At a school play, there is a spotlight above the center of
the floor that covers a lighted area with a radius of 7 feet.
What is the area covered by the spotlight?
Area = 3.14*r²

Because f is analytic variable in D, it is differentiable, and every place in D has a zero derivative. Therefore, f in D must be constant.

A function f must be differentiable in the domain D if it is analytic there. The derivative of f must therefore exist and be continuous in D. The derivative of f must be zero everywhere in D if the absolute value of f, |f(z)|, is constant in D since the derivative of an absolute value of a function is always zero. As a result, f must be constant in D since the derivative of f is zero everywhere in D. Due to the fact that the derivative of |f(z)| is always zero, this holds true regardless of the value of |f(z)|. As a result, the function f(z) is constant in the domain D if f is analytical in D and |f(z)| is constant in D.

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Both cash and credit sales for the period are reported on the income statement of a retail company. So the correct statement is option (b) Both cash and credit sales for the period.

An income statement shows the income, expenses, and profitability of a company for a specified period. It is also sometimes referred to as the Income Statement (P&L) or Income Statement. it shows you:

Revenue from sales of products or services. 

An income statement shows how much revenue a company has generated, what expenses it has spent, and what income it has allocated over a period of time. You can use it to understand your company's business model and understand how your company is competitive within your industry. 

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If C.)Angle B is congruent to angle E then ∆ABC and ∆DEF are congruent by the ASA criterion.

In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.

According to the ASA congruence postulate, two triangles are congruent when two angles and the included side of one triangle are congruent with two angles and the included side of another triangle.

Looking at our diagram, we can see that the angle C and side CB of ABC are, respectively, congruent with the angle F and side FE of DEF.

Since CB is a side of ABC between angles C and B and FE is a side of DEF between angles F and E, angle B must be equal to angle E for these two triangles to be congruent by ASA.

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1. If , _____ then ∆ABC and ∆DEF are congruent by the ASA criterion.

A.) AB=DE

B.)CA=FD

C.)Angle B is congruent to angle E

D.)Angle A is congruent to angle D

Answer: To find the angle θ in a right triangle given the cosine value, we can use the inverse cosine function, also known as arccos.

cos θ = A/H = 22/38

θ = arccos (22/38)

Using a table of trigonometric values or a calculator, we can find the value of arccos (22/38) to be approximately 56.3 degrees.

So, θ = 56.3° (rounded to the nearest degree).

Step-by-step explanation:

After solving (c+3)², the procedure is an illustration of the distributive property of equations, and c = -3.

Perhaps the most basic form has just been assumed to represent the fraction with the least process. How or when to decide mostly on fundamental form. Look for components that are present in both the denominator and the numerator. Sometimes it is possible to determine if a fractional integer is a power of two.

A algebraic theorem is a statement with two equal sides and an equal sign in the center. The degrees earned by each variable are stated in descending order, similar to an equation's general form. A linear equation has the formula x + b = 0. A three distinct linear function has the general form x + b y + c = 0. (or something similar). Mathematics can be categorized as identities or conditioned equations.

given:

[tex]( c+ 3 ) ^ 2[/tex]

= by using formulas of [tex]( a+ b ) ^2[/tex]

by distributive property

[tex]c^2 + 3^2 + 6c[/tex]

[tex]= c^2 + 9 + 6c[/tex]

[tex]c^2 + 3c + 3c + 9 = 0[/tex]

= (c + 3 )(c+ 3)

c = -3

After solving (c+3)², the procedure is an illustration of the distributive property of equations, and c = -3.

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a) Strings of probability 0s and 1s, which reflect the results of the coin flips, make up the sample space, S.

In (b), I A wins equals (1, 01, 001, 0001), (ii) B wins equals (0, 10, 010, 0010), and (iii) (A U B) equals (1, 0, 01, 10, 001, 010, etc.).

(a) Strings of 0s and 1s, which reflect the results of the coin flips, make up the sample space, S. Each 0 denotes a tail, while each 1 denotes a head. Since there is no restriction on the number of coin flips that can take place, the sample space is unlimited. Due to the fact that every result can be expressed as a string of 0s and 1s, it is also a finite collection. (b) I A wins means that A flipped a head before B or C, and the results are 1, 01, 001, 0001, etc. (ii) B wins = 0 (zero), 10 (one), 010 (zero), 0010 (zero), etc., indicating that B flipped a head before A or C. (iii) (A U B) = 1, 0, 01, 10, 001, 010, etc., indicating that either A or B turned over a coin before C. Presumably, A flips first, followed by B, C, then A, and so on.

