Jane set up a lemonade stand to sell lemonade. The total cost of making the lemonade was $1. 50. She sold each cup of lemonade for $0. 25,

Enter an equation that can be used to find the number of cups of lemonade, x, Jane sold if her earnings were $9. 75 after paying out the cost of the lemonade.

Answer:

Step-by-step explanation:

A linear system with 3 equations and 4 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form can be solved.

A mathematical representation of a system based on the application of a linear operator is known as a "linear system" in systems theory. Ordinarily, compared to nonlinear systems, linear systems display much simpler features and properties. The automatic control theory, signal processing, and telecommunications all heavily rely on linear systems as a mathematical abstraction or idealisation.

Linear systems, for instance, are frequently used to model the propagation medium for wireless communication systems. An operator, H, that converts an input, x(t), into an output, y(t), a kind of black box description, can be used to describe a general deterministic system.

The superposition principle, or alternatively both the additivity and homogeneity properties, must be satisfied by a system to be considered linear, and only then.

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The value of x is 3 and solution is extraneous

Here, we have,

Solving equations

Given the equation below;

√8x+1 = 5

Square both sides to have;

8x+1 = 5^2

8x + 1 = 25

Subtract 1 from both sides

8x + 1 = 25 -1

8x = 24

x = 24/8

x = 3

Hence the value of x is 3

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Answer:

18

Step-by-step explanation:

It's the only size that works.

Answer:

NO

Step-by-step explanation:

First, we should simplify both fractions to the most simple form.

4/14=2/7

12/40=6/20=3/10

A proportion means that it is equal. 2/7 does not equal 3/10, so the answer is no.

The sides of the triangle as vectors are,

AB = (3, 1) = 3i + j

BC = (- 2, 3) = - 2i + 3j

CA = (- 1, - 4) = - i - 4j

We have to give that,

Vertices of the triangle are,

A(2, 2), B(5, 3) , and C(3, 6)

Hence, the sides of the triangle as vectors are,

AB = (5, 3) - (2, 2) = (5 - 2, 3 - 2) = (3, 1)

BC = (3, 6) - (5, 3) = (3 - 5, 6 - 3) = (- 2, 3)

CA = (2, 2) - (3, 6) = (2 - 3, 2 - 6) = (- 1, - 4)

Therefore, the sides of the triangle as vectors are,

AB = (3, 1) = 3i + j

BC = (- 2, 3) = - 2i + 3j

CA = (- 1, - 4) = - i - 4j

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Answer:

26.25 cm

Step-by-step explanation:

Take the width and multiply by the scale factor

15 * 7/4

105/4 cm

26.25 cm

The solution to the initial value problem is y(x) = 3x sin x + 2x cos x.

To solve the initial value problem, we can use an integrating factor method. First, we rewrite the given equation as dy/dx + y tan x = 2x sin x. By comparing this form with the standard form of a linear first-order differential equation, we can determine the integrating factor. The integrating factor is given by exp(integral(tan x dx)), which simplifies to cos x.

Multiplying the entire equation by cos x, we get cos x dy/dx + y sin x = 2x sin x cos x. The left-hand side of the equation is now the derivative of (y cos x) with respect to x. Integrating both sides, we have ∫(y cos x) dx = ∫(2x sin x cos x) dx.

Integrating the right-hand side and simplifying, we get y(x) cos x = x sin^2(x) + C, where C is the constant of integration. To find the value of C, we use the initial condition y(0) = 5. Plugging in x = 0 and y = 5 into the equation, we get 5 cos 0 = 0 + C. Simplifying, we find C = 5.

Substituting C back into the equation, we have y(x) cos x = x sin^2(x) + 5. Dividing both sides by cos x, we obtain the final solution y(x) = (x sin^2(x) + 5) / cos x. Simplifying further, we get y(x) = 3x sin x + 2x cos x, which is the solution to the initial value problem.

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The given series is equivalent to the sum of two p-series: ∑n^(-1/2) + ∑n^(-2). Where the first series converges since p1 = 1/2 > 0 and the second series also converges since p2 = 2 > 1.

To start, we can simplify the given series as:

sqrt(n+4)/n^2 = 1

Taking the reciprocal of both sides:

n^2/sqrt(n+4) = 1

Multiplying both sides by sqrt(n+4):

n^2 = sqrt(n+4)

Squaring both sides:

n^4 = n+4

This is a quadratic equation that we can solve using the quadratic formula:

n = (-1 ± sqrt(17))/2

Since we are only interested in positive integer values of n, we take the larger root:

n = (-1 + sqrt(17))/2 ≈ 1.56

Now that we have found the value of n that satisfies the equation, we can rewrite the given series in terms of p-series:

sqrt(n+4)/n^2 = (n+4)^(1/2) / n^2
= (1 + 4/n)^(1/2) / n^2

Using the formula for the p-series:

∑n^-p = 1/1^p + 1/2^p + 1/3^p + ...

