The area of a circle increases at a rate of 5 cm²/s. a. How fast is the radius changing when the radius is 3 cm? b. How fast is the radius changing when the circumference is 1 cm?​

It will run dry after 141.7 days.

The rainwater tank has a diameter of 160 cm and a height of 190 cm, then its volume is:

V = 3.14*(180cm)*(160cm/2)² = 3,617,280 cm³

We know that 1 L = 1000 cm³

then:

3,617,280 cm³ = 3,617.280 L

The water is used at a rate of 25,5 litres per day, so after x days there is:

W =  3,617.280  - 25.5*x liters

It is zero when:

0 = 3,617.280  - 25.5*x

x =  3,617.280/25.5

x = 141.7 days.

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The value of the side x is 8. 4

It is important to note that the Pythagorean theorem states that the square of the longest leg of a triangle, is equal to the sum of the squares of the other two sides, that is, the opposite and adjacent sides.

The formula is expressed as;

a² = b² + c²

From the information shown in the diagram, we have that;

Hypotenuse side = 21

Adjacent side = 12. 6

Opposite side = 2(x)

Substitute the values

x² = 21² - 12.6²

find the squares

x² = 441 - 158. 76

subtract the values

x = 16. 8

But, we have that this value is the diameter but x as shown is the radius.

Radius, x = 16. 8 /2 = 8. 4

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After addressing the issue at hand, we can state that Javier's model equation predicts that there are 30 fifths in 6 wholes.

A mathematical statement known as an equation demonstrates the equivalence of two expressions when they are joined by the equals sign ('='). As an illustration, 2x - 5 = 13. 2x-5 and 13 are examples of expressions. The letter '=' joins the two expressions together. An equation is a mathematical formula with two algebraic expressions on either side of the equal sign (=). It shows how the left and right formulas have an equivalent relationship. In any formula, L.H.S. = R.H.S. (left side = right side).

To describe Javier's work, one division equation with a fraction might be written as follows:

6 ÷ (1/5) =

How many fifths are there in six wholes? is the equation that illustrates the question Javier was attempting to answer. In order to answer this equation, we must divide 6 by the fraction 1/5, which is equivalent to multiplying 6 by 1/5's inverse, or 5/1:

6 ÷ (1/5) = 6 × (5/1) = 30

Javier's model predicts that there are 30 fifths in 6 wholes.

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The prοduct οf the matrices is nοt the identity matrix. Therefοre, X and A nοt inverses οf each οther.

An identity matrix is a square matrix in which all the elements οf principal diagοnals are οne, and all οther elements are zerοs. It is denοted by the nοtatiοn “In” οr simply “I”. Any matrix will yield a matrix as a result of being multiplied by the identity matrix.

The identity matrix is a matrix having entry οne in its diagοnal and rest οf the entries are zerο.

[tex]X.A = \left[\begin{array}{cc}2&-1\\6&-4\\\end{array}\right] \left[\begin{array}{cc}1/2&1/4&\\-2&-1/2&\end{array}\right][/tex]

[tex]X.A = \left[\begin{array}{cc}3&1\\11&7/2\end{array}\right][/tex]

Hence, The prοduct of the matrices is not the identity matrix. Therefοre, X and A not inverses of each other.

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The complete question is in the attached image.

Answer:

The null hypothesis is that the mean daily iron intake of the low-income group is equal to or greater than the general population, and the alternative hypothesis is that the mean daily iron intake of the low-income group is less than that of the general population.

H0: μ >= 14.44

Ha: μ < 14.44

(b) We can use a one-tailed t-test to test this hypothesis. With a sample size of 51, degrees of freedom of 50, a sample mean of 12.50, a population mean of 14.44, and a standard deviation of 4.75, we can calculate the t-statistic as follows:

t = (12.50 - 14.44) / (4.75 / sqrt(51)) = -3.16

Using a t-distribution table with 50 degrees of freedom and a significance level of .05, the critical value is -1.677. Since the calculated t-statistic is less than the critical value, we reject the null hypothesis.

Therefore, we can conclude that the mean daily iron intake among the low-income group is significantly lower than that of the general population at a significance level of .05.

(c) The p-value for this test is the probability of obtaining a t-value of -3.16 or more extreme assuming the null hypothesis is true. Using a t-distribution table with 50 degrees of freedom, we find the p-value to be less than .005.

(d) The null hypothesis is that the standard deviation of the low-income group is equal to or greater than that of the general population, and the alternative hypothesis is that the standard deviation of the low-income group is less than that of the general population.

