What is 1 2/3 as an improper fraction

The area of the region that is bounded by the given curve and lies in the specified sector is A = 2(e^(7π/12) - e^(π/12))

The polar curve r = e^(θ/2) represents a spiral that starts from the origin and gets farther away as it unwinds. We want to find the area of the region that lies inside this spiral and inside the sector defined by the angles θ = π/6 and θ = 7π/6.

To solve the problem, we need to find the points where the curve intersects the sector, which are given by plugging in the values of θ:

r(π/6) = e^(π/12)

r(7π/6) = e^(7π/12)

Then we can set up the integral for the area inside the sector:

A = 1/2 ∫[π/6, 7π/6] (r(θ))^2 dθ

Substituting the equation for r:

A = 1/2 ∫[π/6, 7π/6] e^θ/2 dθ

Using the power rule for integration:

A = 2(e^(7π/12) - e^(π/12))

This is the exact value of the area inside the sector and inside the spiral. If we want a decimal approximation, we can use a calculator or computer software to evaluate it.

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The given statement "Star clusters with lots of bright, blue stars of spectral type O and B are generally younger than clusters that don't have any such stars." is True. The reason for this is that O and B stars are short-lived and burn through their fuel quickly.

The reason for this is that O and B stars burn through their fuel quickly, causing them to exhaust their nuclear fuel and end their lives in a relatively short period, typically within a few tens of millions of years.

On the other hand, stars of lower mass and cooler temperatures, like G and K type stars like our sun, have longer lifetimes and take billions of years to exhaust their nuclear fuel.

Therefore, clusters without any bright, blue stars are likely to have evolved for longer periods, allowing these short-lived stars to have already expired.

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Answer:

it should be true because sum of 3 interior angle of a triangle is 180 degree

Answer:

True.

Step-by-step explanation:

A triangle's angles add up to 180 degrees because one exterior angle is equal to the sum of the other two angles in the triangle. In other words, the other two angles in the triangle (the ones that add up to form the exterior angle) must combine with the third angle to make a 180 angle.

The graph of the linear function 4x + 6y = -12 is given by the image presented at the end of the answer.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

The function for this problem is given as follows:

4x + 6y = -12.

In slope-intercept form, the function is given as follows:

6y = -4x - 12.

y = -2x/3 - 2.

The slope and the intercept are given as follows:

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Answer:

Area = 559.17 square feet

Perimeter = 94.26 ft

Step-by-step explanation:

Make sure all the units are the same and consistent.

r = radius of semi-circle

 = [tex]\frac{Diameter}{2}[/tex]

 = [tex]\frac{18}{2}[/tex] ft

 = 9 ft

Area of composite figure = Area of rectangle + Area of semi-circle:

                                          = [Length × Breadth] + [[tex]\frac{1}{2}[/tex] × (Area of circle)]
                                          = [24 ft × 18 ft] + [[tex]\frac{1}{2}[/tex] × ([tex]\pi r^{2}[/tex])]

                                          = 432 [tex]ft^{2}[/tex] + [[tex]\frac{1}{2}[/tex] × ([tex]\pi 9^{2}[/tex])] [tex]ft^{2}[/tex]

                                          = 432 + [[tex]\frac{1}{2}[/tex] × (3.14) ×(81)]

                                          = 559.17[tex]ft^{2}[/tex]

Perimeter of composite figure =

     Circumference of semi-circle + 3 outer sides of rectangle:

 = [[tex]\frac{1}{2}[/tex] × [tex]2\pi r[/tex]] + [24 + 18 + 24]

 = ( [tex]\pi r[/tex] + 66) ft

 = [(3.14)(9) + 66] ft

 = 94.26 ft

The probability of obtaining a reading less than 0.35° C is approximately 35%.

Accοrding tο the prοbability fοrmula, the likelihοοd οf an event οccurring is equal tο the ratiο οf the number οf favοurable οutcοmes tο the tοtal number οf οutcοmes. Prοbability οf an event οccurring P(E) = The number οf favοurable οutcοmes divided by the tοtal number οf οutcοmes.

The readings at freezing οn a set οf thermοmeters are nοrmally distributed, with a mean (x) οf 0°C and a standard deviatiοn (μ) οf 1.00°C. We want tο knοw hοw likely it is that we will get a reading that is less than 0.35°C.

