Susan us flying a kite behind her house. She drops her string holder, and the kite get s caught in the top of a tree.

If the string makes 44 degree angle with the ground, and the holder is 90 feet from the base of the tree, how tall is the tree, rounded to the nearest whole foot.

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Kami will have $1,413.60 in the account after 4 years.

(1) Depositing $550 at 3% interest for one year generated a $16.50 profit for Abe.

(2) Jessi returned a $1,500 loan with a quarterly 3% rate and paid $1,511.25 in total.

(3) After keeping a deposit worth $418 at 2% for a year, Heath made an $8.36 profit. In 5.5 years, his account balance grew to $463.98.

(4) By depositing $825.50 at 4%, Pablo saw a $33.02 profit within a year.

(5) Kami put down $1,140 earning a $68.40 annual yield thanks to the 6% interest rate. Four years later, her account balance reached $1,413.60.

Interest = 1,140 * 0.06 * 4 = $273.60

Now, add the interest to the principal:

Total Amount = Principal + Interest = 1,140 + 273.60 = $1,413.60

Kami will have $1,413.60 in the account after 4 years.

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The surface area of the cube is 1014 square millimeters.

The formula for the surface area of a cube is a=6s², where s is the length of the side of the cube. Given that the length of each side of the cube is 13 mm, we can substitute this value into the formula and simplify:

a = 6s²

a = 6(13²)

a = 6(169)

a = 1014

The surface area of a cube refers to the total area of all its faces. Since a cube has 6 faces of equal size, we can multiply the area of one face by 6 to obtain the total surface area of the cube. The formula A = 6s² represents this concept, where s is the length of the side of the cube.

Therefore, the surface area of the cube with a side length of 13 mm is 1014 square millimeters.

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Answer: 10

Step-by-step explanation:

6 1/2 + 3 1/2
13/2 + 7/2
20/2
= 10

Step 1: Add the whole numbers

6 + 3 = 9

Step 2: Add the fractions

We can simply add the numerators while keeping the same denominator because the denominators are the same.

1/2 + 1/2 = 2/2 = 1

Step 3: Combine them

The answer is 10 since 9 and 1 are whole numbers. So we simply add them.

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Overall, we add the whole numbers first, then add the fractions. If the denominators are identical, we add the numerators and keep the same denominator. The solution will be displayed as a mixed number/fraction or a whole number. Since there was no fraction at the end, in this case, we simply added the whole numbers.

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The top-written number in a fraction is the numerator. It shows how many parts of the whole you are talking about.

For example, the numerator of the fraction 3/5 is 3, which means there are 3 parts out of a total of 5 equal parts.

The number of parts taken out of the whole is therefore represented by the numerator.

The number written at the bottom of a fraction works as the denominator. It gives the number of equally sized parts of the whole.

As an example, the denominator of the fraction 2/5 is 5, which shows that the entire is divided into four equal parts.

The total number of equal parts that make up the whole is represented by the denominator.

Mixing a full number and a fraction creates a mixed number. The whole number comes first, then a space, and then the fraction is written.

For example, the mixed number 2 1/2 is a whole number and a fraction, with 2 being the whole number. The word "and" between a mixed number's whole and fraction might be removed or included.

A proper fraction, or one that is less than one whole, must make up the fractional part of the mixed number. Quantities that are not whole numbers but instead consist of several whole numbers are represented by mixed numbers.

A number that represents a finished item or thing is said to be a whole number. It is a number that is neither a decimal nor a fraction.

All natural numbers and zero are considered whole numbers. These are positive integers without decimals, fractions, or negative numbers. A few examples of entire numbers include 1, 2, 3, and so forth.

For counting items that cannot be divided into smaller portions, whole numbers are used.

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π^2/6 = Σn=1^∞ 1/n^2

This completes the proof.

To calculate the Fourier series of the function f(x) = 1-x^2, we first extend it to a periodic function on (-∞, ∞) with period 2 by defining it as follows:

f(x) = 1 - x^2, -1 < x ≤ 1

f(x+2) = f(x), for all x in R

Since f is an even function, its Fourier series only contains cosine terms:

f(x) = a0/2 + Σn=1^∞ an cos(nπx/2), -∞ < x < ∞

where an = (2/π) ∫[-1,1] f(x) cos(nπx/2) dx.

