a line segment 42 inches long is divided into 3 portions in a ration of 1;2;4. how many inches longer is the longest portion than the shortest protion

The region inside cardioid r = 1 + cos(θ) and outside the circle r = 3 cos(θ) is equal to [tex]\frac{\pi}{4}[/tex]

Now, According to the question:

We will first draw both of them on the same plane and then find their point of intersection. Then using the given information, we will shade the region whose area we have to find.

The point of intersection of  r = 1 + cos(θ) and  r = 3 cos(θ)

1 + cos(θ)  = 3 cos(θ)

1 = 2 cos(θ)

=> cos(θ) = 1/2

=> θ = [tex]cos^-^1[/tex][tex]\frac{1}{2}[/tex]

=> θ = ±[tex]\frac{\pi }{3}[/tex]

So, two curves intersect at θ = ± [tex]\frac{\pi }{3}[/tex]

Area of the cardioid,

[tex]A_1 = \int\limits^\pi _\frac{\pi }{3} {\frac{1}{2}(1+cos\theta)^2 } \, d\theta= \frac{1}{2} \int\limits^\pi _\frac{\pi }{3} (1+cos^2\theta+2cos\theta) } \, d\theta=\frac{1}{2}[/tex]

          [tex]\int\limits^\pi _\frac{\pi }{3} {(1 + \frac{cos2\theta+1}{2} +2cos\theta)} \, d\theta[/tex]

 => [tex]A_1 = \frac{1}{2}[\theta+\frac{1}{2}(\frac{sin2\theta}{2} +\theta)+2sin\theta ]^\pi _\frac{\pi }{3}[/tex]

=> [tex]A_1= \frac{1}{2}[\frac{3\pi }{2}+0+0-\frac{\pi }{2}-\frac{\sqrt{3} }{8} -\sqrt{3} ][/tex]

=> [tex]A_1= \frac{1}{2} [\pi +\frac{-9\sqrt{3} }{8} ]\\\\A_1 = \frac{\pi }{2} - \frac{9\sqrt{3} }{16}[/tex]

Area of the circle,

[tex]A_2 = \int\limits^\frac{\pi }{2} _\frac{\pi }{3} {\frac{1}{2}(3cos\theta)^2 } \, d\theta \\\\A_2 = \frac{9}{2} \int\limits^\frac{\pi }{2} _\frac{\pi }{3} \frac{cos2\theta+1}{2}d\theta\\ \\A_2 = \frac{9}{4}[\frac{sin2\theta}{2}+\theta ]^\frac{\pi }{2}_\frac{\pi }{3}[/tex]

[tex]A_2 = \frac{9}{4}[0+\frac{\pi }{2} -\frac{\sqrt{3} }{4} -\frac{\pi }{3} ][/tex]

[tex]A_2=\frac{9}{4}[\frac{\pi}{6}-\frac{\sqrt{3} }{4} ][/tex]

[tex]A_2= \frac{3\pi }{8}-\frac{9\sqrt{3} }{16}[/tex]

Area of the shaded region,

[tex]A = A_1-A_2[/tex]

[tex]A = \frac{\pi }{2} - \frac{9\sqrt{3} }{16} - \frac{3\pi }{8}+\frac{9\sqrt{3} }{16}[/tex]

[tex]A = \frac{\pi }{8}[/tex]

This is the area of the shaded region in the first and the second quadrant. We see that by symmetry, the area of the shaded region in the first and the second quadrant is equal to the area of the shaded region in the third and the fourth quadrant.

So, the total area is 2 ×[tex]\frac{\pi}{8}[/tex] = [tex]\frac{\pi}{4}[/tex]

Hence, the region inside cardioid r = 1 + cos(θ) and outside the circle r = 3 cos(θ) is equal to [tex]\frac{\pi}{4}[/tex]

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The principal stresses are the maximum normal stresses in the x, y, and z directions.

These are typically determined using Mohr's circle. The absolute maximum shear stress is the maximum shear stress that can act on a given material, which is equal to one-half the difference between the two principal stresses.

The principal stresses are determined using Mohr's circle and the absolute maximum shear stress is equal to one-half the difference between the two principal stresses.

The principal stresses are the maximum normal stresses in the x, y, and z directions.

These are typically determined using Mohr's circle. The absolute maximum shear stress is the maximum shear stress that can act on a given material, which is equal to one-half the difference between the two principal stresses.