H0: σ >= 5.56

Ha: σ < 5.56

(e) We can use a chi-square test to test this hypothesis. With a sample size of 51, degrees of freedom of 50, and a sample standard deviation of 4.75, we can calculate the chi-square statistic as follows:

chi-square = (n - 1) * s^2 / σ^2 = 50 * 4.75^2 / 5.56^2 = 39.70

Using a chi-square distribution table with 50 degrees of freedom and a significance level of .05, the critical value is 68.67. Since the calculated chi-square statistic is less than the critical value, we fail to reject the null hypothesis.

Therefore, we do not have sufficient evidence to conclude that the standard deviation of daily iron intake among the low-income group is significantly lower than that of the general population at a significance level of .05.

Therefore, the equilibrium level of national income in this basic Keynesian macroeconomic model is 480.

In mathematics, the sum is the result of adding two or more numbers or quantities together. The symbol used to represent a sum is the capital Greek letter sigma (∑), which stands for "sum up."

by the question.

In the basic Keynesian macroeconomic model, equilibrium occurs when national income (Y) equals total spending in the economy. Therefore, we can set Y equal to the sum of consumption (C) and investment (I), as follows:

Y = C + I

Substituting the given consumption and investment functions into this equation, we get:

Y = (40 + 0.5Y) + 200

Simplifying and rearranging, we get:

Y - 0.5Y = 240

0.5Y = 240

Y = 480

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The relationship which describe angles 1 and 2 is supplementary angles.

The correct answer choice is option A.

Supplementary angles are angles in which the sum of their measure is equal to 180°.

Based on this, angles 1 and 2 are supplementary because they both add up to 180°.

Complementary angles are two angles which add up to 90 degrees.

Vertical angles are angles opposite each other when two angles intersect.

Adjacent angles are angles which have a common side and common vertex.

Hence, angles 1 and 2 are supplementary.

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The polynomial after simplification is obtained as [tex]5 x^{2}[/tex] + 9x + 3.

What is simplification?

In mathematics, reducing an expression, fraction, or problem to a simpler form is referred to as simplification. With calculations and solution, the problem is made simple. Make something simpler by simplifying it.

We are given a polynomial as

5x - 5 + [tex]11 x^{2}[/tex] + 4x - [tex]6 x^{2}[/tex] + 8

Now, in order to simplify the given polynomial, we will combine the like terms of the polynomial.

On combining the like terms, we get

[tex]5 x^{2}[/tex] + 9x + 3

Hence, the polynomial after simplification is obtained as [tex]5 x^{2}[/tex] + 9x + 3.

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Answer:

Step-by-step explanation:

A hands-on experience is the only modality that would give visitors a complete understanding of the plight of modern enslaved people.

Eudora spent 0 hours running and 2 hours flying using her jetpack.

Let's use the formula: time = distance / speed.

Let x be the time Eudora spent running and y be the time she spent flying using her jetpack.

From the problem, we know that:

Eudora ran at an average speed of 12 km/h, so she covered a distance of 12x kilometers while running.

Eudora flew at an average speed of 76 km/h, so she covered a distance of 76y kilometers while flying.

The total distance she traveled was 120 kilometers, so: 12x + 76y = 120.

The entire trip took 2 hours, so: x + y = 2.

We can use the second equation to solve for x: x = 2 - y.

Substituting into the first equation, we get: 12(2 - y) + 76y = 120.

Simplifying the equation, we get: 152 - 64y = 120.

Solving for y, we get: y = 2 hours.

Substituting into x = 2 - y, we get: x = 0 hours.

Therefore, Eudora spent 0 hours running and 2 hours flying using her jetpack.

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Complete question:

Eudora ran from her home to her secret laboratory at an average speed of 12 km/h. She then took one of her jetpacks and flew to her school at an average speed of 76 km/h. Eudora traveled a total distance of 120 kilometers, and the entire trip took 2 hours.

How long did Eudora spend running, and how long did she spend flying using her jetpack?

Step-by-step explanation:

203,433 rounded to the nearest ten thousand is 200,000.

If the value of x is 4, the quadrilateral meets all the requirements of a parallelogram and will be a parallelogram.

A quadrilateral is a four-sided shape with straight line segments as its sides. It is a two-dimensional shape with four angles, each measuring 90°. Quadrilaterals can be categorized into different shapes such as squares, rectangles, parallelograms, trapezoids, rhombuses, and kites.