To solve this problem, we must use the z-score formula to standardise the value:

[tex]$Z = \frac{x - \mu}{\sigma}[/tex]

Z = standard score

x = observed value

[tex]\mu[/tex] = mean of the sample

[tex]\sigma[/tex] = standard deviation of the sample

Here

x = 0.35° C

[tex]\mu[/tex] = 0° C

[tex]\sigma[/tex] = 1.00°C

Using the values on the formula:

[tex]$Z = \frac{0.35 - 0}{1}[/tex]

Z = 0.35

The probability of obtaining a reading less than 0.35° C is approximately 35%.

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Mr. Woodstock will need to purchase 144 meters of wire to fully encircle his land. He will need to measure the length of the four sides of the land and add them together. The four sides measure 36 meters + 36 meters + 16 meters + 16 meters, which equals a total of 104 meters. He should buy enough wire to cover an additional 40 meters to account for any extra material he may need. Therefore, he needs to purchase 144 meters of wire for his fencing.

Accοrding tο the half-life fοrmula, we wοuld have tο wait apprοximately 4975 years fοr 1/7 οf the radium tο decay.

Expοnential decay is a mathematical prοcess in which a quantity decreases οver time in a manner prοpοrtiοnal tο its current value. This means that the rate οf decay is prοpοrtiοnal tο the amοunt οf the substance remaining, and as the amοunt οf the substance decreases, the rate οf decay alsο decreases. The fοrmula fοr expοnential decay is οften written as:

N(t) = N₀ *[tex]e^{(-kt)[/tex]

where N(t) is the amοunt οf substance remaining at time t, N₀ is the initial amοunt οf the substance, k is the decay cοnstant, and e is the base οf the natural lοgarithm.

The half-life οf radium is apprοximately 1590 years, which means that after 1590 years, half οf the οriginal radium will have decayed. Therefοre, we can use the half-life fοrmula tο find the amοunt οf time it wοuld take fοr 1/7 οf the radium tο decay:

N = N₀[tex]* (1/2)^{(t/t1/2)[/tex]

where N is the final amοunt (1/7 οf the οriginal amοunt), N0 is the initial amοunt, t is the time elapsed, and t1/2 is the half-life.

We can rearrange this fοrmula tο sοlve fοr t:

t = t1/2 * lοg2(N₀/N)

t = 1590 years * lοg2(7)

t ≈ 4975 years

Therefοre, we wοuld have tο wait apprοximately 4975 years fοr 1/7 οf the radium tο decay.

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Answer: 13,708 ft

Step-by-step explanation:

To find the difference in elevation between the mountain and the valley, we need to subtract the elevation of the valley from the elevation of the mountain:

13,318 ft (mountain) - (-390 ft) (valley) = 13,318 ft + 390 ft = 13,708 ft

Therefore, the difference in elevation between the mountain and the valley is 13,708 ft.

Answer: The difference is 13,708 ft.

Given that a mountain is 13,318 feet above sea level. So the elevation of the mountain is [tex]= +13,318 \ \text{ft}[/tex].

Given that a valley is 390 feet below sea level.

So the elevation of the valley is [tex]= -390 \ \text{ft}[/tex].

So the difference between them is [tex]= 13,318 - (-390) = 13,318 + 390 = 13,708 \ \text{ft}.[/tex]

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Answer:

Step-by-step explanation:

To calculate the interest earned by Linda for the first year, we can use the formula:

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

Where A is the amount after t years, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

For the first year, we have:

A = $50,000(1 + 0.06/1)^(1*1) = $53,000

So, the interest earned by Linda for the first year is:

Interest = $53,000 - $50,000 = $3,000

For the second year, we can use the same formula with t = 2:

A = $50,000(1 + 0.06/1)^(1*2) = $56,180

Interest = $56,180 - $53,000 = $3,180

For the third year, we can use the same formula with t = 3:

A = $50,000(1 + 0.06/1)^(1*3) = $59,468.80

Interest = $59,468.80 - $56,180 = $3,288.80

Now, to calculate the interest earned by Bob for each of the first three years, we can use the formula:

Interest = Prt

Where P is the principal amount, r is the annual interest rate, and t is the time in years.

For the first year, we have:

Interest = $50,0000.061 = $3,000

For the second year, we have:

Interest = $50,0000.061 = $3,000

For the third year, we have:

Interest = $50,0000.061 = $3,000

As we can see, Linda earns more interest than Bob for each year, as her interest is compounded annually, while Bob's interest is simple interest. Therefore, the answer is:

Linda earns more.

