( divided both sides of equation by [tex]2\pi[/tex])

 [tex]r\approx 16cm[/tex]

The wheel's radius is approximately 16cm.

At the initial investment of $1600 compounded quarterly for 7 years with an interest of 4.12%, the amount Alyssa will get is $2,131.72.

The powerful investing principle of compounding entails generating returns on both your initial investment and returns you have already received.

You must reinvest your returns back into your account for compounding to take effect.

Consider an investment of $1,000 that yields a 6% return.

The words under and ground, for instance, are combined to get the word subterranean.

So, we know that:

Initial deposit: $1600

Interest: 4.12%

Compounded quarterly for 7 years.

Then, the amount after 7 years would be:

Using the compounding calculator we know that after 7 years Alyssa will have: $2,131.72

Therefore, at the initial investment of $1600 compounded quarterly for 7 years with an interest of 4.12%, the amount Alyssa will get is $2,131.72.

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Complete question:

Alyssa started a savings account with an initial deposit of $1600. The account earns 4.12% interest compounded quarterly for 7 years. Calculate how much money will be in the account after 7 years, assuming Alyssa

makes no additional deposits or withdrawals.

The mean and standard deviation are used to compare the centers and ranges of two symmetric distributions because they provide a measure of position and variability that are consistent with normally distributed data.

When the data are approximately normal, the mean is a good measure of the center of the distribution, and the standard deviation is a good measure of the spread of the distribution.

By comparing the means and standard deviations of two symmetric distributions, we can know their similarity or difference in terms of central tendency and variability.

On the other hand, a five-digit summary (minimum, Q1, mean, Q3, maximum) is used to compare the center and difference of two skewed or outlier distributions.

The five-digit summary provides a way to summarize the key characteristics of a distribution, and it is more robust for outliers than for mean and standard deviation.

By comparing the median and interquartile range (IQR) of two skewed or outlier distributions, we can tell their similarities or differences in terms of central tendency and variability.

In summary, mean and standard deviation are suitable for comparing centers and ranges of normally distributed data, while five-digit summaries are more suitable for comparing centers and ranges of skewed distributions.

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Step-by-step explanation:

There are  1+ 3+ 4 = 8 parts to divide

80 / 8 =  £ 10  for each part

                          1 part    = 1 x 10  = £ 10

                           3 parts = 3 x 10 = £ 30

                          4 parts = 4 x 10 = £ 40

Answer:

10

Step-by-step explanation:

1+3+4=8

80/8=10

Answer:

---------------------------

Inverse variation equation:

We have:

Substitute initial values and find the value of k:

The equation now becomes:

Find the value of V when P = 5:

Hence the answer is 166.4 in³.

Answer:6

Step-by-step explanation:

there is a law for that:

5*12=x*(x+4)

60=x*x+4x

x*x+4x-60=0

(x+2)*(x+2)-64=0

(x+2)(x+2)=8*8

x+2=8

x=6

Answer:6

Step-by-step explanation:there is a law for that:5*12=x*(x+4)60=x*x+4xx*x+4x-60=0(x+2)*(x+2)-64=0(x+2)(x+2)=8*8x+2=8x=6

The correct number line that shows the solution to the inequality y - 2 < -5 is the first option: "A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the left of the circle is shaded."

What is inequalities?

In mathematics, an inequality is a mathematical statement that indicates that two expressions are not equal. It is a statement that compares two values, usually using one of the following symbols: "<" (less than), ">" (greater than), "≤" (less than or equal to), or "≥" (greater than or equal to).

The inequality can be rewritten as y < -3, which means that y is less than -3. The open circle at -3 indicates that -3 is not included in the solution set, and everything to the left of it, which represents numbers less than -3, should be shaded.

Therefore, the correct number line that shows the solution to the inequality y - 2 < -5 is the first option: "A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the left of the circle is shaded."

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Addie measured her height on her 5th birthday, she was 44 4/5 inches tall. Each year, Addies grew 2/5 inch.  Addie height on her 12th birthday is 47 2/5 inches.

Algebra is a branch of mathematics that deals with mathematical symbols and the rules for manipulating these symbols to solve equations and understand mathematical relationships. In algebra, variables are used to represent unknown values, and equations are used to express relationships between these variables.

Algebra involves the use of mathematical operations such as addition, subtraction, multiplication, and division, as well as the use of exponents, logarithms, and other advanced mathematical concepts. Algebraic equations can be solved using various methods, such as substitution, elimination, and graphing.

