PLS HELP ME WITH THIS 2. QUESTIONS, 50 POINTS

( The first 2 images are from the first question, the other one is from the second)

Two triangles have perimeters that can be expressed as 12x + 12 and 8x - 2,  with a difference of 4x + 14, and have perimeters of 47 and 28 when x = 3.

a. The perimeter of each triangle can be found by adding up the lengths of all three sides:

Triangle 1: (4x+2) + (7x+7) + (x+3) = 12x + 12

Triangle 2: (2x-5) + (x+7) + (5x-4) = 8x - 2

b. To find the difference in perimeter between the larger triangle and the smaller triangle, we can subtract the smaller perimeter from the larger perimeter:

(12x + 12) - (8x - 2) = 4x + 14

c. To find the perimeter for each triangle when x = 3, we can substitute x = 3 into the expressions found in part (a):

Triangle 1: (4(3)+2) + (7(3)+7) + (3+3) = 47

Triangle 2: (2(3)-5) + (3+7) + (5(3)-4) = 28

Therefore, the perimeter of Triangle 1 is 47 units and the perimeter of Triangle 2 is 28 units when x = 3.

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The ratio table shows the costs for different amounts of birdseed. So, the unit rate is $2 per pound in this example.

To find the unit rate in dollars per pound, follow these steps:
1. Look at the given ratio table, which shows the costs for different amounts of bird seed.
2. Identify the cost and the corresponding amount (in pounds) of bird seed for any one row in the table.
3. Divide the cost (in dollars) by the amount (in pounds) to calculate the unit rate.

For example, if the table shows that the cost is $6 for 3 pounds of bird seed, the unit rate would be calculated as follows:

Unit rate = Cost / Amount
Unit rate = $6 / 3 pounds
Unit rate = $2 per pound
So, the unit rate is $2 per pound in this example

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Each plant should produce 576.92 units and 384.61 units respectively to maximize the company's profit.

To maximize the company's profit, we need to find the quantity that maximizes the difference between the total revenue and the total cost. The total revenue is given by:

TR = pq

= (40 - 0.04q)(q1 + q2)

= 40q1 + 40q2 - 0.04[tex]q1^2[/tex]- 0.04[tex]q2^2[/tex] - 0.04q1q2

The total cost is given by:

TC = C1 + C2

[tex]= 6.7 + 0.03q1^2 + 7.9 + 0.04q2^2= 14.6 + 0.03q1^2 + 0.04q2^2[/tex]

The profit is given by:

π = TR - TC

= [tex]40q1 + 40q2 - 0.04q1^2 - 0.04q2^2 - 0.04q1q2 - 14.6 - 0.03q1^2 - 0.04q2^2[/tex]

Simplifying, we get:

π = [tex]40q1 + 40q2 - 0.04q1^2 - 0.04q2^2 - 0.04q1q2 - 14.6 - 0.03q1^2 - 0.04q2^2[/tex]

= [tex]-0.03q1^2 - 0.04q2^2 - 0.04q1q2 + 40q1 + 40q2 - 14.6[/tex]

To maximize profit, we need to take the partial derivatives of the profit function with respect to q1 and q2 and set them equal to zero:

∂π/∂q1 = -0.06q1 - 0.04q2 + 40 = 0

∂π/∂q2 = -0.08q2 - 0.04q1 + 40 = 0

Solving these equations simultaneously, we get:

q1 = 576.92

q2 = 384.61

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The solutions to the system of nonlinear equations are

x = 9 and x =[tex]3^2[/tex]= 9.

To determine the solution to the system of nonlinear equations:

f(x) = g(x)

We can substitute the given expressions for f(x) and g(x) and simplify:

log3(x) + 3 =[tex]log3(x^3) - 1[/tex]

Using the properties of logarithms, we can simplify this equation as follows:

log3(x) + 3 = 3*log3(x) - 1

4 = 2*log3(x)

2 = log3(x)

x =[tex]3^2[/tex]

Therefore, the solutions to the system of nonlinear equations are x = 9 and [tex]x = 3^2 = 9.[/tex]

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i hope this helps you.