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The zeros of the quadratic equation y=4(x+9)²-36 are -6 and -12.

What is a quadratic equation?

Any equation of the form  ax² + bx + c = 0where x is variable and a, b, and c are any real numbers where a ≠ 0 is called a quadratic equation.

Given:

y=4(x+9)²-36

on solving the bracket ,we get

y = 4x² + 324 + 72x - 36

or

y = 4x² + 72x + 288

or

y = 4(x² + 18x + 72)

or

y = 4(x² + 12x + 6x + 72)

or

y = 4(x(x+12)+6(x+12))

or

y = 4(x+6)(x+12)

Therefore , zeroes are -6 and -12.

Hence , the zeros of the quadratic equation y=4(x+9)²-36 are -6 and -12.

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The actual value of $10 in currency A would be of $1.7 in currency B.

$17.

The actual value is obtained applying the proportions in the context of this problem.

The exchange rate is given as follows:

$1 in currency A - $1.7 in currency B.

Hence the value in currency B of $10 in currency A is obtained as follows:

1.7 x 10 = $17.

The problem asks for the value in currency B of $10 in currency A.

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The p=value of this sample is p = 0.076

You should understand that the p value is a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true.

P values are used in hypothesis testing to help decide whether to reject the null hypothesis. The smaller the p value, the more likely you are to reject the null hypothesis.

This is a test of a single population mean since we are dealing with mean

For the null hypothesis,

µ = 73.8

For the alternative hypothesis,

µ ≠ 73.8

Since the number of samples is small and no population standard deviation is given, the distribution is a student's true

Since n = 59,

Degrees of freedom, df = n - 1 = 59 - 1 = 58

t = (x - µ)/(s/√n)

Where

x = sample mean = 76.6

µ = population mean = 73.8

s = samples standard deviation = 8.6

t = (76.6 - 73.8)/(8.6/√58) = 2.8/1.13=  2.4779

We would determine the p-value using the t-test calculator. It becomes

p = 2.4779

Since alpha, 0.01 < than the p-value, 2.4779, then we would fail to reject the null hypothesis. Therefore, At a 1 % level of significance, the sample data showed that there is no significant evidence that μ ≠ 73.8

Therefore, this p-value leads to a decision to accept the null hypothesis

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The solution of the system of equations is equal to (3, 4).

In Mathematics, a point of intersection can be defined as the location on a graph where two (2) lines intersect or cross each other, which is primarily represented as an ordered pair containing the point that corresponds to the x-coordinate (x-axis) and y-coordinate (y-axis) on a cartesian coordinate.

By critically observing the graph of the lines representing the given system of equations, we can reasonably and logically deduce that the correct solution set lies in Quadrant II and it is denoted by the point of intersection of both the x-coordinate (x-axis) and y-coordinate (y-axis), which is given by this ordered pair (3, 4).

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1) The Mathematical curriculum teaches a combination of mathematical concepts, problem solving.

2) Some mathematical instructions that are promoted include understanding of mathematical concepts, computational fluency.

3) Yes, there are specific procedures used to teach basic mathematical instruction.

The Mathematical curriculum can be said to:

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A graph of the linear equation -3x + 5y = 7 in slope-intercept form is shown in the image attached below.

In Mathematics, a graph is a type of chart that is typically used for the graphical representation of data points or ordered pairs on both the horizontal and vertical lines of a cartesian coordinate respectively.

Next, we would rearrange and simplify the given given linear equation in slope-intercept form in order to enable us plot it on a graph:

-3x + 5y = 7

5y = 3x + 7

y = 3x/5 + 7/5

Lastly, we would use an online graphing calculator to plot the given function as shown in the graph attached below.

In conclusion, the slope of this linear equation is equal to 3/5 and it does not represent a proportional relationship.

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Using the midpoint formula, B is: (-9, 1).

The midpoint of a line segment is given by the average of the coordinates of the endpoints. So, the midpoint of AB is (-2,4), and A = (5, 7), we can find B by using the midpoint formula:

M(x, y) = [(x1 + x2)/2, (y1 + y2)/2]

Given:

A(5, 7) = (x1, y1)

B(?, ?) = (x2, y2)

M(x, y)

Plug in the values:

(-2, 4) = [(5 + x2)/2, (7 + y2)/2]

Solve for each coordinates of B:

-2 = (5 + x2)/2

-4 = 5 + x2

-4 - 5 = x2

x2 = -9

4 = (7 + y2)/2

8 = 7 + y2

8 - 7 = y2

y2 = 1

B is: (-9, 1)

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Answer:

111.76

Step-by-step explanation:

My attempt:

To find the number that 95 is 85% of, we can set up an equation using the information given. Let's call this number "x".