To find the Fourier coefficients an, we first calculate a0:

a0 = (2/π) ∫[-1,1] f(x) dx

= (2/π) ∫[-1,1] (1 - x^2) dx

= 4/π

Next, we calculate an for n > 0:

an = (2/π) ∫[-1,1] f(x) cos(nπx/2) dx

= (2/π) ∫[-1,1] (1 - x^2) cos(nπx/2) dx

= 8/[n^3π^3 (1 - (-1)^n)] for n > 0

Therefore, the Fourier series of f is:

f(x) = 2/π - (8/π) Σn=1^∞ [1/((nπ)^2 (1 - (-1)^n))] cos(nπx/2), -∞ < x < ∞

Now, we can use this series to prove that:

Σn=1^∞ 1/n^2 = π^2/6

To do this, we start with the identity:

f(x) = (2/π) Σn=1^∞ [1/((nπ)^2 (1 - (-1)^n))] cos(nπx/2)

Integrating both sides over [-1,1], we get:

2/π ∫[-1,1] f(x) dx = (2/π) Σn=1^∞ [1/((nπ)^2 (1 - (-1)^n))] ∫[-1,1] cos(nπx/2) dx

The integral on the right-hand side is equal to 0 for odd values of n and 2 for even values of n. Therefore, we can simplify the equation as:

1 = (4/π) Σn=1^∞ [1/((nπ)^2)]

Multiplying both sides by (π^2/6), we get:

π^2/6 = Σn=1^∞ 1/n^2

This completes the proof.

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Answer:

385 in ^3

Step-by-step explanation:

11 x 7 x 5

Answer:

Step-by-step explanation:

its 2.82, a little further forward, 82% of the way to number 3

Check the picture below.

The zeroes of polynomial are 43/5 and -3/4.

We have,

4 (3x-2)² -3 (3x-2) (x+5) - 7(x+5)²

simplifying the above expression we get

4 (9x² + 4 -12x ) -3 (3x² + 15x - 2x - 10) - 7(x² + 25 + 10x)

= 36x² - 48x + 16 - 9x² -39x + 30 - 7x² - 175 - 70x

= 20x² -157x -129

Now, solving the quadratic equation we get

x = 43/5 and x= -3/4

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The length 20 inches of the icicle in feet is 1 2/3 feet

From the question, we have the following parameters that can be used in our computation:

Jessica found an icicle 20 inches long.

This means that

Length = 20 inches

To convert inches to feet, we divide the length value by 12

So, we have

Length = 20/12 feet

Evaluate

Length = 1 2/3 feet

Hence, the length is 1 2/3 feet

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G and H in terms of x and y is given by H = [tex]B^d[/tex](a, 1) and G = [tex]B^{d_2}(0, 1)[/tex] , the  set S of all possible values of y is x+y≥0 the Cartesian coordinates systems is S = [-7/5, -3/5].

Choosing a point O of the line (the origin), a unit of length, and an orientation for the line are all steps in choosing a Cartesian coordinate system for a one-dimensional space, or for a straight line. The line "is oriented" (or "points") from the negative half towards the positive half when an orientation determines which of the two half-lines given by O is the positive half and which is the negative half. Then, depending on which half-line contains P, the distance between each point P on the line and O can be specified.

a) O = (0, 0) a = (2, -1)∈R²

G = [tex]B^{d_2}(0, 1)[/tex]

D = [tex]\sqrt{x^2+y^2}[/tex] < 1

So this is a circle until center at (0, 0) and no point on

[tex]x^2+y^2[/tex] and every point inside it

H = [tex]B^d[/tex](a, 1) = {v=(x,y)∈R²: d(a, v)≤1}

b) x-2 + y=1 ≤ 1

x-y ≤4

For, x-2≤0, y+1≥0  we get,

2-x+y+1≤1 y-x ≤-2

For, x-2≤0,   y+1≤0,   2-x-y-1≤1

x+y≥0

c) Therefore,

d(a, (13/5, y) ≤ 1

(13/5 -2) + (y +1) ≤ 1

S = [-7/5, -3/5].

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The manufacturer claims that their batteries, Type A, exceed their competitor, Type B. However, based on the summary statistics collected by the consumer organization, no definitive conclusion can be drawn as to which battery type lasts longer.

The summary statistics for Type A and Type B batteries must be compared to determine which one lasts longer. The statistics to compare include the mean, median, and range of each battery type.

If the mean and median lifespans of Type A are higher than those of Type B, and if the range of Type A is smaller than that of Type B, then it can be concluded that Type A lasts longer.

However, if the statistics show the opposite, or if there is overlap between the ranges of the two types, then no definitive conclusion can be made as to which battery type lasts longer.

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The residuals for the sales data are 106, 95, 86, 96, 89, 89, 89, 93, 83, and 94.

Residuals represent the differences between the observed values and the predicted values of the dependent variable. In a regression analysis, the predicted values are estimated using the regression equation, while the observed values are the actual values of the dependent variable.

To compute the residuals in this case, we need to first use the estimated regression equation to predict the values of annual sales based on years of experience for each salesperson. The estimated regression equation is:

Annual Sales ($1,000s) = -30 + 4 x Years of Experience

Using this equation, we can predict the annual sales for each salesperson based on their years of experience. Then, we can subtract the predicted values from the actual values to obtain the residuals.

For example, for the first salesperson who has one year of experience and annual sales of $80, we can predict their annual sales using the regression equation as:

Annual Sales = -30 + 4 x 1 = -26

The residual for this salesperson is then:

Residual = $80 - (-26) = $106

We can repeat this process for each salesperson and obtain the following table:

Years of Experience Annual Sales ($1,000s) Predicted Annual Sales Residuals

1 80 -26 106

3 97 2 95

4 92 6 86

4 102 6 96

6 103 14 89

8 111 22 89

10 119 30 89

10 123 30 93

11 117 34 83

13 136 42 94

So the residuals for the sales data are 106, 95, 86, 96, 89, 89, 89, 93, 83, and 94.

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The estimated values of a, b, and c are approximately 16.32, 0.9555, and 0.6417, respectively.

Solving for y3 using the method of successive approximation:

We start by setting up the first iteration, with h = dx = 0.1:

y1 = 1 (given)

y2 = y1 + h(x1 + y1) = 1 + 0.1(1+1) = 1.2

y3 = y2 + h(x2 + y2) = 1.2 + 0.1(2+1.2) = 1.44

y4 = y3 + h(x3 + y3) = 1.44 + 0.1(3+1.44) = 1.728

And so on.

After several iterations, the values converge to a particular value. In this case, y3 ≈ 1.6273.

Interpolating f(2.6) using Lagrange interpolation:

We have:

f(2.6) = L1(2.6)y1 + L2(2.6)y2 + L3(2.6)y3

where Li(x) = ∏j≠i (x-xj)/(xi-xj)

Evaluating Li(2.6) for i = 1, 2, 3:

L1(2.6) = (2.6-2)(2.6-3) / ((1-2)(1-3)) = 0.25

L2(2.6) = (2.6-1)(2.6-3) / ((2-1)(2-3)) = -0.5

L3(2.6) = (2.6-1)(2.6-2) / ((3-1)(3-2)) = 0.25

Substituting the given values:

f(2.6) ≈ 0.25(2.7183) - 0.5(7.3891) + 0.25(20.0855) ≈ 8.6082

Therefore, f(2.6) ≈ 8.6082.

Estimating the values of a, b, and c using multiple linear regression:

We can rewrite the model as a linear equation by taking the natural logarithm of both sides:

ln(y) = ln(a) + b ln(x) + c ln(e)

We can then use linear regression techniques to estimate the values of ln(a), b, and c. Using the given data and a statistical software, we obtain the following estimates:

ln(a) = 2.7912

b = 0.9555

c = -0.4462

To obtain estimates for a, b, and c themselves, we exponentiate the values of ln(a) and ln(e):

a ≈ e^2.7912 ≈ 16.32

b ≈ 0.9555

c ≈ 0.6417

Therefore, the estimated values of a, b, and c are approximately 16.32, 0.9555, and 0.6417, respectively.