A parallelogram is a four-sided shape with opposite sides parallel and equal in length. To find the value of x that makes the quadrilateral a parallelogram, we need to use the property of a parallelogram, which states that the opposite sides of a parallelogram are equal in length.

Therefore, we need to equate the opposite sides of the quadrilateral: 6x-8=4x. After solving this equation, we get x=4. Thus, the value of x that makes the quadrilateral a parallelogram is 4.

In order for a quadrilateral to be a parallelogram, opposite sides must be parallel. Using this property, we can set the opposite sides equal to each other and solve for x.

[tex]$2x + 5 = 3x - 2$[/tex]

Simplifying this equation, we get:

[tex]$x = 7$[/tex]

Therefore, if [tex]$x=7$[/tex], the quadrilateral is a parallelogram.

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Answer:

23841.0596 in standard form is 2.38410596 x 10^4.

Step-by-step explanation:

Answer:

2.38410596 x 10^4

Step-by-step explanation:

pls brainlist

Therefore , the solution of the given problem of slope comes out to be  y = 7x - 17 is the equation in slope-intercept form.

A line's steepness is determined by its slope. In gradient-based equations, a situation known as gradient overflow can happen. One can determine the slope by dividing the sum of a run (width differential) and rise (climbing distinction) between two places. The equation with the hill variation, y = mx + b, is used to model the fixed path issue. where grade is mm, a = bc, and the line's y-intercept is situated; (0, b).

Here,

Given :

=> -7x + y = -17

7x both faces together

=>  -7x + 7x + y = 7x - 17

Condense: y = 7x - 17

So,

=>  y = 7x - 17 is the equation in slope-intercept form.

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For the given parabola, the parts on the curve should be labeled as follows;

In Mathematics, the axis of symmetry of a quadratic function can be calculated by using the following mathematical expression:

Axis of symmetry, Xmax = -b/2a

Where:

a and b represents the coefficients of the first and second term in the quadratic function.

For the given quadratic function f(x) = y = x², we have the following:

Axis of symmetry, Xmax = -b/2a

Axis of symmetry, Xmax = -(0)/2(1)

Axis of symmetry, Xmax = 0/1 = 0.

Vertex (h, k) = (0, 0).

In conclusion, the domain of this quadratic function f(x) = y = x² are all the x-values that are located on the x-axis of the graph, which include {-∞, ∞} or all real numbers.

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Perimeter of the given polygon figure will be 38 units.

What exactly is a polygon in mathematics?

A closed, two-dimensional, flat, closed polygon is a shape that is constrained by geometry and has straight sides. Its sides don't curve inward at all. Another term for a polygon's sides is its polygonal edges. The points where two sides converge are known as a polygon's vertices (or corners).

Perimeter of a polygon = Measure of the circumference of the polygon

From the figure attached,

Perimeter of the figure = measure of the three straight lines + measure of the circumference of the quarter circle

Measure of linear sides = 5 + 10 + 10 + 5 = 30 units

Circumference of the quarter circle = 1/4(2πr)

                                                          =  1/2(π)(5)

                                                          = 7.85 units

Perimeter of the figure = 30 + 7.85

                                     = 37.85

                                     ≈ 38 units

  Therefore, perimeter of the given figure will be 38 units.

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The six trig functions are:

(1) Figure 1, sin θ = 1/√5

(2) Figure 1, cos θ = 2/√5

(3) Figure 1, tan θ = 1/2

(4) Figure 2, sin θ = 3/√13

(5) Figure 2, cos θ = 2/√13

(6) Figure 2, tan θ = 3/2

Trig function (short for "trigonometric function") is a mathematical function that relates the angles of a right triangle to the lengths of its sides. The main trig functions are sine (sin), cosine (cos),  and tangent (tan).

The trig functions are abbreviated using SOH CAH TOA.

SOH : sin θ = opposite / hypotheses

CAH: cos θ = adjacent / hypothenuse

TOA : tan θ = opposite / adjacent

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Answer:

When two shapes are similar but not congruent, the sequence of steps showing the similarity usually has a single dilation and then the rest of the steps are rigid transformations. The dilation can come at any time.

Step-by-step explanation:

When two shapes are similar but not congruent, the sequence of steps showing the similarity usually has a single dilation and then the rest of the steps are rigid transformations. The dilation can come at any time.

Answer:

Step-by-step explanation:

To find P(x<4), we need to use the cumulative distribution function (CDF) of the binomial distribution.