Algebra has numerous practical applications in various fields, including science, engineering, economics, and finance. It is used to model and solve real-world problems, analyze data, and make predictions. Algebra is also an essential foundation for more advanced mathematical topics, such as calculus, linear algebra, and abstract algebra.

Addie was 44 4/5 inches tall on her 5th birthday. From her 5th to 12th birthday, she grew for 7 years, so her height increased by 7 * (2/5) = 2.8 inches.

Adding this to her height on her 5th birthday, we get:

44 4/5 + 2.8 = 47 2/5

Therefore, Addie was 47 2/5 inches tall on her 12th birthday.

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The type of hypothesis which posits a difference between groups, but the difference is not specified is (c) nondirectional hypothesis.

The Non-Directional hypothesis explains that there is a difference between two groups without specifying the direction of the difference.

The non-directional hypothesis is also known as a two-tailed hypothesis.

For example, a nondirectional hypothesis might be "there is a difference in intelligence between males and females" without specifying whether males or females have higher intelligence.

A nondirectional hypothesis is appropriate when there is no prior knowledge or expectation about the direction of the difference between two groups.

Therefore, the correct option is (c).

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The given question is incomplete, the complete question is

What type of hypothesis posits a difference between groups, but the difference is not specified?

(a) directional hypothesis

(b) research hypothesis

(c) nondirectional hypothesis

(d) null hypothesis.

The billing period is 31 days, so the average daily balance is $1,437.73.

Algebra is a branch of mathematics that deals with the study of mathematical symbols and the rules for manipulating these symbols to solve equations and understand relationships between variables. Algebra uses letters and symbols to represent quantities and their relationships, and it involves solving equations and inequalities, working with graphs and functions, and manipulating expressions.

Algebraic concepts and techniques are used extensively in many areas of science, engineering, economics, and other fields. It is a foundational subject in mathematics, and is often studied as a prerequisite for more advanced courses in mathematics and other sciences.

To find the average daily balance, we need to calculate the total balance over the billing period and divide it by the number of days in the billing period.

The total balance is:

10 days at $778.12 = $7,781.20

8 days at $1,876.00 = $15,008.00

3 days at $2,112.50 = $6,337.50

10 days at $1,544.31 = $15,443.10

Total balance = $44,569.80

The billing period is 31 days, so the average daily balance is:

$44,569.80 / 31 = $1,437.73

Rounding this to the nearest cent, we get an answer of $1,437.74, which is option C.

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Answer:

888 offices

Step-by-step explanation:

Since there are a total of 24 floors in the building and each floor has a total of 37 offices on it, we have to multiply the number of floors by the number of offices in total:

[tex]24 \times 37 = 888[/tex]

The number of offices in the building amounts to a total of 888, indicating a significant amount of workspace available within the premises.

To calculate the total number of offices in the building, we need to multiply the number of floors by the number of offices on each floor.

Given that the building has 24 floors and 37 offices on each floor, we can use the formula:

Total number of offices = Number of floors * Number of offices per floor

Plugging in the given values, we get:

Total number of offices = 24 * 37 = 888

Therefore, there are 888 offices in the building.

This calculation assumes that each floor has the same number of offices and that there are no variations or exceptions. It also assumes that there are no common areas or other types of spaces that are not considered offices.

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10.6 is the height, of the triangular wall .

What is known as a triangle?

The three vertices of a triangle make it a three-sided polygon. The triangle's angles are formed at a point where the three sides are joined end to end.

                               The triangle's three angles add up to a total of 180 degrees. the three different kinds of triangles that are classified according to the size of their biggest angle. These triangles are the acute, right, and obtuse triangles.

Area = 1/2 * b * h

64 = 1/2 *12 * h

64 * 2/12  = h

    h = 10.66

13) 3y+ 6 = 2x

      3y = 2x - 6

       y = 2/3x - 2

compare with y= mx + c

        slope = 2/3

           c = -2

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As the sample size becomes larger, the sampling distribution of the sample mean approaches a normal distribution. Correct option is D.

This is known as the central limit theorem, which states that the sampling distribution of the sample mean is approximately normal regardless of the underlying population distribution, as long as the sample size is sufficiently large.

The central limit theorem is a fundamental result in statistics and has many practical applications. For example, it allows us to use the normal distribution to make inferences about population parameters based on sample statistics, such as constructing confidence intervals or conducting hypothesis tests.

On the other hand, the binomial distribution describes the number of successes in a fixed number of independent trials with a constant probability of success, while the Poisson distribution describes the number of rare events that occur in a fixed interval of time or space.

The hypergeometric distribution describes the probability of drawing a specified number of objects of interest from a population of known size without replacement. These distributions are not related to the sampling distribution of the sample mean and do not converge to a normal distribution as the sample size increases.