According to the problem, 95 is 85% of x, so we can write this as an equation:

95 = 0.85x

Now we can solve for x by dividing both sides by 0.85:

x = 111.76

So 95 is 85% of 111.76.

Answer:

111.76

Step-by-step explanation:

1.We have, 85% × x = 95 or,  85 100 × x = 95

2. Multiplying both sides by 100 and dividing both sides by 85,

we have x = 95 × 100/85

x = 111.76

The benefits and limitations of quadratic models in real-world applications such as bridge design are listed below.

Quadratic models are often used in real-world applications, such as bridge design, because of their ability to describe complex relationships between variables.

Some benefits of using quadratic models in these applications include:

1. Flexibility: Quadratic models are more flexible than linear models and can better capture non-linear relationships between variables.

2. Accuracy: Quadratic models can provide a more accurate representation of the data compared to linear models, especially when the data exhibits a curved trend.

3. Prediction: Quadratic models can be used to make predictions about the future based on past data. For example, they can be used to predict the load-bearing capacity of a bridge over time.

However, there are also some limitations of using quadratic models in real-world applications such as bridge design:

1. Overfitting: Quadratic models have the potential to overfit the data, leading to poor generalization to new data. This can result in an overly complex model that does not accurately represent the underlying relationships in the data.

2. Computational complexity: Quadratic models can be computationally more complex than linear models and require more advanced optimization techniques to solve.

3. Interpretation: Quadratic models can be difficult to interpret, making it challenging to understand the underlying relationships between variables.

4. Unpredictable behavior: Quadratic models can exhibit unpredictable behavior for certain input values, which may not be suitable for certain applications such as bridge design where the model must be reliable and predictable.

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The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

It is a statistical calculation that provides a way of summarizing how much a set of values deviates from the mean, or average, of those values.

The standard deviation is calculated by first finding the difference between each value in the set and the mean, squaring these differences, summing the squares, and dividing by the number of values in the set. The square root of this result gives the standard deviation.

A low standard deviation indicates that the values in a set are closely clustered around the mean, while a high standard deviation indicates that the values are more spread out. The standard deviation is useful in many applications, including quality control, finance, and social sciences, as it provides a way to quantify the variability of a set of values and to make meaningful comparisons between different sets of data.

We can estimate the answer by using the cumulative distribution function (CDF) of the normal distribution. The CDF gives us the probability that a random variable is less than or equal to a certain value. We would need to use a calculator or software to determine the exact value of the CDF, but we can estimate it using the empirical rule.

Since approximately 68% of regular grade gasoline falls within one standard deviation of the mean, we can estimate that roughly 68% of gasoline is less than 83.75.

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Answer:.

Step-by-step explanation:

.

The Hypergeometric distribution is 0.7717, and the binomial distribution is 0.7763. The probability from the binomial approximation is greater than the probability from hypergeometric distribution.

In statistics, a hypergeometric distribution is a distribution function in which members of two groups are chosen from without being replaced. The absence of replacements in the hypergeometric distribution sets it apart from the binomial distribution. As a result, it is frequently used in random sampling to ensure statistical quality.

Given that,

The number of defective bolts are 5 out of 200.

Thus, the probability of 0 defective bolts is given as:

1. Hypergeometric distribution:

P (X= 0) = 5C 0 (195C 10) / (200C 10)

= 0.7717

2. Binomial approximation:

B(n, m/ N)

P (X= 0) = 10C 0 (0.025)^0 (1-0.025)^10

= 0.7763

Thus, probability from the binomial approximation is greater than the probability from hypergeometric distribution.

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Surface area is, [tex]225\pi (yd^{2})[/tex] or [tex]706.86(yd^{2})[/tex] .

As we may see in the image, the surface area has the shape of a circle.

Formula: [tex]A=\pi r^{2}[/tex]; where "r" is the radius of the circle.

As we can see in the image, the radius of this circle is 15 yd (yards).

[tex]A=\pi (15yd)^{2}\\ \\A=225\pi (yd^{2}) =706.86(yd^{2})[/tex]

Surface area is, [tex]225\pi (yd^{2})[/tex] or [tex]706.86(yd^{2})[/tex] (706.86 square yards).