Step-by-step explanation:

you can solve cos(u) by

cos(u) = adjecent / hypotenes...general formula of cos

cos(u) = √44 / 12

cos(u) = 2√11 / 12 ..... √44 = √4×11 = 2√11

cos(u) = √11 / 6

u = cos^-1 ( √11 / 6 ) ..... divided both aide by cos ( multiple by cos invers )

u = 56.442 .... so we get it's angle

Answer:

[tex]cos(U)=\frac{\sqrt{11} }{6}[/tex]

Step-by-step explanation:

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. Therefore, we have:

cos(U) = adjacent/hypotenuse = TU/SU

We are given that TU = sqrt(44) and SU = 12, so:

cos(U) = sqrt(44)/12

To simplify this expression, we can first factor 44 into 4 * 11, since 4 is a perfect square and a factor of 44:

cos(U) = sqrt(4 * 11) / 12

cos(U) = (sqrt (4) * sqrt (11)) / 12

cos (U) = (2 * sqrt (11)) / 12

Simplifying the fraction by dividing both the numerator and denominator by 2, we get:

cos(U) = sqrt(11)/6

Therefore, the exact value of cos(U) in simplest radical form is sqrt(11)/6

Furthermore, if you want another way to write the answer, dividing by 6 is the same as multiplying by 1/6 so you can do cos (U) = 1/6 * sqrt (11)

Although the other individual was correct that you use inverse trig (cos ^ -1) to find the measure of U, getting an exact answer requires us to leave it in simplest radical form since the number is so large and at best will yield an approximation if you don't keep it in simplest radical form.

The instantaneous rate of change of cost with respect to x when x = 100 is 4.

We can begin by calculating C(100+h) and C(100):

C(100+h) = 0.000002(100+h)^3 + 4(100+h) + 300

C(100+h) = 0.000002(1,000,000 + 300h^2 + 30h^2 + h^3) + 400 + 4h + 300

C(100+h) = 0.000002h^3 + 0.0006h^2 + 4h + 700

C(100) = 0.000002(100)^3 + 4(100) + 300

C(100) = 2 + 400 + 300

C(100) = 702

Therefore,

C(100+h) - C(100) = (0.000002h^3 + 0.0006h^2 + 4h + 700) - 702

C(100+h) - C(100) = 0.000002h^3 + 0.0006h^2 + 4h - 2

Now, we can find the rate of change of cost with respect to x by dividing this expression by h and taking the limit as h approaches 0:

(C(100+h) - C(100))/h = (0.000002h^3 + 0.0006h^2 + 4h - 2)/h

(C(100+h) - C(100))/h = 0.000002h^2 + 0.0006h + 4 - (2/h)

As h approaches 0, the term 2/h approaches infinity, which means the rate of change of cost with respect to x is undefined. However, we can calculate the limit of the expression as h approaches 0 from the left and from the right to see if it has a finite value:

limit (h->0+) ((C(100+h) - C(100))/h) = 4

limit (h->0-) ((C(100+h) - C(100))/h) = 4

Since the left and right limits are equal, the overall limit exists and equals 4. Therefore, the instantaneous rate of change of cost with respect to x when x = 100 is 4.

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Step-by-step explanation:

Let the other rational number be represented by "x". We know that the difference of the two rational numbers is 17/28, which can be written as:

x - (-9/14) = 17/28

Simplifying the left-hand side:

x + 9/14 = 17/28

Multiplying both sides by the least common multiple of 14 and 28, which is 28, we get:

28(x + 9/14) = 28(17/28)

Simplifying:

4(2x + 9) = 17

Expanding:

8x + 36 = 17

Subtracting 36 from both sides:

8x = -19

Dividing by 8:

x = -19/8

Therefore, the other rational number is -19/8.