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The answer is C. 2 km.

To find the distance after the policeman spots the thief where he will catch him, we'll need to use the terms distance, speed, and time.

Given:
- Distance between thief and policeman: 500 m
- Speed of thief: 8 km/hr
- Speed of policeman: 10 km/hr

Step 1: Convert the distance to kilometers (since the speeds are in km/hr).
500 m = 0.5 km

Step 2: Calculate the relative speed of the policeman to the thief.
Relative speed = Speed of policeman - Speed of thief = 10 km/hr - 8 km/hr = 2 km/hr

Step 3: Calculate the time it will take for the policeman to catch the thief.
Time = Distance / Relative speed = 0.5 km / 2 km/hr = 0.25 hr

Step 4: Calculate the distance traveled by the policeman in that time.
Distance traveled = Speed of policeman × Time = 10 km/hr × 0.25 hr = 2.5 km

Step 5: Subtract the initial distance between them to find the distance after the policeman spots the thief.
Distance after spotting = Distance traveled - Initial distance = 2.5 km - 0.5 km = 2 km

So, the policeman will catch the thief at a distance of 2 km after he spots him. The answer is C. 2 km.

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Is due to the fact that the standard error of the sample mean decreases with increasing sample size, leading to a more accurate estimation of the population mean.

To determine whether it is more likely to randomly select one woman with a height less than 64.4 inches or a sample of 21 women with a mean height less than 64.4 inches, we need to calculate the probability in each case.

Case 1: Randomly selecting 1 woman with height less than 64.4 inches

Since the height is normally distributed with a mean of 63.7 inches and a standard deviation of 2.5 inches, we can use the z-score formula to calculate the probability of selecting a woman with height less than 64.4 inches:

z = (64.4 - 63.7) / 2.5 = 0.28

From the standard normal distribution table, we can find that the probability of selecting a woman with a z-score of 0.28 or less is approximately 0.6103. Therefore, the probability of randomly selecting one woman with a height less than 64.4 inches is 0.6103.

Case 2: Selecting a sample of 21 women with mean height less than 64.4 inches

Since we are dealing with a sample mean, we need to use the central limit theorem, which tells us that the distribution of sample means will be approximately normal, with a mean of the population mean (63.7 inches) and a standard deviation of the population standard deviation divided by the square root of the sample size (2.5 / sqrt(21) = 0.545).

Using the same formula as before, we can calculate the z-score for a sample mean of less than 64.4 inches:

z = (64.4 - 63.7) / (2.5 / sqrt(21)) = 1.252

From the standard normal distribution table, we can find that the probability of selecting a sample mean with a z-score of 1.252 or less is approximately 0.8944. Therefore, the probability of selecting a sample of 21 women with mean height less than 64.4 inches is 0.8944.

Conclusion:

Based on the calculated probabilities, we can conclude that it is more likely to select a sample of 21 women with mean height less than 64.4 inches, as the probability of this event is higher than the probability of randomly selecting one woman with height less than 64.4 inches. This is due to the fact that the standard error of the sample mean decreases with increasing sample size, leading to a more accurate estimation of the population mean.

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The value of the unknown angle is 40⁰

A chord of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. More generally, a chord is a line segment joining two points on any curve

In geometry, a circular segment (symbol:  also known as a disk segment, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord.

Circle theorems are properties that show relationships between angles within the geometry of a circle. We can use these theorems along with prior knowledge of other angle properties to calculate missing angles, without the use of a protractor. This has very useful applications within design and engineering.

Angles in the same segment are equal

The two angles marked are seen to be in the same segment and as such they are equal angles

The value of each of the angles is 40⁰

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1. The instantaneous rate of change in the amount after 2 years is [tex]$1,605.64[/tex] per year

2. The instantaneous rate of change in the amount at the time the amount is equal to [tex]$14,101[/tex] is approximately $994.78 per year

[tex]A = P[/tex]× [tex]e^{rt}[/tex]

where P is the principal (initial investment), r is the annual interest rate as a decimal, and t is the time in years.

For this problem, we have P = $10,000, r = 0.067 (6.7% as a decimal), and we want to find the instantaneous rate of change in the amount after 2 years, so t = 2.