Answer:

144 feet

5 seconds

Step-by-step explanation:

A directly proportional distance between s and  t can be written as
s ∝ t²

or in equation form

s(t) = k · t²

k = constant of proportionality

Given s = 16 feet for t = 1 second we get

16 = k (1)²

or

k = 16

and the equation is s(t) = 16t²

In 3 seconds the object will fall
s(3) = 16 (3)²

= 16 x 9

= 144

s(3) = 144

To find time taken to fall 400 feet, substitute for s s(t) = 400

400 = 16t²

16t² = 400

t² = 400/16 = 25

t = ± √25 = ± 5 seconds

Since time is non-negatiive,

t = 5 seconds

Answer:

$108

Step-by-step explanation:

The tax amount is $89.25 x 6% = $5.35.

The tip amount is $89.25 x 15% = $13.39.

So the total cost is $89.25 + $5.35 + $13.39 = $108.

Step-by-step explanation:

a probability is always the ratio of

desired cases / totally possible cases

in our case, where we pull one card out of 52, the totally possible cases are, of course, 52.

now, we need to find the number of desired cases.

8 of diamonds is pulled.

well, that is one specific card. no other card matches this criteria. so, the probability is

1/52 = 0.019230769... ≈ 0.019

heart or spade is selected.

we have 13 cards of heart.

we have 13 cards of spades.

there is no overlap, so, together they are 26.

the probably is then

26/52 = 1/2 = 0.5

number smaller than 7 is pulled.

"smaller than 7" means 1..6.

that means 6 cards of each of the 4 suits = 24 cards.

the probability is then

24/52 = 6/13 = 0.461538462... ≈ 0.462

The y-coordinate of center of mass is 1.2819 kg.

The center of mass, also known as the center of gravity, is a point in a system of objects where the total mass of the objects is concentrated. It represents the balance point of the system and is the point at which the system would balance if it were suspended from that point. The position of the center of mass is dependent on the distribution of mass in the objects making up the system.

The centre of mass of region R is given by

[tex]$\mathrm{\bar{y}=\frac{\int_{R}\rho y_{e}dA}{Mass_{regionR}} \ \ where \ y_{e} \ center \ of \ mass \ of \ element}[/tex]

For the vertical element,

[tex]\mathrm{y_{e}=\dfrac{y}{2}}[/tex]

Find the numerator of the y coordinate of Center of Mass

[tex]$\mathrm{\frac{1}{2}\int_{R}\rho yydx=\frac{\rho}{2} \int_{\frac{-\pi}{3}}^{\frac{\pi}{3}} (3cosx)^2dx=\frac{9\rho}{2} \int_{\frac{-\pi}{3}}^{\frac{\pi}{3}} cos^2xdx=\frac{9\rho}{2} \int_{\frac{-\pi}{3}}^{\frac{\pi}{3}} (\frac{1+cos(2x)}{2})dx}[/tex]

[tex]$\mathrn{\frac{9\rho}{4} \int_{\frac{-\pi}{3}}^{\frac{\pi}{3}} (1+cos(2x))dx=\frac{9\rho}{4} [x+\frac{sin(2x)}{2}]_{\frac{-\pi}{3}}^{\frac{\pi}{3}}}[/tex]

[tex]$\mathrm{\frac{9\rho}{4} \left[\frac{\pi}{3}+\frac{sin(2(\frac{\pi}{3}))}{2}\right ]-\frac{9\rho}{4} \left[\frac{-\pi}{3}+\frac{sin(2(\frac{-\pi}{3}))}{2}\right ]}[/tex]

[tex]$\mathrn{\frac{9\rho}{4} \left[\frac{\pi}{3}+\frac{\sqrt{3}}{4}\right ]-\frac{9\rho}{4} \left[\frac{-\pi}{3}-\frac{\sqrt{3}}{4}\right ]=\frac{9\rho }{4} \left[\frac{2\pi}{3}+\frac{\sqrt{3}}{2}\right ]}[/tex]

Therefore, the y-coordinate of center of mass is

[tex]$\mathrm{\bar{y}=\frac{\dfrac{9\rho }{4}\left [\dfrac{2\pi}{3}+\dfrac{\sqrt{3}}{2}\right ]}{3\rho \sqrt{3}}=\frac{\sqrt{3}}{4}\left [\frac{2\pi}{3}+\frac{\sqrt{3}}{2}\right ]=\frac{3}{8}+\frac{\sqrt{3}\pi}{6}=1.2819}[/tex]

Thus, The y-coordinate of center of mass is 1.2819 kg

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The center of mass of a thin plate of constant density is determined by its area and the distribution of its mass.

