The area of a triangle is 48cm^2. The height of the triangle is 6cm. Calculate the length of the base

Answer:

  (π+2)/(π-1)

Step-by-step explanation:

The sum of an infinite geometric series with first term 'a' and common ratio 'r' is given by the formula ...

  S = a/(1 -r) . . . . . for |r| < 1

The given sum can be decomposed into a constant and a series:

  = -2 +(3 +3/π +3/π² +3/π³ +...)

  = -2 +S . . . where a=3 and r=1/π in the above sum formula

Then the sum is ...

  \(-2+\dfrac{3}{1-\dfrac{1}{\pi}}=-2+\dfrac{3\pi}{\pi-1}=\dfrac{-2(\pi-1)+3\pi}{\pi-1}=\boxed{\dfrac{\pi+2}{\pi-1}}\)

__

Additional comment

There is no way to rationalize the denominator of this fraction, but the numerator can be rationalized by writing it as a mixed number:

  \(=\dfrac{3}{\pi-1}+1\)

Effectively, this is the same as we would have gotten with ...

  1 +S . . . where a=3/π and r=1/π

A recursive formula for the explicit formula include the following: B. A(n) = A(n - 1) - 6; A(1) = 8.

In Mathematics, the nth term of an arithmetic sequence can be calculated by using this expression:

aₙ =  a₁ + (n - 1)d

Where:

Next, we would determine the first term of this arithmetic sequence as follows;

A(n) = 8 + (n − 1)(-6)

A(1) = 8 + (1 − 1)(-6)

A(1) = 8 + (0)(-6)

A(1) = 8

Therefore, the recursive formula based on this explicit formula A(n) = 8 + (n − 1)(-6) is given by;

A(n) = 8 + (n − 1)(-6)

A(n) = A(n − 1) - 6

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Answer:

the first picture would be concave and the second one would be convex

Step-by-step explanation:

Well concave means to go inwards and convex means to go outwards.

Answer: x is greater than or equal to 18

Step-by-step explanation:

Answer:

x ≤ - 2

Step-by-step explanation:

- x/6 ≤ 3

- x/2 ≤ 1

- x ≤ 2

x ≤ - 2

Answer:

396 cm²

Step-by-step explanation:

This may look complex, but it's actually an easy one.

The length of the left hand side is 27 cm, while the length of the right hand side is 9 cm. Subtracting 9 from 27, we have 18 cm

Also, the breadth of the bottom is 24 cm, while the breadth of the top is 10 cm. Subtracting 10 from 24, we have 14 cm.

Now, to find the area, we divide the diagram into 2. Taking the breadth first, we have breadth of 24 cm and it's relatively small height of 9 am.

Area of a rectangle is l * b, = 24 * 9 = 216 cm

Now, we look at the remaining part of the diagram. The length with a height of 18 cm, and a breadth of 10 cm. Area of a rectangle is l * b, = 180 cm

Adding both calculations together, we have 180 + 216 = 396 cm²

The cylindrical shell is subjected to an internal pressure of 70MPa. The shell's inner radius is 10 cm, and the outer radius is 16 cm. The maximum and minimum hoop stresses in the cylindrical shell are determined below.

For an element of thickness dr at a distance r from the center, the hoop stress is given by equation i:

σθ = pdθ...[i]Where, p is the internal pressure.

The thickness of the shell is drThe circumference of the shell is 2πr.

Therefore, the force acting on the element is given by:F = σθ(2πrdr)....[ii]

Let σmax be the maximum stress in the shell. The stress at radius r = a, which is at the maximum stress, is given by:σmax = pa/b....[iii]

Here a = radius of the shell, and b = thickness of the shell.

According to equation [i], the hoop stress at radius r = a is given by:σmax = pa/b....[iii].

Substitute the given values:σmax = 70 × 10^6 × (16 - 10)/(2 × 10) = 56 × 10^6 Pa.

The minimum hoop stress in the shell occurs at the inner surface of the shell. Let σmin be the minimum stress in the shell.σmin = pi/b....[iv].

According to equation [i], the hoop stress at radius r = b is given by:σmin = pi/b....[iv]Substitute the given values:

σmin = 70 × 10^6 × 10/(2 × 10) = 35 × 10^6 Pa.

Therefore, the maximum hoop stress in the shell is 56 × 10^6 Pa and the minimum hoop stress is 35 × 10^6 Pa.

A thick cylindrical shell with an inner radius of 10 cm and an outer radius of 16 cm is subjected to an internal pressure of 70MPa. Maximum and minimum hoop stresses in the cylindrical shell can be determined using equations and the given data. σθ = pdθ is the formula for hoop stress in the cylindrical shell.