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Answer:    10. equal 0.27

                  11. 5.12

                  12. 3.35

Step-by-step explanation:

10. 0.9 * 0.3 = 0.27

11. 1.6 * 3.2 = 5.12

12. 0.5 * 6.7 = 3.35

13. 0.81*7.3 = 5.873

14. 5.6*3.9 = 21.84

15. 1.2 * 0.05 = 0.06

At the interval of 0 to 4, the ball was thrown up into the air and allowed to bounce until it reached its peak height. Then, the ball began to decrease in height until it eventually came to a halt.

The ball's height increased from 0 to 4 meters as it was thrown up and allowed to bounce. Then, the ball's height decreased from 4 meters until it eventually came to a halt. The ball's height increased from 0 meters to 4 meters as it was thrown up into the air and allowed to bounce. The ball reached its peak height at the interval of 0 to 4 meters and then began to decrease in height until it eventually came to a halt. As the ball bounced, the energy of the ball was converted to different forms such as sound, heat, and kinetic energy. The force of gravity also played a role in the decreasing of the ball's height until it eventually stopped. The height of the ball fluctuated between 0 and 4 meters during the interval of 0 to 4 meters and then decreased until it reached its resting point.

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The inverse function of y = 2(x - 5)² is  [tex]y = \pm\sqrt\frac{x}{2} + 5.[/tex]

First to be an inverse function that function needs to one to one function, meaning every different preimage must correspond to a different image.

We can obtain the inverse of a function by switching the variables x and y with their respective positions and solving for y in terms of x.

Let, y = 2(x - 5)².

y/2 = (x - 5)².

Solving for 'x'.

[tex]x - 5 = \pm\sqrt\frac{y}{2}[/tex].

[tex]x = \pm\sqrt\frac{y}{2} + 5.[/tex]

Now, We have to interchange the positions of x and y.

[tex]y = \pm\sqrt\frac{x}{2} + 5.[/tex]

So, The inverse of y = 2(x - 5)² is [tex]y = \pm\sqrt\frac{x}{2} + 5.[/tex]

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Answer:

For example, a spacecraft leaving the surface of Earth needs to be going 7 miles per second, or nearly 25,000 miles per hour to leave without falling back to the surface or falling into orbit.

a. Since both a strong-positive and weak-positive correlation are formally considered to be "growing," this is ambiguous (positive slope). In the same way, negative correlation. Speaking strictly in terms of truth or untrue, I would classify this as false

b. This is once more a little tricky. When "r" is zero, there isn't any evidence of a linear link. It doesn't necessarily follow that the variables don't correlate with one another. The answer is probably True if this is a course on statistics for beginners. If not, False.

c. This is simple to understand. Inverse relationships are indicated by negative coefficients. One grows while the other shrinks (or vice versa). False.

d. The range of the coefficient r is [0, 1] (inclusive), not (0,1). In this case, True if the text suggests [0,1].

e. Simple and accurate

f. Plain and simple, FALSE. Quite a high correlation coefficient

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Simplification of the exponential expressions using laws of exponents gives;

A) 1

B)  ¹/₂

C) 2

D) 2

Some of the laws of exponents states that;

1) When we multiply like bases, we must keep the base the same and add the exponents.

2) When we raise a base with a power to another power, we must keep the base the same and multiply the exponents.

3) When dividing like bases, we must keep the base the same and subtract the denominator exponent from the numerator exponent.

A) [(2 * 3⁻²)³(5 * 3²)²]/(3⁻²)(5 * 2)²

Applying laws of exponents, we have;

[(2² * 3⁻²*³)(5² * 3²*²)]/(3⁻²)(5² * 2²)

= 2² and 5² will cancel out to give;

3⁻⁶ * 3⁴ * 3²

= 3⁻⁶ * 3⁴⁺²

= 3⁻⁶ * 3⁶

= 3⁰ = 1

B) (3³) (4⁰)²(3 * 2)⁻³ (2²)

This can be broken down into;

3³ * 1 * 3⁻³ * 2⁻³ * 2²

= 2⁻¹ = ¹/₂

C) (3⁷ * 4⁷)(2 * 5)⁻³(5²)]/(12⁷ * 5⁻¹ * 2⁻⁴)

= (3⁷ * 4⁷)(2⁻³ * 5⁻¹)/((3⁷ * 4⁷ * 5⁻¹ * 2⁻⁴)

3⁷ * 4⁷ * 5⁻¹ will cancel out to get;

2⁻³/2⁻⁴ = 2

D) ((2 * 3⁻¹) * 2⁰)/(2 * 3⁻¹)

(2 * 3⁻¹) will cancel out to get;

2⁰ = 1

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The most typical or average value among a group of numbers is called the mean. It is a statistical measure of a probability distribution's central tendency along the median and mode. It also goes by the name "anticipated value."