To find the center of mass of this plate, we first need to calculate its area. We can do this by using the formula for the area of a region bounded by the x-axis and a curve y=f(x):

A = ∫[tex]a^{b}[/tex] ydx = ∫-pi/3^pi/3 3cosx dx

A = 3∫-pi/3^pi/3cosx dx

A = 3(sin(pi/3) - sin(-pi/3))

A = 3(sin(pi/3) + sin(pi/3))

A = 6sin(pi/3)

A = 6*(1/2)

A = 3

Now that we know the area of the plate, we can calculate its center of mass. To do this, we need to use the formula for the center of mass of a region bounded by the x-axis and a curve y=f(x):

xcm = 1/A ∫[tex]a^{b}[/tex] xydx

xcm = 1/3 ∫-pi/3^pi/3 x(3cosx) dx

xcm = 1/3 ∫-pi/3^pi/3 3xcosx dx

xcm = 1/3(3(sin(pi/3) - sin(-pi/3)) - 3xsin(pi/3) + 3xsin(-pi/3))

xcm = 1/3(6sin(pi/3) - 3pi/3sin(pi/3) + 3(-pi/3)sin(-pi/3))

xcm = 1/3(6sin(pi/3) - 3pi/3sin(pi/3) + 3pi/3sin(pi/3))

xcm = 0

Therefore, the center of mass of the plate is located at (0, 0).

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Compared with the parent function y= sin(x), the graph of 4 sin[ (π/6)x ] + 1 will be stretched vertically by a scale factor of 4, translated 1 unit up, and with a shorter distance between the peaks.

What is a function?

Each element of X receives exactly one element of Y when a function from one set to the other is used. The sets X and Y are collectively referred to as the function's domain and codomain, respectively. Initially, functions represented the idealized relationship between varied quantities and other variables.

Here, we have

1) The parent function is sin(x)

2) sin(x) has:

Middle line: y = 0

Amplitude: 1 because the function goes from 1 unit up to 1 unit down the middle line.

Period: 2π because the sine function repeats every 2π unit.

y-intercept: (0,0) because sin(0) = 0.

Now look at how these changes in the function reflect on the parameters:

A sin (ωx + B) + C:

That function will have:

amplitude A, because the amplitude is scaled by that factor

Period: 2π / ω, because the function is compressed horizontally by that factor.

It will be translated B units to the left

It will be translated C units up.

And you need

Period = 12 => 2π / ω = 12 => ω = π/6

A = 4

Translate the midline from y = 0 to y = 1 => shift the function 1 unit up => C = 1.

Translate the y-intercept from y = 0 to y = 1, which is already accomplished when you translate the function 1 unit up.

So, this is the function searched

y = A sin (ωx + B) + C = 4 sin[ (π/6)x ] + 1

Now you can check the amplitude, the period, the middle line, and the y-intercept of that y = 4 sin[ (π/6)x ] + 1.

Hence, compared with the parent function y= sin(x), the graph of 4 sin[ (π/6)x ] + 1 will be stretched vertically by a scale factor of 4, translated 1 unit up, and with a shorter distance between the peaks.

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Dustin walk at the rate of 6 miles per hour.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

Dustin can run 2 miles per hour faster than he can walk around the track. If Dustin walks around the track in 20 minutes and runs around the track in 15 minutes.

Let the speed of Dustin walking = x

We know that;

Speed = Distance / Time

For walking;

Speed = Distance / Time

x = Distance / (20/60)

x × 20/60 = Distance

Distance = x/3 miles

For run;

Speed = Distance / Time

⇒ x + 2 = (x/3) / (15/60)

⇒ 3 (x + 2) = 4x

⇒ 3x + 6 = 4x

⇒ 4x - 3x = 6

⇒ x = 6

Thus, Dustin walk at the rate of 6 miles per hour.

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The independent variable, x, represents the number of questions Maya answered correctly, and the dependent variable is the score, because the score depends on the number of questions Maya answered correctly. A function relating these variables is R(x) = [5x].

So R(14) = [70], meaning 14 correct answers will give a score of 70.

What is a variable?

In mathematics, a variable is referred to as the alphanumeric symbol used to represent a number or numerical value. An unknown quantity is represented as a variable in algebraic equations.

The variables can be divided into two groups, including dependent variable and independent variable.

We are given that each question answered correctly on a test would be worth 5 points.

So, the dependent variable is the score as it depends on the number of questions Maya answered correctly.

The independent variable is the number of questions Maya answered correctly.

From this, we get the function as R(x) = [5x].

Substituting 14 in place of x, we get

R(14) = [70]

This means that 14 correct answers will give a score of 70.

Hence, the independent variable, x, represents the number of questions Maya answered correctly, and the dependent variable is the score, because the score depends on the number of questions Maya answered correctly.

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Answer:

9.80

Step-by-step explanation:

x²=14²-10² = 196-100 = 96

x=√96 = √(16*6 ) = 4*√6 =4*2.45=9.80