Therefore, the correct option is D.

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Complete question is:

As the sample size becomes larger, the sampling distribution of the sample mean approaches a .

a. binomial distribution

b. poisson distribution

c. hypergeometric distribution

d. normal distribution

Factor f(x)=3x³ + 7x²2-18x+8 into linear factors is f(x) = (x+4)(3x+7)(x-7/3)

In mathematics, factorization (also known as factoring) is the process of finding the factors of a given mathematical expression or number. A factor is a number or expression that divides another number or expression exactly, leaving no remainder.

For example, the factors of 6 are 1, 2, 3, and 6 because these are the numbers that divide 6 exactly with no remainder. Similarly, the factors of x² - 4 are (x+2)(x-2) because when we multiply these two expressions together, we get x² - 4.

If -4 is a zero of f(x), then x+4 is a factor of f(x) by the factor theorem

f(x) = (x+4)(3x² - 5x - 78)

To factor the quadratic further, we can use the quadratic formula or factoring by grouping:

3x² - 5x - 78 = 0

x = (5 ± √(5² + 4(3)(78))) / (2(3))

x = (5 ± 19) / 6

x = -7/3 or x = 13/3

Therefore, we have:

f(x) = (x+4)(3x+7)(x-7/3)

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The value of the side a is 12. 8 . Option C

From the diagram shown, we have that;

m<A = 45 degrees

c = 17

m<B = 25

Note that the sum of the angles in a triangle is 180, then, we have;

<A + <B + < C = 180

substitute the values

45 + 25 + < C = 180

collect the like terms

<C = 110

Using the sine rule, we have;

sin A/a = sin B/b = sin C/c

Substitute the values, we get;

sin 45/a = sin 110/17

cross multiply the values, we have;

a = 12. 02/0. 9396

a = 12. 79

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Step-by-step explanation:

To find the volume, you have to use the formula:

V = whl

Answer:

y = 3x -7

Step-by-step explanation:

Since we already know that the slope is 3 and it goes through the point (3,2).

We can use - Slope intercept form: y = mx + b to find the y-intercept first.

Using the given info:

m = 3

x = 3

y = 2

Thus,

2 = 3(3) + b

2 = 9 + b

2- 9 = 9- 9  + b

-7 = b

Hence, the y-intercept is -7.

As a result,

y = mx + b        >> plug it in

y = 3x -7

RevyBreeze

The area of Triangle B is 2 times greater than the area of Triangle A.

What is are of triangle?

The territory included by a triangle's sides is referred to as its area. Depending on the length of the sides and the internal angles, a triangle's area changes from one triangle to another. Square units like m2, cm2, and in2 are used to express the area of a triangle.

Area of the triangle = 1/2 h*b

For Triangle A, the area is:

Area_A = 1/2 * b * h

For Triangle B, the area is:

Area_B = 1/2 * b * 2h = b * h

So, the area of Triangle B is twice the area of Triangle A.

Therefore, the area of Triangle B is 2 times greater than the area of Triangle A.

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The required 99% confidence interval representing the true proportion of all college students owns a car lies between the range of 0.3253 and 0.4283.

Sample size n = 319

Students who own a car represents the number of successes x = 120  

Confidence interval =  99%

True proportion p of all college students who own a car.

The formula for the confidence interval for a population proportion is,

p1± zα/2 × √(p1(1-p1)/n)

where p1 is the sample proportion,

zα/2 is the z-score corresponding to the desired level of confidence interval 99%.

First, we find the sample proportion,

p1 = x/n

    = 120/319

    ≈ 0.3768

z-score corresponding to a 99% confidence level.

This is a two-tailed test,

Split the alpha level evenly between the two tails.

α/2 = (1 - 0.99) / 2

     = 0.005,

The z-score that encloses 0.005 in each tail of the standard normal distribution.

Using a standard normal table ,

zα/2 = 2.576.

Substituting the values into the formula, we get,

p1 ± zα/2 × √(p1(1-p1)/n)

= 0.3768 ± 2.576×√(0.3768(1-0.3768)/319)

= 0.3768 ± 0.0515

99% confidence interval for the true proportion p of all college students who own a car is,

0.3253 ≤ p ≤ 0.4283

Therefore, 99% confidence interval that the true proportion of all college students who own a car lies between 0.3253 and 0.4283.

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The probability that the respondent has a pet given that the respondent has had a pet is approximately 0.519, or 51.9%.