Steel rods are produced by a firm. The lengths of the steel rods have a mean of 97 cm and a standard deviation of 1 cm, and they are regularly distributed. 15 steel rods are packaged together for shipping. determine the likelihood that the average length of a bundle of steel rods chosen at random is less than 140.7 cm.

mean is N (M) (109,1.6)

Looking for P(M 108.7) to P(z ???) conversion utilizing N(0,1) z-statistic transformation

P(z [M - population mean]/[SD/square root(sample size)]) = P(z [108.7-109)/[1.6/sqrt(27)] = P(z -0.974] = 0.165.

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Answer: 176 BLUE marbles

Explanation:

First --> Total marbles ==> 264 / 0.60 = 440

Second --> Hence, the total number of BLUE marbles is 440 - 264 = 176

and there is 176 BLUE marbles

Answer:

660

Step-by-step explanation:

first, 264 yellow marbles.

the percentage of yellow is 60.

If total percentage is 100 then, blue is 100-(yellow percentage)

that is: 40 percent

the next sentence.

total marbles of blue is:

divide both sides by 0.4

x= 660.

Stratified sampling is used when the population is heterogeneous.

What is Stratified sampling?

A method of probability sampling (where all members of the population have an equal chance of being included) Population is divided into 'strata' (sub-populations) and random samples are drawn from each. This increases representativeness as a proportion of each population is represented.

The sampling method described in the study is a stratified sampling method.

In stratified sampling, the population is first divided into subgroups or strata based on some relevant characteristic, such as size in this case.

Then, a simple random sample is taken from each stratum.

In this study, the sample is chosen by separating all cars by size and selecting 10 of each size grouping, this is the definition of stratified sampling.

Stratified sampling is useful when the population is heterogeneous and you want to ensure that the sample is representative of specific subgroups within the population.

This method allows for a more precise estimation of population characteristics and can reduce sampling error.

Hence, Stratified sampling is used when the population is heterogeneous.

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The length of something like the curve is 8.083, going per the supplied statement.

Curve, An abstract phrase used to characterize the trajectory of a continually moving point in mathematics (see continuity). Usually, an equation will create such a route. The term may also be used to describe a solid line or a collection of connected line segments. The IS curve displays permutations of borrowing costs and production levels where anticipated expenditure and income are equal. The many interest and income configurations along which the commodity is in homeostasis are represented by the IS Curve.

[tex]\begin{aligned}& r(t)=\left\langle 8 t, t^2, \frac{1}{12} t^3\right\rangle \\& r^{\prime}(t)=\left\langle 8,2 t, \frac{t^2}{4}\right\rangle\end{aligned}[/tex]

[tex]\begin{aligned}& \left|\lambda^{\prime}(t)\right|=\sqrt{64+4 t^2+\frac{t^4}{16}}=\sqrt{1 / 16\left(t^2+32\right)^2} \\& \left|r^{\prime}(t)\right|=\frac{1}{4}\left(t^2+32\right)\end{aligned}[/tex]

Length of arc = [tex]\int_a^b\left|r^{\prime}(t)\right| d t[/tex]

[tex]\begin{aligned}& L=\frac{1}{4} \int_0^1\left(t^2+32\right) d t \\& L=\frac{1}{4}\left[\frac{t^3}{3}+32 t\right]_0^1=\frac{1}{4}\left[\frac{1}{3}+32\right]\end{aligned}[/tex]

L = 8.083

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Answer:

The measure of the third angle of the triangle is 45°.

Step-by-step explanation:

What is a triangle?

A triangle is a polygon with 3 sides.

Given that, the measures of the two angles of a triangle are 50° and 85°.

Let the measure of the third angle be x.