Part 1:

To find the instantaneous rate of change, we need to take the derivative of the function A(t) with respect to t:

[tex]dA/dt = Pre^{rt}[/tex]

At[tex]t = 2[/tex], we have:

[tex]A(2) = $10,000e^{0.0672}[/tex]

[tex]= $11,868.94[/tex]

[tex]dA/dt = $10,0000.067e^{0.067}[/tex]×[tex]2)[/tex]

[tex]= $1,605.64[/tex]

So the instantaneous rate of change in the amount after 2 years is $1,605.64 per year

Part 2:

To find the time at which the amount in the account is $14,101, we need to solve the equation A = $14,101 for t:

[tex]$14,101[/tex][tex]= $10,000[/tex] × [tex]e^{0.067t}[/tex]

Dividing both sides by $10,000:

[tex]1.4101 = e^{0.067t}[/tex]

Taking the natural logarithm of both sides:

[tex]ln(1.4101) = 0.067t[/tex]

Solving for t:

[tex]t = ln(1.4101)/0.067[/tex]

≈ [tex]3.5 years[/tex]

So the time at which the amount in the account is $14,101 is approximately 3.5 years.

To find the instantaneous rate of change at this time, we need to evaluate the derivative at t = 3.5:

[tex]dA/dt = $10,0000.067e^{0.067}[/tex]×[tex]3.5)[/tex]

≈ [tex]$994.78[/tex]

So the instantaneous rate of change in the amount at the time the amount is equal to $14,101 is approximately $994.78 per year

Compound interest is the interest you earn on interest. This can be illustrated by using basic math: if you have $100 and it earns 5% interest each year, you'll have $105 at the end of the first year. At the end of the second year, you'll have $110.25

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Answer:

To determine the base of a triangle, we need more information. The base of a triangle is one of its sides, typically denoted as the side opposite to the triangle's vertex or apex. In order to find the base, we need to know either the length of the other side and the angle between them, or the height of the triangle along with the length of one of its sides.

If you have additional information about the triangle, such as the length of another side or the height, please provide that information so that we can help you find the base.

Step-by-step explanation:

A. Theoretical probability of rolling each value for both Diamond and Trevor is 16.67%.

The experimental probability of rolling each value for Diamond is 16.95% for one, 33.20% for two, and 49.84% for three.

The experimental probability of rolling each value for Trevor is 69.38% for one, 24.45% for two, and 6.17% for three. Based on the Law of Large Numbers, Diamond's dice can be assumed to be fair, but Trevor's dice cannot be assumed to be fair.

The theoretical probability of rolling each value for both Diamond and Trevor is 1/6 or 16.67%. To find the experimental probability for Diamond, we divide the number of times each value was rolled by the total number of rolls and multiply by 100%. For example, the experimental probability of rolling one is (217/1280) x 100% = 16.95%.

Based on the Law of Large Numbers, which states that the sample mean will approach the population mean as the sample size increases, we can reasonably assume that Diamond's dice is fair.

However, Trevor's dice cannot be assumed to be fair because the experimental probability of rolling each value is significantly different from the theoretical probability, indicating a potential bias in the dice.

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The value of the output is independent of the value of the input.

In the graph, the input is the x value (x-axis) and the output is the y value (y-axis).

Looking at the graph, you notice the y values are constant (the same) while the x values changes.

What this means is that whatever the value of the input (x value), the value of the output (y value) will remain the same. That is the value of the output is independent of the value of the input.

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Answer: Here i will explain it to you and give an example

Here's an example: let's say you have three positive integers, 5, 12, and 13. To check if they form a Pythagorean triple, you can compute 5^2 + 12^2 = 25 + 144 = 169, which is equal to 13^2. Since the equation holds, the three numbers 5, 12, and 13 form a Pythagorean triple.

In fact, this is a well-known Pythagorean triple, because it is one of the smallest triples, and it is frequently used in geometry and mathematics. The triple (5, 12, 13) satisfies the Pythagorean theorem and represents the lengths of the sides of a right triangle.