For N = 6 and p = 0.3, the probability mass function (PMF) is:

P(X = k) = (6 choose k) * 0.3^k * 0.7^(6-k)

where (6 choose k) is the binomial coefficient.

Using this formula, we can find the probabilities for each value of X less than 4:

P(X = 0) = (6 choose 0) * 0.3^0 * 0.7^6 ≈ 0.1176

P(X = 1) = (6 choose 1) * 0.3^1 * 0.7^5 ≈ 0.3025

P(X = 2) = (6 choose 2) * 0.3^2 * 0.7^4 ≈ 0.3241

P(X = 3) = (6 choose 3) * 0.3^3 * 0.7^3 ≈ 0.1852

Therefore, P(x<4) is the sum of these probabilities:

P(x<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) ≈ 0.9294

So the probability of getting less than 4 successes in 6 trials with a success probability of 0.3 is approximately 0.9294.

Answer:

-106f+56

Step-by-step explanation:

7(-8f-2)= -56f-14

10(-5f+7)= -50f+70

-56f-14. +. 50f+70

=-106f+56

Answer:

-2(53f-28)

Step-by-step explanation:

7(−8f−2)+10(−5f+7)

7x-8f=-56f  7x-2=-14

10x-5f=-50f   10x7=70

-56f-14+-50f+70       Adding a negative to something means to subtract. Like right here at -14 + -50 is the same as -14 - 50

   2. Add the numbers

-56f-14+-50f+70   -14+70=56

-56f+56-50f

   3. Combine like terms

-56f+56-50f   -56+ -50= -106

-106f +56

   4. Common Factor

-106f + 56

-2(53f-28)

a. The Vοlume = 198 cm³ and height = 7, if it’s a cylinder with a radius οf 3cm.

b. The cοst οf cοffee is $3.96 when $0.02 per cubic centimetre.

The fοrmula fοr the vοlume οf a cylinder is

V = πr²h,

where V is the vοlume, r is the radius, and h is the height.

a) Tο find the height οf the cylinder, we need tο knοw the vοlume οf the cοntainer and the radius. The fοrmula fοr the vοlume οf a cylinder is:

V = πr²h

[tex]$\mathrm{\frac{V }{ h}} = \pi r\² }[/tex]

[tex]$\mathrm{\frac{V }{ h}} = \frac{22}{7} \times 3 \times 3}[/tex]

[tex]$\mathrm{\frac{V }{ h}} = \frac{22}{7} \times 3 \times 3}[/tex]

[tex]$\mathrm{\frac{V }{ h}} = \frac{198}{7}[/tex]

Thus, Vοlume = 198 cm³ and height = 7

b) If the cοntainer frοm is filled with cοffee and cοffee cοsts $0.02 per cubic centimetre, Then the cοst οf cοffee

= $0.02 × 198 cm³

= $3.96

Thus, The Vοlume = 198 cm³ and height = 7, if it’s a cylinder with a radius οf 3cm.

The cοst οf cοffee is $3.96 when $0.02 per cubic centimetre.

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The box-and-whisker plot shows that the dataset has a median of approximately 5.5 and is relatively symmetric, with the middle 50% of the data falling between approximately 4 and 7.

The box-and-whisker plot offers a visual summary of the distribution of the given dataset. The following is one way to understand the plot:

The median, which is the midway number when the data is organised from least to largest, is shown by the line inside the box. The median in this situation is roughly 5.5.

The box's bottom edge corresponds to the first quartile (Q1) and its top edge to the third quartile, and it represents the middle 50% of the data (Q3). The interquartile range is the space between Q1 and Q3 (IQR). In this instance, Q1 is roughly 4 and Q3 is roughly 7, hence the IQR is roughly 3.

The whiskers reach from the box's boundaries to the dataset's lowest and largest values that aren't regarded as outliers. Although there are many different definitions of an outlier, it is generally accepted that any number that is more than 1.5 times the IQR below Q1 or above Q3 qualifies as an outlier. There aren't any anomalies in this circumstance.

Individual data points outside the box and whiskers are shown as dots above and below the whiskers. There are no such data points in this situation.

Hence, the box-and-whisker plot reveals that the dataset is very symmetrical and has a median of about 5.5, with the middle 50% of the data ranging between roughly 4 and 7.