This formula calculates the hoop stress for an element of thickness dr at a distance r from the center.

For the cylindrical shell in question, the force acting on the element is F = σθ(2πrdr).

Let σmax be the maximum stress in the shell. According to equation [iii], the stress at the radius r = a, which is the maximum stress, is σmax = pa/b.σmax is calculated by substituting the given values.

The maximum hoop stress in the shell is 56 × 10^6 Pa according to this equation.

Similarly, σmin = pi/b is the formula for minimum hoop stress in the shell, which occurs at the inner surface of the shell.

The minimum hoop stress is obtained by substituting the given values into equation [iv].

The minimum hoop stress in the shell is 35 × 10^6 Pa.As a result, the maximum and minimum hoop stresses in the cylindrical shell are 56 × 10^6 Pa and 35 × 10^6 Pa, respectively.

Thus, the maximum hoop stress in the shell is 56 × 10^6 Pa and the minimum hoop stress is 35 × 10^6 Pa. These results are obtained using equations and given data.

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Answer:

12t - 1

Step-by-step explanation:

The rate at which the diameter decreases is  -0.16cm/min

Diameter:

In geometry, diameter refers the straight line that goes from one side to the other side of a circle, passing through the center.

Given,

Here we need to find if a snowball melts so that its surface area decreases at a rate of 1cm^2/min, find the rate at which the diameter decreases when the diameter is 10 cm. Finish solving the problem. Write an equation that relates the quantities.

Here  we obtained the equation that relates the surface area S and the diameter D of a sphere given by

=> S = πD²

Now, we have differentiate the equation as,

=> dS/dt = d/dt (πD²)

=> dS/dt = π d/dt (D²)

​=> dS/dt = 2πD dD/dt

Now, Solving for dD/dt, which is the unknown for this problem, we have

=> dD/dt ​= 1/2πD​ x dS​/dt

Now, substituting the given values, the rate of change of the diameter when D=10 cm and dS/dt as (-1) is

=> dD/dt = 1/2 x π x 10 (-1)

=> dD/dt = -1/20π

=> dD/dt = -0.16cm/min

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Answer:

? = -611.3

I used a calculator.

hope this helps

The graph of the inequality y ≥ (1/2)x - 1 and x - y > 1 is attached. Shannon's graph is correct.

An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.

Inequalities are used for the non equal comparison of numbers and variables.

Given the inequalities:

y ≥ (1/2)x - 1     (1)

and

x - y > 1     (2)

The graph of the inequality is attached. Shannon's graph is correct.

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Answer:

No

Step-by-step explanation:

Hope this helps :)

Answer:

send me link

Step-by-step explanation:

i need link to help

Answer:

I wrote answers to what I can see below. I'm happy to walk you through the rest of the problem if you can write out the rest in a comment, or LMK if you post a new pic/question.

Step-by-step explanation:

Step 1 answers:

What do you want to know: How many bicycles, how many unicyles?

What do you know?

A bicycle has 1 seat and 2 wheels

a unicycle has 1 seat and 1 wheel

there are a total of 18 seats and 28 wheels

Step 2 answers:

let b = number of bicycles

let u = number of unicycles

The second number in an ordered pair is the y-coordinate and corresponds to a number on the y-axis.

It follows from the task content that the second number in an ordered pair represents what coordinates as required to be determined.

Recall that an ordered pair of coordinates usually takes the form; (x, y) in which case, x which is the first number represents the x-coordinate while y, which is the second number represents the y-coordinate.

On this note, the two blanks in the given sentence to be completed are best filled with; "second number" and y-axis respectively.

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The equation √( x - 1 ) - 5 = x - 8 has 2 valid and no extraneous solution , a valid solution is x = 2 or 5.

Equation is equal to,

√( x - 1 ) - 5 = x - 8

Add 5 on both the side of the equation we get,

⇒ √( x - 1 ) - 5 + 5 = x - 8 + 5

⇒ √( x - 1 ) = x - 3

Squaring both the sides of the equation we get,

⇒ ( √x - 1 )² = ( x - 3 )²

⇒ x - 1 = x² - 6x + 9

⇒ x² - 6x - x +  9 + 1 = 0

⇒ x² -7x + 10 =0

Factorize it using splitting method to get the solution,

⇒ x² - 2x - 5x + 10 =0

⇒ x( x -2 ) - 5 ( x - 2 ) =0

⇒ ( x - 2 ) ( x - 5 ) = 0

⇒  x = 2 or x = 5

Therefore, equation has 2 valid solution and no extraneous solution.

valid solution for x = 2 or 5.