We can see that the given series is equivalent to:

(1 + 4/n)^(1/2) / n^2 = n^(-2) * (1 + 4/n)^(1/2)
= n^(-p1) + n^(-p2)

Where p1 is the smaller value and p2 is the larger value of p that make up the two p-series.

We can find p1 and p2 by comparing the exponents of n on both sides of the equation:

p1 = 1/2
p2 = 2

Therefore, the given series is equivalent to the sum of two p-series:

∑n^(-1/2) + ∑n^(-2)

Where the first series converges since p1 = 1/2 > 0 and the second series also converges since p2 = 2 > 1.

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Sasha's elevation now, in miles is -8/3miles.

If Sasha takes a break at every 2/3 of a mile and has hiked down to the fourth rest station, we can calculate her elevation change based on the given information.

At each rest station, Sasha descends by 2/3 of a mile. Since she has reached the fourth rest station, she has descended by 4 * (2/3) = 8/3 miles.

Now, to determine her elevation change, we need to consider the direction of the hike. If Sasha is descending, her elevation change will be negative, indicating a decrease in elevation.

The correct answer is: -8/3

This answer represents a descent of 8/3 miles, which corresponds to a decrease in elevation.

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Sample Mean: 3.25

Standard Deviation: 2.29

CI: 1.36 to 5.14

To compute the sample mean, add up all the values and divide the sum by the number of values in the data set. In this case, the data set is 3, 4, 6, 8, 2, 1, 0, 2.

Summing up the values: 3 + 4 + 6 + 8 + 2 + 1 + 0 + 2 = 26.

There are 8 values in the data set, so divide the sum by 8: 26 / 8 = 3.25.

Therefore, the sample mean is 3.25 cups of coffee per day.

To compute the sample standard deviation, we need to find the square root of the variance. The variance is the average of the squared differences from the mean.

First, calculate the squared differences from the mean for each value:
(3 - 3.25)^2 = 0.0625
(4 - 3.25)^2 = 0.5625
(6 - 3.25)^2 = 7.5625
(8 - 3.25)^2 = 20.0625
(2 - 3.25)^2 = 1.5625
(1 - 3.25)^2 = 5.0625
(0 - 3.25)^2 = 10.5625
(2 - 3.25)^2 = 1.5625

Next, find the average of these squared differences:
(0.0625 + 0.5625 + 7.5625 + 20.0625 + 1.5625 + 5.0625 + 10.5625 + 1.5625) / 8 = 5.25.

Finally, take the square root of the variance to find the sample standard deviation:
sqrt(5.25) ≈ 2.29.

Therefore, the sample standard deviation is approximately 2.29 cups.

To construct a 95% confidence interval for the mean number of cups of coffee consumed among all undergraduates, we can use the formula:

CI = sample mean ± (t * (sample standard deviation / sqrt(n))),

where t is the t-value for the desired confidence level, sample standard deviation is the calculated standard deviation, sqrt(n) is the square root of the sample size, and the sample mean is the calculated mean.

For a 95% confidence level, with a sample size of 8, we can find the t-value from a t-table. The t-value for a 95% confidence level with 8 degrees of freedom is approximately 2.306.

Plugging in the values, the confidence interval is:

CI = 3.25 ± (2.306 * (2.29 / sqrt(8))).

Calculating the lower limit:
3.25 - (2.306 * (2.29 / sqrt(8))) ≈ 1.36.

Calculating the upper limit:
3.25 + (2.306 * (2.29 / sqrt(8))) ≈ 5.14.

Therefore, the 95% confidence interval for the mean number of cups of coffee consumed among all undergraduates is approximately 1.36 to 5.14 cups per day.

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Answer:

5/45

Step-by-step explanation:

The 5th equivalent fraction to 1/9 will be 5/45.

What is Multiplication?

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

Given that;

The expression is,

''5th equivalent fraction to 1/9''

Now,

Solve the fraction as;

''5th equivalent fraction to 1/9''

= 5/5 x 1/9

= (5 x 1) / (5 x 9)

= 5 / 45

Thus, The 5th equivalent fraction to 1/9 will be 5/45.

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The equation describing the intersection of the sphere with the plane z = 9 is (x+8)^2 + (y-4)^2 = 15.

To find the equation of the sphere centered at (-8,4,8) with radius 4 and the intersection with the plane z = 9.