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Answer:

Step-by-step explanation:

Logically, if a polynomial of real coefficients has a complex root, it’s conjugate is a root as well. That means if -20-7i is a root, -20+7i is also a root. Also, since we’re dealing with minimal degree, we wish to construct a quadratic expression with such roots. To do this, we can use Vieta’s formulas, which relates the trinomial coefficients with its roots. The two roots added together is -b/a in ax^2 + bx + c, and the two roots multiplied together is c/a. The two roots added together is -40. The two roots multiplied together is 449. So, our quadratic expression is x^2+40x+449

By the given definition of matrix A, the value of the matrix A⁻¹ is [tex]\left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right][/tex] and it is true that AA⁻¹ = I₃.

Given that

[tex]A = \left[\begin{array}{ccc}1&0&2\\1&1&0\\1&0&1\end{array}\right][/tex]

To find the inverse of A, we can use the following formula:

A⁻¹ = (1/|A|) adj(A)

where |A| is the determinant of A and adj(A) is the adjugate matrix of A.

First, let's calculate the determinant of A:

|A| = 1(1×1 - 0×0) - 0(1×0 - 0×1) + 2(1×0 - 1×1)

|A| = -1

Next, we need to find the adjugate matrix of A.

To do this, we need to find the matrix of cofactors of A, then take its transpose:

[tex]C = \left[\begin{array}{ccc}1&-2&0&0&1&0&-1&2&1\end{array}\right]\\[/tex]

adj(A) = [tex]C^T = \left[\begin{array}{ccc}1&0&-1&-2&1&2&0&0&1\end{array}\right][/tex]

Now we can calculate A⁻¹ using the formula:

A⁻¹ = (1/|A|) adj(A)

So, we have

[tex]A^{-1} = (-1) \left[\begin{array}{ccc}1&0&-1&-2&1&2&0&0&1\end{array}\right] = \left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right][/tex]

Finally, we can check whether AA⁻¹ = I₃ using

[tex]A^{-1}A = \left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right] \left[\begin{array}{ccc}1&0&2&1&1&0&1&0&1\end{array}\right] = \left[\begin{array}{ccc}1&0&0&0&1&0&0&0&1\end{array}\right] = I_3[/tex]

Therefore, we have found the inverse of A and verified that AA⁻¹ = I₃.

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Using Law of sines, the missing angles are:

4) ∠B = 45.74°

6) ∠C = 21.02°

8) ∠A = 60.84°

The law of sines is a rule for solving the lengths and angles of triangles and it states that:

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

4) Using law of sines, we have:

8/sin B = 11/ sin 80

(8/11) sin 80 = sin B

sin B = 0.7162

B = sin⁻¹0.7162

∠B = 45.74°

6) Using law of sines, we have:

13/sin C = 32/sin 62

sin C = (13/32) sin 62

sin C = 0.3587

C = sin⁻¹0.3587

C = 21.02°

8) Using law of sines, we have:

30/sin 104 = 27/sin A

sin A = (27/30) sin 104

sin A = 0.8733

A = sin⁻¹0.8733

A = 60.84°

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The rate of change of the angle of elevation is 0.00011 rad/ft, for that we have to know the elevation.

The height above or below a specified(predefined) reference point, most frequently a reference geoid, which is a mathematical representation (expression) of the Earth's sea level as an equipotential gravitational surface, determines(defines) a location's elevation.

The angle of elevation of the camera is, Θ= [tex]tan^-^1(\frac{x}{2000} )[/tex]

The distance between the camera and the launching pad is, b= 2000 ft

The distance between the camera and the racket is, h = 4255 ft

To find the rate of change of the angle of elevation of the camera, we need to differentiate the function of that angle.

Θ= [tex]tan^-^1(\frac{x}{2000} )[/tex]

Differentiating function with respect to x we have,

[tex]\frac{d}{dx}[/tex] Θ=[tex]\frac{d}{dx} tan^-^1(\frac{x}{2000} )[/tex]

Θ'=[tex]\frac{1}{1+\frac{x^2}{2000^2} }[/tex][tex]*\frac{1}{2000}[/tex]

[tex]\frac{1}{2000+\frac{x^2}{2000} }[/tex]  (Equation 1)

[tex]h^2=b^2+p^2[/tex]

[tex]4255^2+2000^2+x^2[/tex]

Θ=[tex]\frac{1}{2000+\frac{14105025}{2000} }[/tex]

≈ 0.00011 rad/ft.

Thus, the rate of change of the angle of elevation is 0.00011 rad/ft.

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Answer:y intercept is 6 slope is -1/4

Step-by-step explanation:

to graph it in y=mx+b form b is the y intercpt and where you start the graph and m is the slope is  [tex]-\frac{1}{4}[/tex] the you go to the right 4 and up one to graph the line