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The above question is incomplete, the complete question is :

Consider this equation.

√( x - 1 ) - 5 = x - 8

The equation has blank and blank. A valid solution for x is blank.

Answer:So the radius of the cylinder is 2.65 cm.

A cylinder can be defined as a solid figure that is bound by a curved surface and two flat surfaces. The surface area of a cylinder can be found by breaking it down into 2 parts:

1.  The two circles that make up the caps of the cylinder.

2.  The side of the cylinder, which when "unrolled" is a rectangle.

The area of each end cap can be found from the radius r of the circle, which is given by:

A = πr2

Thus the total area of the caps is 2πr2.

The area of a rectangle is given by:

A = height × width

The width is the height h of the cylinder, and the length is the distance around the end circles, or in other words the perimeter/circumference of the base/top circle and is given by:

P = 2πr

Thus the rectangle's area is rewritten as:

A = 2πr × h

Combining these parts together we will have the total surface area of a cylinder, and the final formula is given by:

A = 2πr2 + 2πrh

where:

π  is Pi, approximately 3.142

r  is the radius of the cylinder

h  height of the cylinder

By factoring 2πr from each term we can simplify the formula to:

A = 2πr(r + h)

The lateral surface area of a cylinder is simply given by: LSA = 2πr × h.

Example 1: Find the surface area of a cylinder with a radius of 4 cm, and a height of 3 cm.

Solution:

SA = 2 × π × r2 + 2 × π × r × h

SA = 2 × 3.14 × 42 +  2 × 3.14 × 4 × 3

SA = 6.28 × 16 + 6.28 × 12

SA = 100.48 + 75.36

SA = 175.84

Surface area = 175.84 cm2

Example 2: Find the surface area of the cylinder with a radius of 5.5cm and height of 10cm.

Solution:

The radius of cylinder = 5.5 cm.

The height of cylinder = 10 cm.

The total surface area of the cylinder is therefore:

TSA = 2πr(r+h)

TSA = 11π (5.5+10)

TSA = 170.5 π

TSA = 535.6 cm2

Example 3: Find the total surface area of a cylindrical tin of radius 17 cm and height 3 cm.

Solution:

Again as in the previous example:

TSA = 2πr(r+h)

TSA = 2π× 17(17+3)

TSA = 2π×17×20

TSA = 2136.56 cm2

Example 4: Find the surface area of the cylinder with radius of 6 cm and height of 9 cm.

Solution:

The radius of cylinder: r = 6 cm

The height of cylinder: h = 9 cm

Total surface area of cylinder is therefore:

TSA = 2πr(r + h)

TSA = 12π (6+9)

TSA = 180 π

TSA = 565.56 cm2

Example 5: Find the radius of cylinder whose lateral surface area is 150 cm2 and its height is 9 cm.

Solution:

Lateral surface area of cylinder is given by:

LSA = 2πrh

Given that:

LSA = 150cm2

h = 9cm

π is the constant and its value = 3.14

Substitute the values in the formula and find the value of r by isolating it from the equation:

LSA = 2πrh

150 = 2× π × r × 9

r = 150 / (2×9× π)

r = 2.65cm

So the radius of the cylinder is 2.65 cm.

The lateral surface area of cylinder is 62.8 square units, total surface area is 87.92 square units and 62.8 cubic units is the volume.

a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.

Let us consider the radius of the cylinder is 2 and height of cylinder is 5.

Lateral surface area = 2πrh

Plug in the values of radius and height

=2×3.14×2×5

=62.8 square units.

Total surface area TSA= 2πr(h+r)

=2×3.14×2(5+2)

=2×3.14×2×7

=87.92 square units

Volume of cylinder= πr²h

=3.14×4×5

=62.8 cubic units

Hence, the lateral surface area of cylinder is 62.8 square units, total surface area is 87.92 square units and 62.8 cubic units is the volume.

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The maximum likelihood estimator of σ^2 is the sample variance, computed as the sum of squared deviations divided by the sample size n.

To determine the maximum likelihood estimator (MLE) of σ^2 (the variance) for a sample x1, ..., xn from a normal population with mean μ and variance σ^2, we can use the likelihood function.

The likelihood function L(μ, σ^2) is defined as the joint probability density function (PDF) of the sample values, given the parameters μ and σ^2. Since the samples are assumed to be independent and identically distributed (i.i.d.), we can write the likelihood function as:

L(μ, σ^2) = f(x1; μ, σ^2) * f(x2; μ, σ^2) * ... * f(xn; μ, σ^2),

where f(xi; μ, σ^2) is the PDF of each sample value xi.