To calculate the probability that a respondent has a pet given that they have had a pet, we can use Bayes' theorem:

P(Have a pet | Have had a pet) = P(Have had a pet | Have a pet) * P(Have a pet) / P(Have had a pet)

We are given that:

P(Have a pet) = 0.41

P(Have had a pet) = 0.79

We don't know P(Have had a pet | Have a pet), but we can use the fact that anyone who has a pet has also had a pet, so:

P(Have had a pet | Have a pet) = 1

Plugging in these values, we get:

P(Have a pet | Have had a pet) = 1 * 0.41 / 0.79

= 0.519 or 51.9%.

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Answer:

  y = (x +2)² -3

Step-by-step explanation:

You want the math operators that are needed to properly write the equation of the given parabola in vertex form.

The vertex is the low point on the curve. Its coordinates are (-2, -3).

The vertex form of the equation for a parabola is ...

  y = a(x -h)² +k

You want the equation for (h, k) = (-2, -3). It is ...

  y = a(x +2)² -3

The operators are plus (+) and minus (-).

Answer:

[tex]y = a(x+2)^2-3[/tex]

Explanation:

You are adding the 2 to move the whole function two times to the left and you are subtracting the 3 so that the minimum of the quadratic function is -3.

Answer:

Let’s start by defining the variables:

Let x be the number of rings manufactured per day.

Let y be the number of chains manufactured per day.

The total number of rings and chains manufactured per day is at most 24. Therefore, we have:

x + y ≤ 24

It takes 1 hour to make a ring and 30 minutes to make a chain. Therefore, we have:

1x + 0.5y ≤ 16

The profit on each ring is Rs. 300 and that on each chain is Rs. 190. Therefore, the total profit can be calculated as:

Total Profit = 300x + 190y

We want to maximize the total profit. This can be done by solving the above equations using linear programming.

The solution to this problem is x = 12 and y = 12. Therefore, the firm should manufacture 12 rings and 12 chains per day to earn the maximum profit

Answer:

To divide fractions, we can use the reciprocal (or multiplicative inverse) of the second fraction and then multiply the two fractions.

So to solve ⅚ ÷ ⅔, we can write it as:

⅚ ÷ ⅔ = ⅚ × (⅔)⁻¹

where (⅔)⁻¹ is the reciprocal of ⅔, which is 3/2.

So we have:

⅚ ÷ ⅔ = ⅚ × (⅔)⁻¹ = ⅚ × 3/2

We can simplify this expression by canceling out a common factor of 2 in the numerator and denominator:

⅚ ÷ ⅔ = ⅚ × (⅔)⁻¹ = ⅚ × 3/2 = (⅚ × 3) / 2 = 9/12

Therefore, another way to solve ⅚ ÷ ⅔ is the equation:

⅚ ÷ ⅔ = 9/12

By the angle-angle-side theorem, triangles ∠ACE and ∠ECA are congruent, and we can conclude that:

AC = EC.

What is the congruent angle?

When two parallel lines are cut by a transversal, the angles that are on the same side of the transversal and in matching corners will be congruent.

Since AB || DE and BC = DC, we can use the alternate interior angles theorem to conclude that angle BCA is congruent to angle DCE:

∠BCA = ∠DCE

We also know that angles ∠BAC and ∠ACD are supplementary:

∠BAC + ∠ACD = 180 degrees

Since AB || DE, we can use the corresponding angles theorem to conclude that ∠BAC is congruent to  ∠ECD:

∠BAC = ∠ECD

Substituting this into the equation above, we get:

∠ECD + ∠ACD = 180 degrees

Simplifying this equation, we get:

∠ECA = 180 degrees - ∠ACD

Also, ∠DCE is supplementary to ∠ACD, so:

∠DCE + ∠ACD = 180 degrees

Substituting this into the equation above, we get:

∠ECA = ∠DCE

Since angles ∠ECA and ∠DCE are congruent, we can use the vertical angles theorem to conclude that angles ∠ACE and ∠AEC are congruent:

∠ACE = ∠AEC

Therefore, by the angle-angle-side theorem, triangles ∠ACE and ∠ECA are congruent, and we can conclude that:

AC = EC.

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Number of students having one pet=27

A fraction is a number that represents a part of a whole or a ratio of two quantities. It is written in the form of one integer, called the numerator, written above a line, and another integer, called the denominator, written below the line. The denominator represents the total number of equal parts into which the whole has been divided, while the numerator represents the number of those parts that are being considered.

For example, the fraction 3/4 represents three out of four equal parts of a whole. The number 3 is the numerator, and 4 is the denominator.

Fraction of students having 1 pet=1/2

Number of students having one pet=53×1/2=26.5≈27

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