Step 1: Find the equation of the sphere. The general equation of a sphere is (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a, b, c) is the center of the sphere and r is the radius. In this case, the center is (-8, 4, 8) and the radius is 4. So, we have:

(x+8)^2 + (y-4)^2 + (z-8)^2 = 16

Step 2: Find the intersection of the sphere with the plane z = 9. Since the plane is given by z = 9, we can substitute 9 for z in the equation of the sphere:

(x+8)^2 + (y-4)^2 + (9-8)^2 = 16

This simplifies to:

(x+8)^2 + (y-4)^2 + 1 = 16

Now, move the constant term to the other side of the equation:

(x+8)^2 + (y-4)^2 = 15

So, the equation describing the intersection of the sphere with the plane z = 9 is (x+8)^2 + (y-4)^2 = 15.

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To determine the percent success, divide the actual score by the theoretical score, and then multiply the result by 100 to convert the value to a percentage.

We define the actual score, theoretical score, and explain how to calculate the percent success on an exam.
Actual score:

The actual score refers to the number of points a student has earned on an exam.

It represents the student's performance on the test, taking into account the correct and incorrect answers.
Theoretical score:

The theoretical score is the maximum number of points a student can earn on an exam.

This represents a perfect performance, where the student answers all questions correctly.
Calculating percent success:

To determine the percent success, divide the actual score by the theoretical score, and then multiply the result by 100 to convert the value to a percentage.
a. Divide the actual score by the theoretical score: (actual score) / (theoretical score)
b. Multiply the result by 100: (result from step a) * 100
c. The final value is the percent success.

For example, if a student has an actual score of 80 and the theoretical score is 100, the percent success would be calculated as follows:
a. 80 / 100 = 0.8
b. 0.8 * 100 = 80
c. The percent success is 80%.

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The length of the microorganism that measure 5μm is equivalent to 0.005 mm

It is the transformation of a value expressed in one unit of measurement into an equivalent value expressed in another unit of measurement of the same nature.

To solve this problem the we have to convert the units with the given information.

1mm is equal to 1000 μm

5μm * (1 mm/1000μm) = (5*1) / 1000 = 5/1000 = 0.005 mm = 5x10^-3 mm

The length of the microorganism that measure 5μm is equivalent to 0.005 mm

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Based on the given information, the compass will lag behind the airplane during a turn from the south. This means that the compass reading will be slower to update and indicate the correct heading compared to the actual direction of the airplane. Therefore, the first statement "The compass exaggerates the rate of turn" is false.

During a turn, the compass needle is affected by the magnetic field of the Earth, and it takes some time for the needle to align with the new heading. This lag in the compass reading is due to the inertia of the compass needle and the damping effect of the compass mechanism. As the airplane turns, the compass needle will gradually catch up and align with the new heading. Therefore, the second statement "The compass will lag behind the airplane" is true.

The third statement "The compass will match the exact heading" is false because, as mentioned earlier, the compass will lag behind the airplane during a turn. It will eventually align with the correct heading, but it will not match the exact heading instantaneously.

In summary, during a turn from the south, the compass will lag behind the airplane, but it will eventually align with the correct heading. It does not exaggerate the rate of turn, and it does not immediately match the exact heading.

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The slope of the tangent line to the polar curve r = 2 - sin(θ) at the point specified by θ = π/3 is -1/2.

To find the slope of the tangent line to the polar curve r = 2 - sin(θ) at the point specified by θ = π/3, we need to first find the derivative of r with respect to θ, and then evaluate it at θ = π/3.

Differentiating both sides of the polar equation with respect to θ, we get:

dr/dθ = d/dθ(2 - sinθ)

dr/dθ = -cosθ

So, the derivative of r with respect to θ is -cosθ.

Evaluating this derivative at θ = π/3, we get:

dr/dθ|θ=π/3 = -cos(π/3) = -1/2

Therefore, the slope of the tangent line to the polar curve r = 2 - sin(θ) at the point specified by θ = π/3 is -1/2.

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Answer: 3/4

Step-by-step explanation:

You have to find one single number that goes into both the numerator and the denominator. In this case, it was 8.

24/8 = 3

32/8 = 4

Answer:

1st one is 3/4

second one is 32

I cant really see the third one

fourth one is $1.03

Step-by-step explanation:

Answer:

43.98

Step By Step Explanation:

Answer:14 and a weird R symbol lol

Step-by-step explanation:

The vertex form of the equation is y = (x - 3)² - 4, which corresponds to option OD.

To convert the given equation from standard form to vertex form, we need to complete the square.