Step-by-step explanation:  Three positive numbers form a Pythagorean triple if they satisfy the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In other words, if a, b, and c are the lengths of the sides of a triangle such that c is the length of the hypotenuse (the longest side) and a and b are the lengths of the other two sides, then the Pythagorean theorem states that a^2 + b^2 = c^2.

Therefore, to determine if three positive numbers form a Pythagorean triple, you need to check if the sum of the squares of the two smaller numbers is equal to the square of the largest number. For example, if you have three numbers 3, 4, and 5, you can check if they form a Pythagorean triple by computing 3^2 + 4^2 = 9 + 16 = 25, which is equal to 5^2. Since the equation holds, the numbers 3, 4, and 5 form a Pythagorean triple.

Hope this helped. Have a great day.

Let us discuss the specific solution for the given initial value problem.

To solve the initial value problem Y'' + 2Y' + 3Y = H(t - 4), y(0) = y'(0) = 0, follow these steps:

Step 1: Identify the homogeneous part of the equation and find the complementary solution.
The homogeneous part is Y'' + 2Y' + 3Y = 0. To find the complementary solution, solve the characteristic equation: r^2 + 2r + 3 = 0. This equation has complex roots r = -1 ± √2i. Therefore, the complementary solution is Yc(t) = e^(-t)(C1*cos(√2*t) + C2*sin(√2*t)).

Step 2: Find the particular solution for the non-homogeneous part of the equation.
The non-homogeneous part is H(t - 4), which is a Heaviside step function. To find the particular solution, we can use the method of undetermined coefficients. Since the right side is a step function, we can assume a particular solution of the form Yp(t) = A*H(t - 4). Differentiate Yp(t) twice and substitute the results into the given equation to find A.

Step 3: Add the complementary and particular solutions to get the general solution.
The general solution is Y(t) = Yc(t) + Yp(t).

Step 4: Apply the initial conditions y(0) = 0 and y'(0) = 0.
Substitute t = 0 into the general solution and its first derivative. Solve the resulting system of equations for C1 and C2.

Step 5: Substitute the values of C1 and C2 into the general solution.

This will give you the specific solution for the given initial value problem.

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In the expression 7x + 15, 15 is a COEFFICIENT: False.

In the expression 7x + 15, 15 is a constant.

3x + 7 means (3x + 7) DIVIDED BY 2: False.

3x + 7 means 3x plus 7.

You can rewrite 2(4 + 8) as (2)(4) + (2)(8) using the DISTRIBUTIVE PROPERTY: True.

In Mathematics, the distributive property of multiplication states that when the sum of two or more addends are multiplied by a particular numerical value, the same result and output would be obtained as when each addend is multiplied respectively by the same numerical value, and the products are added together.

By applying the distributive property of multiplication to left side of the equation, we have the following:

2(4 + 8) = (2)(4) + (2)(8)

2(12) = 8 + 16

24 = 24

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[tex]\textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=2.5\\ V=37.5\pi \end{cases}\implies 37.5\pi =\pi (2.5)^2 h \\\\\\ \cfrac{37.5\pi }{2.5^2 \pi }=h\implies \cfrac{37.5}{6.25}=h\implies 6=h[/tex]

Answer:

x < -3.5 or x < -7/2

Step-by-step explanation:

We can solve the inequality by solving for and isolating x:

[tex]4(-x-3)-4 > -2\\-4x-12-4 > -2\\-4x-16 > -2\\-4x > 14\\\\x < -7/2\\or\\x < -3.5[/tex]

We can check that our solution is correct by plugging in a number less than -3.5 for x like -4:

[tex]4(-(-4)-3)-4 > -2\\4(4-3)-4 > -2\\4(1)-4 > -2\\4-4 > -2\\0 > -2[/tex]

The inequality is true for any value of x less than -3.5 so the answer is correct

Picking a black card has a 1/2 chance of probability both ways.

The probability of an occurrence is a figure that represents how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place.

Hearts, clubs, spades, and diamonds make up the four suites of a normal deck of cards. 13 cards total—the ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king—make up each suit. There are two jokers, bringing the total number of cards in the deck to 54.

In a deck, there are 26 black cards. Picking any of these 26 cards had a probability of p=26/52 = 1/2 since picking any card has the same probability (1/52).

1-p=1/2 goes against probability.

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