Since the population standard deviation is unknown, we use a t-distribution to find the critical value. The degrees of freedom for the t-distribution is n-1, where n is the sample size. In this case, n = 7, so the degrees of freedom is 7-1 = 6. The critical value for a t-distribution with 6 degrees of freedom and a significance level of α = 0.200 (two-tailed) can be found using a t-table or calculator. The critical value is approximately ±1.94.

Answer: Specifically, the pickup truck has lost about 56% of its value compared to the initial value, while the sedan has lost about 58% of its value.

Step-by-step explanation:

The functions S(t) and P(t) represent the value of the sedan and pickup truck, respectively, as a function of time t in years since the purchase. Let's analyze each function:

For S(t)=24,400(0.82)^t, the coefficient 24,400 represents the initial value or starting point of the function. This means that the value of the sedan at the time of purchase was $24,400.

The base 0.82 represents the rate of depreciation or decrease in value of the sedan over time. Specifically, the sedan's value decreases by 18% per year (100% - 82%).

For P(t)=35,900(0.71)^t/2, the coefficient 35,900 represents the initial value or starting point of the function.

This means that the value of the pickup truck at the time of purchase was $35,900. The base 0.71 represents the rate of depreciation or decrease in value of the pickup truck over time.

Specifically, the pickup truck's value decreases by approximately 29% every two years, since the exponent is divided by 2.

Comparing the two functions, we can see that the initial value of the pickup truck was higher than the initial value of the sedan.

However, the rate of depreciation of the pickup truck is greater than that of the sedan. This means that the pickup truck will lose its value at a faster rate than the sedan.

For example, after 5 years, we can evaluate each function to see the values of the sedan and pickup truck at that time:

S(5) = 24,400(0.82)^5 ≈ $10,373.67

P(5) = 35,900(0.71)^(5/2) ≈ $15,864.48

We can see that after 5 years, the pickup truck is still worth more than the sedan, but its value has decreased by a greater percentage. Specifically, the pickup truck has lost about 56% of its value compared to the initial value, while the sedan has lost about 58% of its value.

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We can estimate that the expected number of even numbers picked in 35 picks is 15.

To estimate the expected number of even numbers picked in 35 picks, we need to first understand the probability of picking an even number in one pick. Out of the seven given numbers, there are three even numbers (2, 14, 8) and four odd numbers (7, 3, 11, 5). Therefore, the probability of picking an even number in one pick is 3/7.

To find the expected number of even numbers picked in 35 picks, we can multiply the probability of picking an even number in one pick (3/7) by the number of picks (35).

Expected number of even numbers picked = (3/7) x 35 = 15

Therefore, we can estimate that the expected number of even numbers picked in 35 picks is 15. This means that if we were to repeat the process of picking a card and replacing it 35 times, we would expect to pick 15 even numbers on average.

It is important to note that this is an estimate and the actual number of even numbers picked may vary. However, this estimation gives us a good idea of what to expect on average.

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Taking the data into consideration, we can calculate and conclude that the temperature at midnight was -4, through the use of an equation.

According to the prompt, the temperature is now -7. We also know that, since midnight, the temperature tripled and then rose 5 degrees.If we let x be the temperature at midnight, then we can set up the following equation:

3x + 5 = -7

Subtracting 5 from both sides, we get:

3x = -12

Dividing by 3, we get:

x = -4

Therefore, the temperature at midnight was -4 degrees. With that in mind, we conclude we have correctly answered this question.

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The equation of the circle graphed is given as follows:

(x + 1)² + (y + 3)² = 36.

The equation of a circle of center [tex](x_0, y_0)[/tex] and radius r is given by:

[tex](x - x_0)^2 + (y - y_0)^2 = r^2[/tex]

The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle.

The coordinates of the center of the circle are given as follows:

(-1, -3).

The radius of the circle is given as follows:

r = 6 units.

Then the equation of the circle is given as follows:

(x + 1)² + (y + 3)² = 36.

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The measure of angle PQR is 20 degrees.