The vertex form of a parabola's equation is y = a(x-h)² + k, where (h, k) represents the vertex of the parabola.

Given equation: y = x² - 6x + 14

Move the constant term to the right side:

y - 14 = x² - 6x

Complete the square by adding and subtracting the square of half the coefficient of x:

y - 14 + 9 = x² - 6x + 9 - 9

Group the terms and factor the quadratic:

(y - 5) = (x² - 6x + 9) - 9

Rewrite the quadratic as a perfect square:

(y - 5) = (x - 3)² - 9

Simplify the equation:

y - 5 = (x - 3)² - 9

Move the constant term to the right side:

y = (x - 3)² - 9 + 5

Combine the constants:

y = (x - 3)² - 4

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For a normal distribution ,a positive value of z simply means that the observation or sample mean is above the population mean this implies

none of the options provided is completely accurate.

All the observations must have had positive values,

It is not necessarily true for a positive value of z.

The value of z indicates how many standard deviations away from the mean a particular observation or sample mean is.

A positive value of z simply means that the observation or sample mean is above the population mean.

The area corresponding to the z is either positive or negative,

It is also not accurate.

The area under the normal curve corresponds to probabilities, not positive or negative values.

The area to the right of the mean corresponds to positive z-values, and the area to the left of the mean corresponds to negative z-values.

The sample mean is smaller than the population mean,

It is a possibility when the z-value is negative, indicating that the sample mean is below the population mean.

The sample mean is larger than the population mean,

It is a possibility when the z-value is positive, indicating that the sample mean is above the population mean.

Therefore, the sample mean can be either larger or smaller than the population mean depending on the direction of the z-value.

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Answer: 25°

Step-by-step explanation:

So we know that the right angle equals 90°.

And we know that all the angles of a triangle have to add up to 180°.

Also angle x and the exterior angle that is 115° are supplementary, so they have to add up to 180°. So we have to subtract 180 - 115 = 65.

So our equation for y is:

y + 65 + 90 = 180

y + 155 = 180

y = 25

So y = 25°

Hope this helps!!! :)

The probability helps us to know the chances of an event occurring. The probability of winning a small prize in the carnival is 35.714%.

The probability helps us to know the chances of an event occurring.

\(\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}\)

Given that there is a total of 56 ducks floating in the pond, out of which 20 ducks are marked as small-prize winners. Therefore, the probability of winning a small price can be written as:

\(\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}\\\\\\\rm Probability=\dfrac{\text{Nnmber of small price ducks}}{\text{Total number of ducks}}\\\\\\Probability = \dfrac{20}{56} = 0.35714 = 35.714\%\)

Hence, the probability of winning a small prize in the carnival is 35.714%.

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Answer:

Distance is 8.49 units

Step-by-step explanation:

Answer:

d= √  72

Step-by-step explanation:

d=√  (1−7)2+(8−2)2

d=√  (−6)2+(6)2

d=√  36+36

d=√  72

d=8.485281

Answer:

80 m

Step-by-step explanation:

La velocidad inicial de la pelota es 0 m/s.

La altura del árbol es s.

El tiempo necesario es de 4 segundos.

Podemos aplicar una de las ecuaciones de movimiento de Newton:

\(s = ut + \frac{1}{2}gt^2\)

donde u = velocidad inicial

t = tiempo empleado

g = aceleración debido a la gravedad

Por lo tanto:

\(s = 0 * 4 + (1/2 * 10 * 4^2)\\\\s = 0 + 80 \\\\s = 80 m\)

El árbol tiene 80 metros de altura.

Answer:

HI!

Step-by-step explanation:

BYE!

Sorry :(

The regression equation for the given relationship is y = x + 5.

Given, two points (0,5) and (5,10) fall on the regression line for a perfect positive linear relationship.

The formula for a linear regression model is represented as;Y = a + bxwhere,a is the y-intercept.b is the slope of the line.x is the independent variable.

Here, the slope can be found as;`b = (y₂ - y₁) / (x₂ - x₁)``b = (10 - 5) / (5 - 0) = 1`Now, the intercept can be found by substituting the slope in the formula of the regression model;`y = a + bx``5 = a + 1(0)``a = 5

`Therefore, the regression equation for the given relationship is y = x + 5.

Summary:The regression equation for the perfect positive linear relationship between the given two points (0,5) and (5,10) is y = x + 5.

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Answer: B

Step-by-step explanation:

\(\frac{2}{7k}(k-7)\)

\(\frac{2k-14}{7k}\)

\(\frac{2k}{7k}-\frac{14}{7k}\)

\(\frac{2}{7}-\frac{2}{k}\)