Since the sum of the interior of a triangle is 180°, it follows,

50°+85°+x=180°

135° +x = 180°

x = 180° - 135°

x = 45°

Hence, the measure of the third angle of the triangle is 45°.

Answer:

28 cups

Step-by-step explanation:

[tex]\frac{2}{1}[/tex] = [tex]\frac{x}{14}[/tex]

1 x 14 = 14, so 2 x 14 = 28

Using sine law, the length of t is 11.34 units

A fundamental conclusion of trigonometry that connects a triangle's sides and angles is known as the law of sines. It claims that the following relationship holds for any triangle with sides a, b, and c and opposite angles a, b, and c:

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

In other words, each of the triangle's three sides has a length that is equal to the sine of the angle opposite it. Any triangle, correct or not, can be related to another triangle in this way.

When some of the information is available, it is common to utilize the Law of Sines to find the triangle's unknown side lengths or angle measurements. In addition to the Law of Cosines, it can also be employed alone.

Using sum of angle of triangle;

V + U + T = 180

v + 100 + 36 = 180

v + 136 = 180

v = 180 - 136

v = 44°

8 / sin 44 = t / sin 100

t = 8 sin100 / sin44

t = 11.34

The value of t is 11.34 units

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Answer:

47

Step-by-step explanation:

Volume of the sphere to the nearest cubic cm will be 803.84 cm^3

A sphere has a spherical, symmetrical form. It is a three-dimensional solid with equal distances between each surface point and the center. On the basis of its radius, it has a surface area and a volume. There are no faces, corners, or edges on it.

The collection of all points in three-dimensional Euclidean space that are separated by a certain amount of space, or "radius," is known as a sphere (the "center"). The diameter is equal to twice the radius, and antipodes are pairs of locations on the sphere that are on opposing sides of a diameter.

Volume of sphere[tex]=\frac{4}{3} \pi r^3[/tex]

As we have the value of radius = r=4 cm

We have the value of total scoops = 3 mm

So, by putting the value in the above expression of the volume of the sphere, we will get it as:

[tex]V =\frac{4}{3} \times 3.14(4)^3 \times 3 \\=3.14 \times 256 \\=803.84 \mathrm{~cm}^3[/tex]

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The correct question should be:

Let's Scream for Ice Cream serves 3 scoops of ice cream in its signature cone. Each scoop is a sphere with a radius of 4 centimeters. To the nearest cubic centimeter, what is the total volume of ice cream served per cone?

The experimental probability of a cube tossed 60 times, landing on a number greater than 4 is the option;

18 over 60

A probability is the fraction or proportion of the times an event will occur when an experiment or task is repeated. It is the measure of the liklyhood an event will occur

Probability compares the number of times an event occurs to the number of times all possible event outcomes occurs.

The formula for the experimental probability is presented as follows;

[tex]Probability = \frac{The \ sum \ of \ the \ times \ the \ event \ takes \ place}{The \ sum \ of \ the \ trials}[/tex]

The number of times the outcome from tossing the coin is greater than 4 obtained by adding the frequencies when for the outcomes of 5 and 6 are;

Number of outcomes greater than 4 = 10 + 8 = 18

The sum of all trials = 12 + 13 + 11 + 6 + 10 + 8 = 80

Therefore, the experimental probability = 18/60

The correct option is; 18 over 60

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Answer: B: 18/60 (18 over 60)

Step-by-step explanation:

Answer:

  a ≠ 0

Step-by-step explanation:

You want to know the restriction applicable to 2(y+a)/a(x-3).

A rational expression is undefined when its denominator is zero. The restrictions are applied so that no denominator is zero.

Here, the expression is ...

  [tex]\dfrac{2(y+a)}{a}(x-3)[/tex]

The only denominator is 'a'. It cannot be zero.

The restriction is a ≠ 0.

__

Additional comment

If we assume this is a relation in which x- and y- are the only variables, there is no restriction on the relation as written.

Perhaps you intend that (x-3) be a denominator factor. Then the relation would have the restriction x≠3. Such an expression would need to be written as ...

  2(y+a)/(a(x-3)) . . . . . . using parentheses around the denominator

<95141404393>

Answer:

[tex]-8 \leq x \leq -3[/tex]

Step-by-step explanation:

[tex]-8 \leq x \leq -3[/tex] this is the expression you are looking for.