We have,

The measure of angle PQR can be found by using the property that the measure of an inscribed angle is half the measure of its intercepted arc.

Given that the measure of arc QR is 40 degrees, we can conclude that the measure of angle PQR is half of that, which is:

The measure of angle PQR = 1/2 x Measure of arc QR

= 1/2 x 40 degrees

= 20 degrees

Therefore,

The measure of angle PQR is 20 degrees.

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It will take 6 minutes for the water hose to fill the 2-gallon bucket at a rate of 1/3 gallon per minute.

To determine the time required to fill a 2-gallon bucket using a water hose that fills at a rate of 1/3 gallon per minute, you can use a simple calculation.

First, identify the fill rate of the hose, which is 1/3 gallon per minute. Now, consider the bucket's capacity, which is 2 gallons. To find out how many minutes it takes to fill the bucket, divide the total capacity of the bucket by the fill rate:

Time (minutes) = Bucket capacity (gallons) / Fill rate (gallons per minute)

In this case:

Time (minutes) = 2 gallons / (1/3 gallons per minute)

To solve this, you can multiply the numerator and denominator by the reciprocal of the fill rate:

Time (minutes) = 2 gallons * (3 minutes per gallon)

Time (minutes) = 6 minutes

So, it will take 6 minutes for the water hose to fill the 2-gallon bucket at a rate of 1/3 gallon per minute.

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Using the compound interest formula, the interest Brian will earn after 6 years is: $2,347.22.

We can use the formula for compound interest to calculate the interest earned by Brian:

A = P (1 + r/n)^(nt)

where:

A = the amount after t years

P = the principal amount (initial investment)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time in years

In this case, P = $9,083, r = 2.90% or 0.029, n = 4 (since interest is compounded quarterly), and t = 6.

So,

A = 9083(1 + 0.029/4)^(4*6)

= $11,430.22

The interest earned will be the difference between the amount after 6 years and the initial investment:

Interest = A - P = $11,430.22 - $9,083 = $2,347.22

Therefore, Brian will earn $2,347.22 in interest after 6 years.

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Answer: 17 or C

Step-by-step explanation:

24 times 17 is 408 and 648 - 408 is 240 ounces

847 cubic feet of cement will be needed for the job.

To find the amount of cement needed for the cistern, we need to calculate the difference in volume between the outer and inner cylinders.

First, let's find the volume of the outer cylinder:
Outer radius (R) = (Inner diameter + 2 * Wall thickness) / 2 = (10 + 2 * 1) / 2 = 6 feet
Outer height (H) = 20 feet
Outer cylinder volume (V1) = π * R^2 * H = 3.14 * 6^2 * 20 = 3.14 * 36 * 20 ≈ 2260.96 cubic feet

Next, let's find the volume of the inner cylinder:
Inner radius (r) = Inner diameter / 2 = 10 / 2 = 5 feet
Inner height (h) = Outer height - 2 * Wall thickness = 20 - 2 * 1 = 18 feet
Inner cylinder volume (V2) = π * r^2 * h = 3.14 * 5^2 * 18 = 3.14 * 25 * 18 ≈ 1413.72 cubic feet

Finally, subtract the inner cylinder volume from the outer cylinder volume to find the amount of cement needed:
Cement volume = V1 - V2 ≈ 2260.96 - 1413.72 ≈ 847.24 cubic feet

To the nearest cubic foot, approximately 847 cubic feet of cement will be needed for the job.

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The slope of the line is 3/5 and the y-intercept is 4/7.

The equation of the line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.

In this form, the slope of the line tells us how steeply the line is rising or falling, while the y-intercept tells us where the line crosses the y-axis.

In the given equation y = 3/5x + 4/7, we can see that the coefficient of x, 3/5, is the slope of the line. This means that for every 1 unit increase in x, the line will increase by 3/5 units in y.

A positive slope means that the line is rising from left to right, while a negative slope means that the line is falling.

We can also see that the constant term, 4/7, is the y-intercept of the line. This tells us that the line crosses the y-axis at the point (0, 4/7). In other words, when x is 0, y is equal to 4/7.

So, to summarize, the slope of the line y = 3/5x + 4/7 is 3/5, which means the line rises 3 units for every 5 units it moves to the right.

The y-intercept is 4/7, which tells us the line crosses the y-axis at the point (0, 4/7).

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To determine the total cost of the laptop each month if you make no payments, we need to calculate the balance on the credit card after each month, including the simple interest charged.

Starting balance = $550

Month Balance Interest Total Cost
0 $550 $0 $550
1 $616 $66 $616 + $66 = $682
2 $689.92 $73.92 $689.92 + $73.92 = $763.84
3 $770.15 $80.23 $770.15 + $80.23 = $850.38
4 $857.09 $86.94 $857.09 + $86.94 = $944.03
5 $951.19 $94.10 $951.19 + $94.10 = $1045.29

To calculate the balance for each month, we multiply the previous balance by 1.12, which represents the 12% interest charged. For example, for month 1, the balance is $550 * 1.12 = $616.

To calculate the interest charged each month, we subtract the previous balance from the new balance. For example, for month 1, the interest charged is $616 - $550 = $66.

To calculate the total cost each month, we add the new balance to the interest charged.

For example, for month 1, the total cost is $616 + $66 = $682.

Note that if you make no payments on the credit card, the balance will continue to grow each month due to the interest charged.

It is always advisable to make at least the minimum payment each month to avoid high interest charges and potential late fees.

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Answer is 64.05% probability that if we chose a student at random

To find the probability that a randomly chosen student would not like the school lunches, we need to find the complement of the probability that they do like the school lunches.

The proportion of 9th graders in the sample is 45%, so the proportion of 10th graders is 100% - 45% = 55%.

Of the 9th graders, 31% said they liked the school lunches, so the proportion that did not like them is 100% - 31% = 69%.

Of the 10th graders, 42% said they liked the school lunches, so the proportion that did not like them is 100% - 42% = 58%.

So, the probability that a randomly chosen student would not like the school lunches is:

(0.45 * 0.69) + (0.55 * 0.58) = 0.6405 or 64.05% (rounded to two decimal places).

Therefore, there is a 64.05% probability that if we chose a student at random, they would not like the school lunches.

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Answer:

The answer is -8

Step-by-step explanation:

Here's a key

()= Do first

/ = Divide

If something is outside the () and there is no symbol then multiply.

[tex]\sf y =\dfrac{2}{5}x-4[/tex]

Step-by-step explanation:

        To find the equation of the required line, first we need to find the slope of the given line in the graph.

          Choose two points from the graph.

         (0 ,4)    x₁ = 0 & y₁ = 4

         (2,-1)     x₂ = 2 & y₂ = -1

[tex]\sf \boxed{\sf \bf Slope=\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

           [tex]\sf = \dfrac{-1-4}{2-0}\\\\=\dfrac{-5}{2}[/tex]

       [tex]\sf m_1=\dfrac{-5}{2}[/tex]

[tex]\sf \text{Slope of the perpendicular line m = $\dfrac{-1}{m_1}$}[/tex]

                                                    [tex]\sf = -1 \div \dfrac{-5}{2}\\\\=-1 * \dfrac{-2}{5}\\\\=\dfrac{2}{5}[/tex]

[tex]\boxed{\sf slope \ intercept \ form \ : \ y = mx + b}[/tex]

Here, m is slope and b is y-intercept.

Substitute the m value in the above equation,

                    [tex]\sf y =\dfrac{2}{5}x + b[/tex]

    The line is passing through (5 , -2),

                     [tex]\sf -2 = \dfrac{2}{5}*5+b[/tex]

                     -2 =  2 + b

                -2 - 2 = b

                      b = -4

Slope-intercept form:

                [tex]\sf y = \dfrac{2}{5}x-4[/tex]

The simple interest earned on $1,272 deposited in a bank for 20 years at 3% annual interest rate is $764.40.

To calculate the simple interest earned on a principal amount deposited in a bank, we use the formula:

Simple Interest = (Principal) x (Rate) x (Time)

where the rate is the annual interest rate, and the time is the number of years the money is deposited.

In this case, the principal is $1,272, the rate is 3%, and the time is 20 years.

Plugging in these values into the formula, we get:

Simple Interest = (1272) x (0.03) x (20)

Simple Interest = $764.40

Therefore, the simple interest earned on $1,272 deposited in a bank for 20 years at 3% annual interest rate is $764.40.

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The solution of the composite function, (f + g)(x)  is 8x + 7

A function relates input and output. In other words, a function is a special relationship among the inputs (independent variable) and their outputs (dependent variable).

A composite function is a function that depends on another function.

Therefore, let's solve the composite function

f(x) = x + 3

g(x) = 7x + 4

Hence,

(f + g)(x) can be solved as follows:

(f + g)(x)  = f(x) + g(x)

(f + g)(x)  = x + 3 + 7x + 4

combine like terms

(f + g)(x)  = x + 7x + 3 + 4

(f + g)(x)  = 8x + 7

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The type of galaxy matched to its description.

Elliptical galaxy: B) Forms a perfect sphere or an ellipse and is flattened to some degree

Galaxy: D) Is a collection of several billion stars and interstellar matter isolated in space

Lens galaxy: A) Has a large flattened core

Spiral galaxy: C) Has a central core from which curved arms spiral outward

Elliptical galaxy: An elliptical galaxy is a type of galaxy that typically forms a perfect sphere or an ellipse shape. It is characterized by its smooth and featureless appearance, lacking the distinct spiral arms seen in spiral galaxies. Elliptical galaxies often have a flattened shape due to their rotation and gravitational interactions with other galaxies.

Galaxy: A galaxy refers to a vast collection of stars, interstellar gas, dust, and dark matter, all held together by gravity. Galaxies come in various shapes and sizes, and they can contain billions or even trillions of stars. They are the building blocks of the universe and are distributed throughout the cosmos.

Lens galaxy: A lens galaxy is a type of galaxy that has a large flattened core. It gets its name from the gravitational lensing effect it produces. Gravitational lensing occurs when the gravitational field of the lens galaxy bends and distorts the light from objects behind it, creating a lens-like effect.

Spiral galaxy: A spiral galaxy is a type of galaxy that has a central core or bulge from which curved arms spiral outward. These arms are made up of stars, gas, and dust, and they give spiral galaxies their distinct appearance. Spiral galaxies often have a flattened disk shape with a central bulge and extended arms that can stretch out across the disk.

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The expression for the area of the rectangle inscribed in a circle of radius 9, in terms of n, is:

Area = n * √(81 - n^2)

To find an expression for the area of a rectangle inscribed in a circle of radius 9, we need to use the terms "rectangle" and "circle."

First, let's denote the length of the rectangle as "n" and the width as "w." Since the rectangle is inscribed in a circle with a radius of 9, its diagonal is equal to the diameter of the circle, which is 2 * 9 = 18.

Using the Pythagorean theorem for the right triangle formed by half of the diagonal, the length, and the width, we can write the equation:

(1/2 * 18)^2 = n^2 + w^2
81 = n^2 + w^2

Now, we need to express w in terms of n. To do this, we'll isolate w from the equation:

w^2 = 81 - n^2
w = √(81 - n^2)

The area of the rectangle can be calculated as the product of its length and width:

Area = n * w
Area = n * √(81 - n^2)

So, the expression for the area of the rectangle inscribed in a circle of radius 9, in terms of n, is:

Area = n * √(81 - n^2)

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The number that is equal to the place values, 7 hundred thousands 4 thousands 3 tens and 6 ones, is 704,036

From the question, we are to determine the number that is equal to the given place values

From the given information, the given place value is

7 hundred thousands 4 thousands 3 tens and 6 ones

Now, we will write each of the values in figures

7 hundred thousands = 700,000

4 thousands = 4,000

3 tens = 30

6 ones = 6

To determine the number that is equal to the place values, we will sum all the digits

700,000 + 4,000 + 30 + 6

704,036

Hence,

The number that is equal to the place value is 704,036

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The measure of angle JML is 180 degrees because in a circle, an angle formed by two chords intersecting inside the circle.

In a circle, the measurement of angle formed by two chords intersecting inside the circle is half the sum of the arcs intercepted by the angle. Using this property, we can find the measure of angle JML.

Since angle JKL is a right angle, its intercepted arc is the diameter of the circle. Therefore, its measure is 180 degrees.

By the same property, we know that angle JML is half the sum of the arcs intercepted by it. The arcs intercepted by angle JML are arcs JL and KM.

Since angle JKL is a right angle, arc JL is also 180 degrees.

Since J, K, L, and M are concyclic, we know that angle JKM is supplementary to angle JLM. Therefore, arc KM is the supplement of arc JL and has measure 360 - 180 = 180 degrees.

Thus, the sum of the intercepted arcs is 180 + 180 = 360 degrees, and angle JML is half of this, so its measure is 180 degrees.

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To find the interval of convergence of the power series, we can use the ratio test:

lim (n->inf) |((n+1)!(x+5)^(n+1)) / (1.3.5....(2n+1))| / |(n!(x+5)^n) / (1.3.5....(2n-1))|

= lim (n->inf) |(x+5)(2n+1)| / (2n+2) = |x+5| lim (n->inf) (2n+1)/(2n+2) = |x+5|

So the series converges if |x+5| < 1, and diverges if |x+5| > 1. Thus the interval of convergence is (-6, -4).

To check for convergence at the endpoints, we can use the limit comparison test with the divergent series:

1/1.3 + 1/1.3.5 + 1/1.3.5.7 + ... = sum (2n-1) terms = inf

At x = -6, we have:

sum (n=0 to inf) (n!(-1)^n)/(1.3.5....(2n-1)) = 1/1 - 1/1.3 + 1/1.3.5 - 1/1.3.5.7 + ... = inf

Since the series diverges at x = -6, the interval of convergence is (-6, -4] using set notation.

At x = -4, we have:

sum (n=0 to inf) (n!(1)^n)/(1.3.5....(2n-1)) = 1/1 - 1/1.3 + 1/1.3.5 - 1/1.3.5.7 + ... = 1 - 1/3 + 1/15 - 1/105 + ...

This is an alternating series that satisfies the conditions of the alternating series test, so it converges. Thus the interval of convergence is (-6, -4] using set notation, or [-6, -4) using interval notation.
To find the interval of convergence of the power series, we'll use the Ratio Test, which states that if the limit L = lim(n→∞) |aₙ₊₁/aₙ| < 1, then the series converges. Here, the series is given by:

Σ(n!(x+5)^n) / 1 . 3 . 5 ... (2n-1)

Let's find the limit L:

L = lim(n→∞) |(aₙ₊₁/aₙ)|
 = lim(n→∞) |((n+1)!(x+5)^(n+1))/(1 . 3 . 5 ... (2(n+1)-1)) * (1 . 3 . 5 ... (2n-1))/(n!(x+5)^n)|

Now, simplify the expression:

L = lim(n→∞) |(n+1)(x+5)/((2n+1))|

For the series to converge, we need L < 1:

|(n+1)(x+5)/((2n+1))| < 1

As n approaches infinity, the above inequality reduces to:

|x+5| < 1

Now, to find the interval of convergence, we need to solve for x:

-1 < x + 5 < 1
-6 < x < -4

The interval of convergence is given by the interval notation (-6, -4). To check the endpoints, we need to substitute x = -6 and x = -4 back into the original series and use other convergence tests such as the Alternating Series Test or the Integral Test. However, the power series will diverge at the endpoints, as the terms do not approach 0. Therefore, the interval of convergence remains (-6, -4).

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