Find the angle of a sector with area 30 cm2 and radius 12 cm.

The number of people who like Coke is 30

Since the Venn diagram visually represents the differences and the similarities between two concepts.

Given that group of 74 people, if the ratio of children who like Coke only and Pepsi only is 3:2

Also , the number of people, who like both drinks is 17 and who do not like both is 7

Total number of children who like Coke only and Pepsi only are 5x

5x = 74 - 17 - 7

5x = 50

x = 10

Thus, the number of people who like Coke  is 3x means

3 (10) = 30

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The measure of angle a is determined as 30 degrees.

The measure of angle a is calculated by applying the following circle theorem as follows;

If line AB is the diameter of the circle, then angle ACB will be equal to 90 degrees.

The value of a is calculated as;

a + 2a = 90 (complementary angles sum up to 90 degrees since angle ACB is equal to 90 degrees)

3a = 90

divide both sides by 3;

3a/3 = 90 / 3

a = 30

Thus, the measure of angle a is calculated by applying the following circle theorem for complementary angles.

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A 90% confidence interval estimate of the mean weight loss for all such subjects is (3.333, 0.867).

Given that,

Total number of adults who used the program = 41

So, n = 41

Mean weight loss = 2.1 lb

So, x = 2.1

Standard deviation = 4.8 lb

So, s = 4.8

Confidence interval is,

CI = x ± z (s / √n)

For 90% confidence level, z = 1.645

CI = 2.1 ± 1.645 (4.8 / √41)

   = (2.1 + 1.645 (4.8 / √41), 2.1 - 1.645 (4.8 / √41))

   = (3.333, 0.867)

Hence the confidence interval is  (3.333, 0.867).

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Answer:

The answer is D. Cone

Step-by-step explanation:

A tables on sale for $237.60 which is 33% of the regular price .The regular price of the table is $720.

To determine the regular price of the table, we can use the given information that the sale price is $237.60, which represents 33% of the regular price.

Let's assume the regular price of the table is represented by the variable "x". We can set up the equation:

33% of x = $237.60

To solve for "x", we need to convert the percentage to a decimal by dividing it by 100:

0.33 * x = $237.60

Now, we can isolate "x" by dividing both sides of the equation by 0.33:

x = $237.60 / 0.33

Calculating this expression, we find:

x ≈ $720

Therefore, the regular price of the table is approximately $720.

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well, the profit is clearly just 490 - 350 = 140.

Now, if we take 350(origin amount) to be the 100%, what is 140 off of it in percentage?

[tex]\begin{array}{ccll} Amount&\%\\ \cline{1-2} 350 & 100\\ 140& x \end{array} \implies \cfrac{350}{140}~~=~~\cfrac{100}{x} \\\\\\ \cfrac{5}{2} ~~=~~ \cfrac{100}{x}\implies 5x=200\implies x=\cfrac{200}{5}\implies x=40[/tex]

1. The volume of the solid generated by rotating the region bounded by y = x([tex]e^x[/tex]), x = 0, x = 1, and y = 0 about the y-axis is approximately 1.305 cubic units.

2. The volume of the solid generated by rotating the region bounded by the curve y = arctan(x), x = 0, x = 1, and y = 0 about the y-axis is π (1 + ln(2)) cubic units.

1. To find the volume of the solid generated by rotating the region bounded by the curve y = x(e^x), x = 0, x = 1, and y = 0 about the y-axis, we can use the method of cylindrical shells.

The volume of a cylindrical shell is given by the formula:

dV = 2πx × f(x) × dx

Integrating this formula from x = 0 to x = 1 will give us the total volume of the solid.

So, let's set up the integral:

V = ∫[0,1] 2πx × (x(e^x)) dx

To solve this integral, we need to use integration by parts. Let's define u = x and dv = x(e^x) dx. Then, we can find du and v:

du = dx

v = ∫x(e^x) dx

Using integration by parts, we find:

v = x(e^x) - ∫e^x dx

= x(e^x) - e^x

Now, we can substitute these values back into the integral:

V = ∫[0,1] 2πx × (x(e^x) - e^x) dx

Expanding and simplifying:

V = ∫[0,1] 2π(x^2e^x - xe^x) dx

Integrating each term separately:

V = 2π ∫[0,1] (x^2e^x - xe^x) dx

= 2π [(x^2 - 2x + 2)e^x] |[0,1]

Evaluating the limits:

V = 2π [(1^2 - 2(1) + 2)e^1 - (0^2 - 2(0) + 2)e^0]

= 2π (1 - 2 + 2e - 2)

= 2π (2e - 3)

Therefore, the volume of the solid generated by rotating the region bounded by y = x(e^x), x = 0, x = 1, and y = 0 about the y-axis is 2π (2e - 3).

2. To find the volume of the solid generated by rotating the region bounded by the curve y = arctan(x), x = 0, x = 1, and y = 0 about the y-axis, we can use the method of cylindrical shells.

The volume can be calculated using the formula:

V = 2π ∫[a,b] x × f(x) dx,

where f(x) is the function representing the curve, and [a,b] is the interval of integration.

In this case, the interval of integration is [0,1], and the function f(x) = arctan(x).

So, the volume can be computed as:

V = 2π ∫[0,1] x × (arctan(x)) dx.

To evaluate this integral, we can use the technique of integration by parts. Let's define u = arctan(x) and dv = x dx. Then, we have du = (1 / (1 + x^2)) dx and v = (1/2)x^2.

Applying integration by parts:

V = 2π [(x^2 × arctan(x))/2 - ∫[0,1] (x^2 / (2(1 + x^2))) dx]

V = π [(x^2 × arctan(x)) - ∫[0,1] (x^2 / (1 + x^2)) dx].

To evaluate the integral on the right side, we can use the substitution method. Let's substitute u = 1 + x^2, then du = 2x dx. Rearranging for x^2:

x^2 = u - 1.

Substituting back into the integral:

V = π ∫[0,1] ((u - 1) / u) du

V = π ∫[0,1] (1 - (1/u)) du

V = π [u - ln(u)] |[0,1]

V = π [(1 + ln(2)) - (0 - ln(1))].

Simplifying further:

V = π (1 + ln(2)).

Therefore, the volume of the solid generated by rotating the region bounded by the curve y = arctan(x), x = 0, x = 1, and y = 0 about the y-axis is π (1 + ln(2)) cubic units.

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Check the picture below.

so we really have four congruent triangles with a base of hmmm 8.9 and a height of 4.4.

[tex]\stackrel{ \textit{four congruent triangles} }{4\left[\cfrac{1}{2}(\underset{b}{8.9})(\underset{h}{4.4}) \right]} ~~ \approx ~~ \text{\LARGE 78.3}~in^2[/tex]

The equation of the line in General form is 2x + y + 1 = 0.

The equation of the line in general form is in the form Ax + By + C = 0, where A, B, and C are constants.

Let's consider the given equation: -2x - y - 1 = 0

To rewrite it in general form, we can rearrange the equation to isolate the x and y terms on the left side:

-2x - y = 1

To eliminate the negative coefficient of x, we can multiply the entire equation by -1:

2x + y = -1

Now, we have the equation in general form: 2x + y + 1 = 0

Therefore, the equation of the line in general form is 2x + y + 1 = 0.

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The correct way to punctuate the sentence is:

"Hannah found a lost puppy in her yard and wanted to keep it; however, she knew she needed to look for its owners first."

This option uses a semicolon (;) to separate the independent clauses "Hannah found a lost puppy in her yard and wanted to keep it" and "she knew she needed to look for its owners first." The semicolon is appropriate in this case because the two clauses are closely related and the second clause expands upon the information provided in the first clause.

Additionally, the word "however" is properly placed after the semicolon, indicating the contrast between Hannah's desire to keep the puppy and her recognition of the need to find its owners. The sentence is correctly structured and punctuated to convey the intended meaning and maintain clarity for the reader.

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Answer: x=log      (64^-1/6)

                         10

Step-by-step explanation:  I took the test.

Hope this helps :)

Please mark me as brainliest.

The quotient is -2x - 2. Matching the expressions with their respective quotients:

(x²-3x - 18) + (x-6) → x² - 2x - 24

(x²-5x-3) + (x²+2x+1) → 2x² - 3x - 2

(x³ + 2x² - 1) / (x² + x + 1) → -2x - 2

Therefore, the expressions are matched with their respective quotients

Let's simplify each expression and match them with their corresponding quotient:

(x²-3x - 18) + (x-6)

Simplifying the expression:

x² - 3x - 18 + x - 6 = x² - 2x - 24

(x²-5x-3) + (x²+2x+1)

Simplifying the expression:

x² - 5x - 3 + x² + 2x + 1 = 2x² - 3x - 2

(x³ + 2x² - 1) / (x² + x + 1)

To divide these polynomials, we need to perform polynomial long division:

         x + 1

  ___________________

x² + x + 1 | x³ + 2x² - 1

- (x³ + x² + x)

----------------

x² - 1

- ( x² + x + 1)

----------------

-2x - 2

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The two triangles can be proved as:

ΔABD ≅ ΔBAC by SSS congruency

We have,

- Side-Side-Side (SSS) Congruence.

The three sides of one triangle are equal to the corresponding three sides of another triangle.

Now,

ΔAED ≅ ΔBEC

This means,

AD = BC

And,

AC ≅ BD

Now,

In ΔABD and ΔBAC,

AB = AB  ( common )

AD = BC (ΔAED ≅ ΔBEC)

AC ≅ BD (given)

So,

ΔABD ≅ ΔBAC ( SSS )

Thus,

ΔABD ≅ ΔBAC by SSS congruency

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The mathematical function that models Autumn's situation of saving $500 that she keeps at home and does not add to the savings is G(x) = 500 + 0x.

A mathematical function is an equation, expression, rule, or law defining the relationship between the independent variable and the dependent variable.

There are many types of mathematical function, including:

The amount that Autumn saved = $500

The amount that Autumn adds to the savings = $0

Let the number of periods that Autumn saves the above amount = x

G(x) = 500 + 0x

Thus, based on the linear function above, the amount that Autumn saved would remain $500 after many periods.

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The linear function that relates y to x is given as follows:

y = -0.5x + 57.

The slope indicates that the height decays 0.5 feet per minute. The y-intercept indicates that the descent starts at a cruising altitude of 57 feet.

The slope-intercept equation for a linear function is presented as follows:

y = mx + b

The coefficients m and b represent the slope and the intercept, respectively, and are explained as follows:

From the table, when x = 0, y = 57, hence the intercept b is given as follows:

b = 57.

When x increases by 10, y decays by 5, hence the slope m is given as follows:

m = -5/10

m = -0.5.

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The cost of one plate is $4.30 and the cost of one cup is $2.00.

Let's assume the cost of one plate is p dollars and the cost of one cup is c dollars.

From the given information, we can set up two equations based on the total costs:

Equation 1: 5p + 4c = 26.30 (since 5 plates and 4 cups cost $26.30)

Equation 2: 6p + 5c = 32.00 (since 6 plates and 5 cups cost $32.00)

We have a system of two equations with two unknowns. We can solve this system of equations using various methods, such as substitution or elimination. Let's use the elimination method.

Multiply Equation 1 by 5 and Equation 2 by 4 to eliminate the variable "c" when we add the equations together:

25p + 20c = 131.50 (5 times Equation 1)

24p + 20c = 128.00 (4 times Equation 2)

Subtract Equation 2 from Equation 1:

25p - 24p = 131.50 - 128.00

p = 3.50

Now, substitute the value of p into Equation 1 to find the value of c:

5(3.50) + 4c = 26.30

17.50 + 4c = 26.30

4c = 26.30 - 17.50

4c = 8.80

c = 2.20

Therefore, the cost of one plate is $3.50 and the cost of one cup is $2.20.

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The volume of the solid generated by rotating the region bounded by the curve y = x(e^x), x = 0, x = 1, and y = 0 about the y-axis is 2π[e - e^1 + 1]   and the volume of the solid generated by rotating the region bounded by the curve y = arctan(x), x = 0, x = 1, and y = 0 about the y-axis is πtan^2(1).

1) To find the volume of the solid generated by rotating the region bounded by the curve y = x(e^x), x = 0, x = 1, and y = 0 about the y-axis, we can use the method of cylindrical shells.

First, let's express the equation in terms of x: x = ln(y)/y. Now we can set up the integral to calculate the volume. The volume can be obtained by integrating the product of the circumference of each cylindrical shell and its height over the interval [0, 1]:

V = ∫[0,1] 2πx(e^x) dx.

Using integration by parts, we can evaluate this integral as follows:

V = 2π ∫[0,1] x(e^x) dx

 = 2π [x(e^x) - ∫[0,1] e^x dx]

 = 2π [x(e^x) - e^x] | [0,1]

 = 2π[(1(e^1) - e^1) - (0(e^0) - e^0)]

 = 2π[e - e^1 + 1].

2) To find the volume of the solid generated by rotating the region bounded by the curve y = arctan(x), x = 0, x = 1, and y = 0 about the y-axis, we can again use the method of cylindrical shells.

Let's express the equation in terms of x: x = tan(y). Now we can set up the integral to calculate the volume. The volume can be obtained by integrating the product of the circumference of each cylindrical shell and its height over the interval [0, 1]:

V = ∫[0,1] 2πtan(y) dy.

Using the substitution method, we can evaluate this integral as follows:

Let u = tan(y), then du = sec^2(y) dy.

When y = 0, x = tan(0) = 0, and when y = 1, x = tan(1).

Now, the integral becomes:

V = 2π ∫[0,1] u du

 = 2π [u^2/2] | [0,1]

 = 2π[(tan(1))^2/2 - 0]

 = πtan^2(1).

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The component form of vector v is (-5, 6). Hence, the correct answer is option D: (-5, 6).

To determine the component form of vector v from the given graph, we need to analyze the horizontal and vertical displacements.

Looking at the graph, we can see that the vector starts at the origin (0,0) and ends at a point that is approximately (-5, 6).

The horizontal displacement is the change in the x-coordinate, which is -5 units.

The vertical displacement is the change in the y-coordinate, which is 6 units.

Therefore, the component form of vector v is (-5, 6).

Hence, the correct answer is option D: (-5, 6).

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Answer:

37.68 ft²

Step-by-step explanation:

radius = 4 feet

Sector angle = 3 ft/2

convert to degree = 3 pi/2 × 180/ pi = 270°

(pi = 3.14)

Area of sector = Angle of sector/Angle of circle × pi × r²

= 270/360 × 3.14 × 4²

= 0.75 × 3.14 × 16

= 37.68 ft²

The equation of the perpendicular line passing through (0, -1) is y = -9x - 1.

To find the equation of a line that is perpendicular to the given line y = (1/9)x + 2 and passes through the point (0, -1), we can use the fact that perpendicular lines have slopes that are negative reciprocals of each other.

The given line has a slope of 1/9. To find the slope of the perpendicular line, we take the negative reciprocal of 1/9, which is -9.

Using the slope-intercept form of a linear equation, y = mx + b, where m represents the slope and b represents the y-intercept, we can substitute the slope and the coordinates of the given point (0, -1) into the equation.

y = -9x + b

Since the line passes through the point (0, -1), we can substitute the x-coordinate as 0 and the y-coordinate as -1 into the equation:

-1 = -9(0) + b

-1 = b

Therefore, the y-intercept (b) of the perpendicular line is -1.

Putting it all together, the equation of the line that passes through (0, -1) and is perpendicular to y = (1/9)x + 2 is:

y = -9x - 1

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The solution is:  The difference in minutes of median collection times before and after the well was installed is: 25

Here, we have,

Before--

Clearly after looking at the whisker-box plot we could observe that the middle line in the box represents the median of the data.

Hence we have :

Median(M)=53

After--

Similarly looking at the box-whisker plot we could observe that the line in the box is at 28.

Hence, Median of the box after installing the well is(M'): 28

Hence,

The difference in Median is calculated as:

M'-M

= 53-28

= 25

Hence, the difference in minutes of median collection is:

25 minutes.

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Answer: A box plot

Step-by-step explanation:

A box plot (also called a box-and-whisker plot) is an appropriate graph to represent the interquartile range. The box plot displays the five-number summary of a dataset: the minimum value, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum value. The box in the middle of the plot represents the interquartile range (IQR), which is the difference between the third and first quartiles. The whiskers extend from the box to the smallest and largest values that are not outliers.

Box plots are useful for comparing the distribution of multiple datasets and identifying any outliers. They provide a visual summary of the central tendency, variability, and skewness of a dataset.

Answer:

V = 326.3 m ^ 3

Step-by-step explanation:

First we look for the area of the base of the prism:

A = (1/2) * (b) * (h)

Where,

b: base of the triangle

h: height of the triangle

For the base we have:

b = 10 * sine (30)

b = 5

For the height we have:

h = 10 * sine (60)

h = 8.7

Substituting we have:

A = (1/2) * (5) * (8.7)

A = 21.75

The prism volume will be:

V = A * H

Where,

H: prism height

V = (21.75) * (15)

V = 326.25

Round to the nearest tenth:

V = 326.3 m ^ 3

Answer:

the volume is:

V = 326.3 m ^ 3

The maximum value of P is 81, and it occurs at (9,0)

Given the function P = 9x - 5t and the constraints x ≥ 0; 5x + 8y ≤ 40; 9x + 4y ≥ 36, the task is to maximize P.We will first plot the feasible region.

5x + 8y ≤ 405y ≤ 40 - 8yy ≤ (40/5) - (8/5)x or y ≤ - (8/5)x + 8This is an equation of a straight line.

To plot the line, we need to find two points on the line.We take x = 0, then y = 8. And,

taking y = 0, we get x = 5 Hence the graph of the inequality is:

Similarly, 9x + 4y ≥ 369x ≥ 36 - 4yy ≥ (36/9) - (4/9)x or y ≥ - (4/9)x + 4

This is the equation of another straight line. We take x = 0, then y = 4. And, taking y = 0,

we get x = 9Hence the graph of the inequality is:We can now graphically represent the feasible region in the following manner:

The corner points of the feasible region are A(0,0), B(5,0), C(6,3), D(9,0),

The value of P at these points is:A(0,0) ⇒ P = 0B(5,0) ⇒ P = 45C(6,3) ⇒ P = 45D(9,0) ⇒ P = 81

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hello

the answer to the question is:

x³ - 3x² + 3x - 1 > (3/2)x(x - 1)² ---->

(x - 1)³ > (3/2)x(x - 1)² ----> x - 1 > (3/2)x ---->

(1/2)x < - 1 ----> x < 2

We can write k(x) as the composition of g(x) and f(x).

k(x) = g(f(x))

A composition of two functions means that we need to evaluate one function in the other one.

Here we have the functions:

f(x)=  x²

g(x) = √(x - 1)

h(x) = 2x + 3

And we know that:

k(x) = √(x² - 1)

So we have a square root, then we need to evaluate g(x), and the argument is a square, then we need to evaluate in f(x), the composition is:

k(x) = g(f(x))

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Answer:

a

Step-by-step explanation:

sin A = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{3}{5}[/tex]

cos A = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{4}{5}[/tex]

tan A = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{3}{4}[/tex]

then

sin A = [tex]\frac{3}{5}[/tex]

To write a function to describe the scenario where Mary has 75 pieces of candy in a jar and eats 3 pieces every day, we can use the following formula:y = 75 - 3xHere, y represents the number of pieces of candy Mary will have left after x days.

The function is a linear equation, where the slope is -3, indicating that Mary eats 3 pieces of candy every day, and the y-intercept is 75, indicating that she starts with 75 pieces in the jar.

To use this function, we can plug in different values for x, representing the number of days since Mary started eating candy, and get the corresponding value of y, representing the number of pieces of candy she has left.

For example, after one day, x = 1, so:y = 75 - 3(1)y = 72This means that Mary will have 72 pieces of candy left after one day. Similarly, after two days, x = 2, so:y = 75 - 3(2)y = 69This means that Mary will have 69 pieces of candy left after two days.

We can continue to plug in different values for x to get the corresponding values of y.The function can also be represented graphically as a straight line with a negative slope.

The y-axis represents the number of pieces of candy Mary has left, and the x-axis represents the number of days since she started eating candy. The line starts at (0, 75) and goes down by 3 units for every unit of x, as shown in the following graph:

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