In a normal distribution, the PDF is given by:

f(xi; μ, σ^2) = (1 / √(2πσ^2)) * exp(-((xi - μ)^2) / (2σ^2)).

Taking the logarithm of the likelihood function (log-likelihood) can simplify the calculations:

log L(μ, σ^2) = log f(x1; μ, σ^2) + log f(x2; μ, σ^2) + ... + log f(xn; μ, σ^2).

Now, we maximize the log-likelihood function with respect to σ^2. To find the maximum, we take the derivative with respect to σ^2, set it equal to zero, and solve for σ^2.

d/d(σ^2) [log L(μ, σ^2)] = 0.

This derivative calculation can be quite involved, but it leads to the following MLE of σ^2:

σ^2_MLE = (1 / n) * Σ(xi - μ)^2,

where Σ(xi - μ)^2 is the sum of squared deviations of the sample values from the mean.

Therefore, the maximum likelihood estimator of σ^2 is the sample variance, computed as the sum of squared deviations divided by the sample size n.

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Based on the equation y = 0.13x + 46, the correct statement is:

The equation indicates that for every additional unit (mile) in the independent variable (flight distance), the dependent variable (ticket price) increases by the coefficient 0.13, which represents 13 cents.

Therefore, the equation suggests a linear relationship between flight distance and ticket price, with a constant increase of 13 cents per mile.

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Answer:

A.indicatint that the poem is about everyday events.

Step-by-step explanation:

This indicates that the poem is about everyday events. Thus the option A is cprrect.

The words of the poem tell us about the ordinary moring functions and way its indicated speaks about the everyday routine work that the author had to go through. Since the poem sets in moring routing it tells about the ordinary household chores and work.

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Answer:

46.4miles/hour

d=46.4t

Step-by-step explanation:

Let's say d=at, where d = distance, a = the unit rate, and t = any given time.

We're given that Emma takes 1.25 hours to drive 58 miles. Plugging that into our formula of d=at, we get:

58 = 1.25a.

So, a = 46.4.

The formula now becomes d=46.4t.

Answer:

The number of weeks and total number of books increase together at a constant rate.

As the number of weeks increases by 1, the total number of books increases by 1.

The common difference between the number of weeks and total number of books is 6.

Doubling the number of weeks also doubles the total number of books.

Step-by-step explanation:

Answer:

x = 80

Step-by-step explanation:

Vertical angles are congruent to each other.

30 + 50 = x

80 = x

Answer:

length=18cm,and width=9cm

Step-by-step explanation:

Let x = length

Then width = (1/2)x

Perimeter = 2(length) + 2(width) = 54

length + width = 27

So, x + (1/2)x = 27

(3/2)x = 27

x = 27(2/3) = 18 (1/2)x = 9

Answer: length = 18 cm and width = 9 cm

Answer:

Width:6 inches, Length:12 inches, Area(if needed):72 sqaure inches

Step-by-step explanation:

Put the width as w, we can write evering in terms of w.

Now:

Empirical, A knowledge of statistics is essential for anyone who wants to make informed decisions based on empirical evidence.

A knowledge of statistics helps us make decisions based on empirical evidence. Empirical evidence is data that has been collected through observation and experimentation, and statistics is a method for analyzing and interpreting this data. By using statistical tools and techniques, we can draw conclusions from empirical evidence and make informed decisions based on this information.

Statistics plays a crucial role in modern society, from medicine and public policy to business and finance. A knowledge of statistics enables individuals to understand and analyze data, to make informed decisions based on empirical evidence, and to communicate findings to others. Statistics helps us to quantify uncertainty and variability, to identify patterns and trends, and to make predictions based on past data. By using statistical methods, we can determine the probability of certain outcomes, assess risks and benefits, and evaluate the effectiveness of interventions or policies. For example, in medical research, statistics is used to test the effectiveness of new treatments or therapies, to determine the safety of drugs, and to identify risk factors for diseases. In public policy, statistics is used to analyze demographic trends, to assess the impact of social programs, and to inform decision-making around issues such as climate change or public health.

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The simplified expressions `[(8x+7)^(17/6)]/[(8x+7)^(5/3)]` and `(8x+7)^(-11/6)` for the expressions `(8x+7)^(5/2)/(8x+7)^(-5/3)` and `(8x+7)^(-5/2) (8x+7)^(2/3)` respectively.