Answer:

Linda earns $9550.8 interest and bob earns $9000 interest

Step-by-step explanation:

Linda takes compound interest: C.I. = Principal (1 + Rate)Time − Principal

interest= 50,000(1+6/100)³

=59550.8 - 50000

Linda earns $9550.8 interest in 3 years.

bob takes simple interest: S.I = prt/100

interest = 50,000*6*3/100

Bob earns $9000 in 3 years.

thus, Linda earns more interest than bob.

Answer: We are given that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C.

To find the probability of obtaining a reading between 0°C and 1.08°C, we need to calculate the z-scores for these values using the formula:

z = (x - mu) / sigma

where x is the value we are interested in, mu is the mean, and sigma is the standard deviation.

For x = 0°C, we have:

z1 = (0 - 0) / 1.00 = 0

For x = 1.08°C, we have:

z2 = (1.08 - 0) / 1.00 = 1.08

Using a standard normal table or a calculator, we can find the probability of obtaining a z-score between 0 and 1.08.

Using a standard normal table or a calculator, we find that the probability of obtaining a z-score between 0 and 1.08 is 0.3583.

Therefore, the probability of obtaining a reading between 0°C and 1.08°C is 0.3583, rounded to 4 decimal places.

Step-by-step explanation:

Answer:

The value of the derivative at (-2/3, 2√3/3) is zero.

Step-by-step explanation:

Given function:

[tex]f(x)=-3x\sqrt{x+1}[/tex]

To differentiate the given function, use the product rule and the chain rule of differentiation.

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Product Rule of Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{7 cm}\underline{Differentiating $[f(x)]^n$}\\\\If $y=[f(x)]^n$, then $\dfrac{\text{d}y}{\text{d}x}=n[f(x)]^{n-1} f'(x)$\\\end{minipage}}[/tex]

[tex]\begin{aligned}\textsf{Let}\;u &= -3x& \implies \dfrac{\text{d}u}{\text{d}{x}} &= -3\\\\\textsf{Let}\;v &= \sqrt{x+1}& \implies \dfrac{\text{d}v}{\text{d}{x}} &=\dfrac{1}{2} \cdot (x+1)^{-\frac{1}{2}}\cdot 1=\dfrac{1}{2\sqrt{x+1}}\end{aligned}[/tex]

Apply the product rule:

[tex]\implies f'(x) =u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}[/tex]

[tex]\implies f'(x)=-3x \cdot \dfrac{1}{2\sqrt{x+1}}+\sqrt{x+1}\cdot -3[/tex]

[tex]\implies f'(x)=- \dfrac{3x}{2\sqrt{x+1}}-3\sqrt{x+1}[/tex]

Simplify:

[tex]\implies f'(x)=- \dfrac{3x}{2\sqrt{x+1}}-\dfrac{3\sqrt{x+1} \cdot 2\sqrt{x+1}}{2\sqrt{x+1}}[/tex]

[tex]\implies f'(x)=- \dfrac{3x}{2\sqrt{x+1}}-\dfrac{6(x+1)}{2\sqrt{x+1}}[/tex]

[tex]\implies f'(x)=- \dfrac{3x+6(x+1)}{2\sqrt{x+1}}[/tex]

[tex]\implies f'(x)=- \dfrac{9x+6}{2\sqrt{x+1}}[/tex]

An extremum is a point where a function has a maximum or minimum value.

From inspection of the given graph, the maximum point of the function is (-2/3, 2√3/3).

To determine the value of the derivative at the maximum point, substitute x = -2/3 into the differentiated function.

[tex]\begin{aligned}\implies f'\left(-\dfrac{2}{3}\right)&=- \dfrac{9\left(-\dfrac{2}{3}\right)+6}{2\sqrt{\left(-\dfrac{2}{3}\right)+1}}\\\\&=-\dfrac{0}{2\sqrt{\dfrac{1}{3}}}\\\\&=0 \end{aligned}[/tex]

Therefore, the value of the derivative at (-2/3, 2√3/3) is zero.

Answer:

Angle AIC is vertical.

Step-by-step explanation:

Defn of vertical angles

Answer: 45.7

Explanation:

Divide the area value by 10000

The value of h(x) using exponents are as follows:

For -1, the value of h(x)=1/10

For 0, the value of h(x) = 1

For 1, the value of h(x) = 10

For 2, the value of h(x) = 100

For 3, the value of h(x) = 1000

The exponent of a number tells us how many times the original value has been multiplied by itself. For instance, 2×2×2×2 can be expressed as [tex]2^{4}[/tex] the result of 4 times multiplying 2 by itself. Thus, 4 is referred to as the "exponent" or "power," while 2 is referred to as the "base."

Generally speaking, [tex]x^{n}[/tex] denotes that x has been multiplied by itself n times. Here x is the base and n is the power.

Now here, as we put the value of x in the equation, h(x) we can get the value of h(x) for each value of x.

So,

For -1, the value of h(x)=1/10

For 0, the value of h(x) = 1

For 1, the value of h(x) = 10

For 2, the value of h(x) = 100

For 3, the value of h(x) = 1000

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The equation of the line passing through (2,5) with a slope of 3 is y = 3x - 1.

This question is incomplete, the complete question is:

What is the equation of line passing through: (2,5), and with a slope = 3​?

The equation of a line in slope-intercept form is expressed as:

y = mx + b

Where m is the slope and b is the y-intercept.

Given that, the point (2, 5) and the slope of the line is 3.

We can use the point-slope form of the equation of a line to find the equation in slope-intercept form:

y - y1 = m(x - x1)

Where x1 and y1 are the coordinates of the given point ( 2,5 ) and m is slope 3.

Substituting the given values, we get:

y - y1 = m(x - x1)

y - 5 = 3(x - 2)

Expanding and rearranging, we get:

y - 5 = 3x - 6

y = 3x - 1

Therefore, the equation of the line is y = 3x - 1.

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Answer:

 (c) ∠STU

Step-by-step explanation:

Transversal UT between parallel sides RU and ST creates alternate interior angles RUT and STU. These are congruent.

 ∠STU has the same measure as ∠RUT

_____

The figure shown is a trapezoid, not a parallelogram.

Answer:

Step-by-step explanation:

2

Answer:

258 m/s

250 m/s

Step-by-step explanation:

s = 2t³

s(5) = 2(5)³ = 250

s(8) = 2(8)³ = 1024

average = (s(8) - s(5))/(8 - 5) = (1024 - 250)/3 = 258 m/s

s(5) = 2(5)³ = 250 m/s

Answer: 56

Step-by-step explanation:

One possible method to estimate the population of snapping turtles in the area is by using the mark and recapture method, also known as the Lincoln-Petersen index.

According to this method, the population size can be estimated by dividing the number of marked individuals in the second sample by the proportion of marked individuals in the combined sample. In other words:

Estimated population size = (Number of individuals in sample 1 × Number of individuals in sample 2) / Number of marked individuals in sample 2

Using the information provided in the problem, we can fill in the formula as follows:

Estimated population size = (15 × 15) / 4

Estimated population size = 56.25

Rounding to the nearest whole number, we get an estimated population size of 56 snapping turtles in the area.

The equation which correctly relates the measure of diameter and radius of a circle is (c) r = d/2.

The Diameter (d) of a circle is defined as the distance across the circle through its center. The radius (r) of a circle is defined as the distance from the center of the circle to any point on the circle.

We know that the radius of the circle is half of diameter, because it extends from the center to the edge of the circle, while the diameter extends all the way across the circle.

So, we can express the relationship between d and r as:

⇒ d = 2r

To solve for r, we can divide both sides by 2:

We get,

⇒ r = d/2

Therefore, The correct equation is Option (c) r = d/2.

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The given question is incomplete, the complete question is

Which of the following correctly relates the measures of the diameter (d) and radius (r) of a circle?

(a) d = r/2

(b) r = 2d

(c) r = d/2

(d) d = 2/r

Answer:

10 in.

Step-by-step explanation:

V = LWH

100 = L × 2.5 × 4

L = 10

Answer: Yes;

Step-by-step explanation: The function is linear in that the X values are increasing by 2 and the Y values are increasing by 3.  

Answer:

Not linear

Step-by-step explanation:

X | -7 | -5 | -3 | -1 | 0

Y | 11 | 14 | 17 | 20 | 23

-5 - (-7) = +2

14 - 11 = +3

-3 - (-5) = +2

17 - 14 = +3

-1 - (-3) = +2

20 - 17 = +3

0 - (-1) = +1

23 - 20 = +3

There is a linear function graphed that would pass through the first 4 points, but not the last one;

You can discern this as there is a common pattern we can identify, everytime the x-value increases by 2 the y-value increases by 3, except with the last coordinates;

The rate of change, also known as the gradient, is the change in x divided by the change in y (Δx/Δy);

In the case of the first 4 points, the rate of change would be 3/2 or 1.5;

This simply means when x increases by 1, y increases by 1.5, IF the function is linear;

However, as mentioned the last point doesn't fit the pattern;

There, we can see the y-value increases by 3 when the x-value only increases by 1;

This means the point (0, 23) isnt on the graph of the function, the points (0, 21.5) and (1, 23), on the other hand, would be.

The calculated value of the indicated missing length x in the right triangle is 12

Given the right triangle

We can start by calculating the value of x using the following equivalent ratio

x : 9 = 25 - 9 : x

Evaluate the difference

This gives

x : 9 = 16 : x

Next, we express the equivalent ratio as a fraction

So, the ratio becomes

x/9 = 16/x

Cross multiply the equation to calculate x

So, we have the following

x * x = 9 * 16

Evaluate the product

x² = 144

Take the square root of both sides

So, we have the solution to be

y = 12

Hence, the value of x is 12

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The area of the shaded region as a percentage of the area of the sector ONQ= 60.3%

The shape of an equilateral triangle is an equilateral triangle.

The word "Equilateral" is formed by combining two words. H. "Equi" means equal, "lateral" means side.

Equilateral triangles are also called regular polygons or equilateral triangles because all sides are equal.

In geometry, an equilateral triangle is a triangle with all sides of equal length.

Three sides are equal, so three angles on the same side are equal. Therefore, it is also called an equilateral triangle with each angle of 60 degrees.

Like other types of triangles, equilateral triangles have formulas for area, perimeter, and height. 

According to our question-

AB=OA=BO= 9CM

ONQ-AOB/ONQ*100

PUTTING VALUES

60.3%

Hence, The area of the shaded region as a percentage of the area of the sector ONQ= 60.3%

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Therefore , the solution of the given problem of standard deviation comes out to be option C with n = 1,000 and p near to 1/2 is the right response.

Statistics uses variance as a way to quantify difference. The image of the result is used to compute the average deviation between the collected data and the mean. Contrary to many other valid measures of variability, it includes those pieces of data on their own by comparing each number to the mean. Variations may be caused by willful mistakes, irrational expectations, or shifting economic or business conditions.

Here,

The following algorithm determines the standard deviation of the sampling distribution of a sample proportion p:

=> √((p*(1-p))/n)

where n is the sample size, and p is the population percentage.

For the sampling distribution of a sample proportion p,

the pair of sample number n and population proportion p that would result in the highest standard deviation is:

=>n =1,000, and p is almost half.

Because p=1/2

yields the highest possible value of the expression (p*(1-p)), a bigger sample size will result in a smaller standard deviation.

The standard deviations will be lower for the other choices, which have smaller sample sizes or extreme values of p.

Therefore, (C) with n = 1,000 and p near to 1/2 is the right response.

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[tex] \huge \: \tt \green{Answer} [/tex]

Olivia and kieran share ratio 2 : 5

[tex] \texttt{olivia's share \: of \: money = £42 }= \frac{2}{7} \\ [/tex]

Total Amount of Money = Olivia's share of money × Reciprocal of olivia's share

[tex] \tt \: = > 42 \times \frac{7}{2} \\ \\ = > 147[/tex]

Kieran's share of Money =

[tex] = > 147 \times \frac{5}{7} \\ \\ = > \sf{ \pink{£105}}[/tex]

i) Find the mean, median, and mode of the frequency table as follows:

ii) The average that justifies the teacher's statement congratulating the class that 'over three quarters were above average' is the average mark of 10, which is 5.

The mean refers to the average or the quotient of the total values divided by the number of items.

The median is the middle value in the data, which occurs with marks 8 for the 13th and 14th students.

The mode is the value that occurs most frequently, which is 3 which occurs 6 times.

Frequency Table:

Mark   Frequency  Cumulative Frequency

3              6                            18 (0 + 3 x 6)

4              3                            30 (18 + 4 x 3)

5              1                            35 (30 + 5 x 1)

6              2                           47 (35 + 6 x 2)

7              0                           47 (47 + 7 x 0)

8              5                           87 (47 + 8 x 5)

9              5                         132 (87 + 9 x 5)

10            4                         172 (132 + 10 x 4)

Mean = 6.6 (172/26)

Median = 8

Mode = 3

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