Answer:

Step-by-step explanation:  

              12 < x ≤-8

Answer:

y^2 + 5y - 750 = 0

y(y + 5) = 750

750 - y(y - 5) = 0

Step-by-step explanation:

Answer:

x= 108/61; y=-163/61

Step-by-step explanation:

if the density of the metal cube is 2.3 x 10⁻³ kilograms/micro liters, 2968 cm. is the length of one side of a cube if the mass is 88 milligrams

The amount of digits that contribute to the correctness of a value—often a measurement—are known as significant figures. The first digit that is non-zero is where we begin counting significant figures.

The number of significant single digits (0 through 9 inclusive) in an expression's coefficient is referred to as the significant figures in scientific notation. The amount of accuracy or assurance an expert or scientist uses to state a quantity is indicated by the number of significant figures in a statement.

The given mass of the metal cube is 2.3 × 10⁻³kg/micro-liters or 2300000 gm/liter

The density of the metal cube is 88 mg or 0.088 gm.

Now, we know that density, d = mass/volume.

So, volume of the metal cube,

V = mass/density

V = (2300000/0.088)

V = 26136363.7 L

Now, we know that volume of a cube is a³ where a is the length of a side of the cube. So, a³ = 26136363.7 L or 26136363.7 dm³.

So, a = 296.8 dm or 2968 cm.

So, the length of one side of the cube is 2968 cm.

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Answer: 2x + 3y + 1

Step-by-step explanation:

That is your answer

Please give brainliest

Answer:

c. 2

Step-by-step explanation:

The valueof v

-9v - 9 = -27

= -9v = -27 + 9

= -9v = -18

= -9v/-9 = -18/-9

v = 2.

Note: negative will cancel negative.

After 6 months of birth the weight will become 6.95 kg.

A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set.

from given condition, a baby weigh 3. 35 kilogram at birth and  the baby weight constantly increase every two month by 1. 2 kilogram.

so, the weight after 2 months of birth is 3.35 kg + 1.2 kg = 4.55 kg

and the weight after 4 months of birth is 4.55 kg + 1.2 kg = 5.75 kg

and the weight after 6 months of birth is 5.75 kg + 1.2 kg = 6.95 kg

Therefore, after 6 months of birth the weight will become 6.95 kg.

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12 13 14 15 16 17 18 19 20 at the fisher farm, the weights of zucchini squash are normally distributed,  then Weight represents the top 10% of the zucchinis is 5.9

The formula for calculating a z-score is z = (x-μ)/σ,

where x is the raw score

μ is the population mean

σ is the population standard deviation.

We're looking for:

Weight is 10% of zucchinis at the top.

Top 10% = 90% of the population

90th percentile Z score is 1.282.

Hence

1.282 = x - 5 / 0.7

1.282 × 0.7 = x - 5

0.8974 = x - 5

x = 5 + 0.8974

x = 5.8974

5.9 ounces, roughly.

The top 10% of the zucchinis have a weight of 5.9 ounces.

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There are three main types of solutions found in mathematics for a given problem or equation: No solution, Unique solution, and Infinitely many solutions.

Solutions in mathematics are a set of values that can be assigned to the variables of an equation or a system of equations so that it holds true. In other words, it is a set of values that satisfies the equation or system of equations.

Types of solutions:

No solution: In this situation, there is no set of values that can be assigned to the variables of an equation or system of equations so that it holds true.

Unique solution: In this situation, there is only one set of values that can be assigned to the variables of an equation or system of equations so that it holds true.

Infinitely many solutions: In this situation, there are infinitely many sets of values that can be assigned to the variables of an equation or system of equations so that it holds true. This situation arises when the given equation is not independent.

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An equation that models the numbers is x, x+1, x + 2

The term equation is known as a condition on a variable such that two expressions in the variable have equal value

Here we know that Chloe is trying to find three consecutive, positive integers such that six times the largest is equal to the twice sum of the two smaller integers.

Then the given parameters are written as the three consecutive positive numbers Chloe is trying to find are x, x+1, x + 2

Here we have the conditions of the three positive numbers are written as,

=> 6 × (x + 2) = 2 × (x + x + 1)

Now, we have to check if the three consecutive numbers can be 4, 5, and 6 by substituting the values as written as,

=> x + 2 = 6

=> x + 1 = 5

Then the value of x is 4

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