To simplify the given expression, we use the following rules of exponents;
Product rule; `(a^n)(a^m) = a^(n+m)`
Quotient rule; `a^n/a^m = a^(n-m)`.Given `(8x+7)^(5/2) / (8x+7)^(-5/3)`.Using the product rule; `8x+7 = (8x+7)^(1)`(8x+7)^(5/2+1)` when multiplied `5/2 + 1`
`= (8x+7)^(12/6+5/2)` when multiplied `12/6 + 5/2`
`= (8x+7)^(17/6)`
Using the quotient rule, `(8x+7)^(-5/3) = 1 / (8x+7)^(5/3)`
The answer is, `[(8x+7)^(17/6)]/[(8x+7)^(5/3)]`
`= (8x+7)^(17/6 - 5/3)`
`= (8x+7)^(1/6)`
2. We are given the expression `(8x+7)^(-5/2) (8x+7)^(2/3)`
Here, we use the product rule; `(a^n)(a^m) = a^(n+m)`
`= (8x+7)^(-5/2 + 2/3)`
`= (8x+7)^(-15/6 + 4/6)`
`= (8x+7)^(-11/6)`Therefore, the answer is `(8x+7)^(-11/6)`.To simplify the given expressions `(8x+7)^(5/2)/(8x+7)^(-5/3)` and .`(8x+7)^(-5/2) (8x+7)^(2/3)`, we use the rules of exponents.

The product rule states that when we multiply two expressions with similar bases, we can add their exponents.

Similarly, the quotient rule states that when we divide two expressions with similar bases, we can subtract their exponents.Given the first expression `(8x+7)^(5/2)/(8x+7)^(-5/3)`, we can apply the product rule.

Thus, we write `(8x+7)^(5/2) (8x+7)^(1)` since `8x+7` is the common base.

This is equivalent to `(8x+7)^(5/2+1)` which can be simplified further to `(8x+7)^(12/6+5/2)` and then to `(8x+7)^(17/6)`.To simplify the second expression `(8x+7)^(-5/2) (8x+7)^(2/3)`, we can apply the product rule again.

Thus, we write `(8x+7)^(-5/2 + 2/3)` which is equivalent to `(8x+7)^(-15/6 + 4/6)`. We can simplify this expression to `(8x+7)^(-11/6)`.

In conclusion, we have the simplified expressions `[(8x+7)^(17/6)]/[(8x+7)^(5/3)]` and `(8x+7)^(-11/6)` for the expressions `(8x+7)^(5/2)/(8x+7)^(-5/3)` and `(8x+7)^(-5/2) (8x+7)^(2/3)` respectively.

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Answer:

n = \(\sqrt{32}\)

Step-by-step explanation:

The 2 right triangles inside the larger right triangle are similar , so corresponding sides are in proportion

\(\frac{8}{n}\) = \(\frac{n}{4}\) ( cross- multiply )

n² = 32 ( take the square root of both sides )

n = \(\sqrt{32}\)

(a) Total number of trips = 270

(b) There are 33 different visits that could make.

We have the information:

There are 18 major sea islands in the queen Elizabeth islands of Canada. There are 15 major lakes in Saskatchewan, Canada.

Let us say that A indicates the major sea Islands and B indicates the major lakes.

Taking both the cases as events, we will use multiplication principle.

Based on the information, the event A occurs in m ways and the event B will occur in n ways. So, the two events will occur in m×n ways.

(a) A × B = 18 × 15

Total number of trips = 270

(b) |A| = 18

|B| = 15

A ∩B = θ         (as we cannot visit both an island and lake)

Addition Principle: There are |A| + |B| ways to choose an element form

A ∪ B , when A and B are the finite sets with A ∩B = θ

|A ∪ B| = |A| + |B| = 18 + 15 = 33

Thus, There are 33 different visits that could make.

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The English translation of the question is

A reservoir was full at the end of May. In the month of June 3/10 of its reserves were consumed and at the end of July only half remained. What fraction of the reservoir was consumed during the month of July?

Answer:

1/5 of the Reservoir will be consumed in the month of July.    

Step-by-step explanation:

We are given

Full reservoir at the end of month May. In the month of June 3/10 of the reservoir was consumed.

Amount present in the Reservoir at the end of the month June will be = 1 -3/10

                                                                                                          = 7/10

Now this 7/10 of the Reservoir will be used for the next month which is July

It is given that at the end of July only 1/2 of the Reservoir is remained that means the amount that was left at the end of the Month June will be used for the month July. So the amount consumed in the month of July will be

    Amount consumed  =  7/10 - 1/2

                                     = 2/10 = 1/5

Therefore 1/5 of the reservoir will be consumed in the month of July.    

Answer:

In this section we will discuss the method of graphing an equation in two variables. In other words, we will sketch a picture of an equation in two variables.

Step-by-step explanation: