How is the logarithmic function f(x) = log4(x) related to the exponential function
g(x) = 4x?

Anna should pay £3666.67 in 2 and a half years from now to meet her obligation. It means simple interest is £3666.67.

To solve this problem, we can use the formula for simple interest:

I = Prt

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Given that Anna borrowed 20000 at 5% for half a year, we can calculate the interest as:

I = Prt = 20000 x 0.05 x 0.5 = 500

So, the total amount Anna needs to pay at maturity is 20000 + 500 = 20500.

Anna plans to pay 8000 on maturity, 2000 in 10 months, and 5000 in 16 months. Therefore, the remaining amount she needs to pay is:

20500 - 8000 - 2000 - 5000 = 5500

To find out how much she needs to pay in 2 and a half years from now, we need to first calculate the time in years:

2 and a half years = 2.5 years

The total time elapsed from the time of borrowing is:

0.5 year (at borrowing) + 2.5 years = 3 years

Using the formula for simple interest, we can calculate the principal amount that will accrue an interest of 5500 in 3 years at 5% per annum:

I = Prt

5500 = P x 0.05 x 3

P = 3666.67

Therefore, Anna should pay £3666.67 in 2 and a half years.

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Answer:

-8/5f - 15

or

-1.6f - 15

Step-by-step explanation:

-2.3f + 0.7f - 12 - 3 =

-8/5f - 15

or

-1.6f - 15

Answer:

[tex] \sf \: -1.6f - 15 [/tex]

Step-by-step explanation:

Now we have to,

→ Simplify the given expression.

The expression is,

→ -2.3f + 0.7f - 12 - 3

Let's simplify the expression,

→ -2.3f + 0.7f - 12 - 3

→ (-2.3f + 0.7f) - 12 - 3

→ (-1.6f) + (-12 - 3)

→ -1.6f + (-15)

→ -1.6f - 15

Hence, the answer is -1.6f - 15.

(1) The probability of getting tails and a 3 is the probability of following the path T-3, which is 1/2 x 1/4 = 1/8.

(ii) The probability of getting heads and an even number is the probability of following the path H-2 or H-4, which is 1/2 x 1/2 + 1/2 x 1/2 = 1/2.

(iii) The probability of not getting heads and an even number is the probability of following the paths T-2 or T-4, which is 1/2 x 1/2 + 1/2 x 1/2 = 1/2.

This is a classic river crossing puzzle. To safely get all three across take the goat across, leave the goat, take the lion across, take the cabbage across,  return with the empty boat, Take the goat across to reunite with the cabbage and lion

Here's one possible solution:

By following these steps, you will have successfully transported all three items across the river without violating the rules.

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The radius and height that maximize the volume of the cone with a slant height of 12 are both 6√2.

To maximize the volume of a cone with a slant height of 12, we need to find the dimensions (radius and height) that give the maximum volume. Use this formula to get the volume of a cone:

[tex]V = 1/3 * \pi * r^2 * h[/tex]

where r and h represents radius ad height respectively.

From the problem statement, we know that the slant height s of the cone is 12. We can use the Pythagorean theorem to relate the height h, radius r, and slant height s:

[tex]s^2 = r^2 + h^2[/tex]

Substituting s = 12, we get:

[tex]12^2 = r^2 + h^2[/tex]

[tex]144 = r^2 + h^2[/tex]

We want to maximize the volume V, so we need to express it in terms of just one variable. We can use the equation relating r, h, and s:

[tex]s^2 = r^2 + h^2[/tex]

Solving for [tex]h^2[/tex], we get:

[tex]h^2 = s^2 - r^2[/tex]

Substituting s = 12, we get:

[tex]h^2 = 144 - r^2[/tex]

Now we can express the volume V as a function of just one variable, r:

[tex]V = 1/3 * \pi * r^2 * h[/tex]

[tex]V = 1/3 * \pi * r^2 * \sqrt{(144 - r^2)[/tex]

To find the maximum volume, we need to take the derivative of V with respect to r, set it equal to zero, and solve for r:

[tex]dV/dr = 1/3 * \pi * (2r * \sqrt{(144 - r^2)} - 2r^3 / \sqrt{(144 - r^2)})[/tex]

[tex]dV/dr = \pi r / 3 * (2\sqrt{(144 - r^2)} - 2r^2 / \sqrt{(144 - r^2)})[/tex]

Setting dV/dr = 0, we get:

[tex]2\sqrt{(144 - r^2)} - 2r^2 / \sqrt{(144 - r^2)} = 0[/tex]

[tex]2(144 - r^2) - 2r^2 = 0[/tex]

[tex]288 - 4r^2 = 0[/tex]

[tex]r^2 = 72[/tex]

So the radius that maximizes the volume is [tex]r = \sqrt{(72)} = 6\sqrt{(2)[/tex]. To find the height h, we can use the equation [tex]h^2 = 144 - r^2[/tex]:

[tex]h^2 = 144 - (6\sqrt{(2)})^2 = 72[/tex]

[tex]h = \sqrt{(72)} = 6\sqrt{(2)[/tex]

Therefore, the dimensions (radius and height) that maximize the volume of the cone with a slant height of 12 are [tex]r = 6\sqrt{(2)[/tex] and [tex]h = 6\sqrt{(2)[/tex].

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The complete question is:

What are the dimensions (radius and height) of all right circular cones with a slant height of 12 that maximize the cone's volume? The cone's slant height is the distance from the outer edge of the base to the vertex.

Answer: "A 265 pizzas"

Step-by-step explanation:

STEP 1: Divide the number of pizzas by the number of days (5565/21).

This should give you the answer 265.

Answer:

12×(5+25)-2

this is the answer to the question

The simplified expression is  [tex]\frac{1-4i\sqrt3}{7}[/tex].

What is simplification?

Simplifying an expression is just another way to say solving a math problem. When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do. For example, take this expression: 4 + 6 + 5.

Here the given expression is

=> [tex]\frac{2-i\sqrt3}{2+i\sqrt3}[/tex]

Multiply and divide by [tex]2-i\sqrt3[/tex] then

=> [tex]\frac{2-i\sqrt3}{2+i\sqrt3}\times\frac{2-i\sqrt3}{2-i\sqrt3}[/tex]

=> [tex]\frac{(2-i\sqrt3)^2}{2^2-(i\sqrt3)^2}[/tex]

=> [tex]\frac{4+i^2\sqrt3^2-4i\sqrt3}{4-i^2\sqrt3^2}[/tex]

=> [tex]\frac{4-3-4i\sqrt3}{4+3}[/tex]

=> [tex]\frac{1-4i\sqrt3}{7}[/tex]

Hence the simplified expression is  [tex]\frac{1-4i\sqrt3}{7}[/tex].

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Answer:

[tex]\textsf{(C)} \quad -\dfrac{1}{4}[/tex]

Step-by-step explanation:

To differentiate an equation that contains a mixture of x and y terms, use implicit differentiation.

To find dy/dx for x² + 3y² = 8 + 2xy, differentiate each term with respect to x.

Begin by placing d/dx in front of each term of the equation:

[tex]\dfrac{\text{d}}{\text{d}x}x^2+\dfrac{\text{d}}{\text{d}x}3y^2=\dfrac{\text{d}}{\text{d}x}8+\dfrac{\text{d}}{\text{d}x}2xy[/tex]

Differentiate the terms in x only (and constant terms):

[tex]\implies 2x+\dfrac{\text{d}}{\text{d}x}3y^2=0+\dfrac{\text{d}}{\text{d}x}2xy[/tex]

Use the chain rule to differentiate terms in y only.

In practice, this means differentiate with respect to y, and place dy/dx at the end:

[tex]\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=0+\dfrac{\text{d}}{\text{d}x}2xy[/tex]

[tex]\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}}{\text{d}x}2xy[/tex]

Use the product rule to differentiate the term in x and y.

[tex]\textsf{Let}\;u(x)=2x \implies u'(x)=2[/tex]

[tex]\textsf{Let}\;v(x)=y \implies v'(x)= 1\dfrac{\text{d}y}{\text{d}x}[/tex]

Therefore:

[tex]\implies \dfrac{\text{d}}{\text{d}x}[u(x) \cdot v(x)]=u(x) \cdot v'(x)+v(x) \cdot u'(x)[/tex]

[tex]\implies \dfrac{\text{d}}{\text{d}x}2xy=2x \cdot 1 \dfrac{\text{d}y}{\text{d}x}+y\cdot 2[/tex]

[tex]\implies \dfrac{\text{d}}{\text{d}x}2xy=2x \dfrac{\text{d}y}{\text{d}x}+2y[/tex]

So the final differentiated equation is:

[tex]\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=2x \dfrac{\text{d}y}{\text{d}x}+2y[/tex]

Rearrange the resulting equation to make dy/dx the subject:

[tex]\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=2x \dfrac{\text{d}y}{\text{d}x}+2y[/tex]

[tex]\implies 6y\dfrac{\text{d}y}{\text{d}x}-2x \dfrac{\text{d}y}{\text{d}x}=-2x+2y[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}(-2x+6y)=-2x+2y[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{-2x+2y}{-2x+6y}[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{2(-x+y)}{2(-x+3y)}[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{-x+y}{-x+3y}[/tex]

To find d²y/dx², differentiate again using the quotient rule and implicit differentiation.

[tex]\textsf{Let}\;u(x)=-x+y \implies u'(x)=-1+1\dfrac{\text{d}y}{\text{d}x}[/tex]

[tex]\textsf{Let}\;v(x)=-x+3y \implies v'(x)=-1+3\dfrac{\text{d}y}{\text{d}x}[/tex]

Therefore:

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\text{d}}{\text{d}x}\left[\dfrac{u(x)}{v(x)}\right]=\dfrac{v(x)\cdot u'(x)-u(x) \cdot v'(x)}{[v(x)]^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\text{d}}{\text{d}x}\left[\dfrac{-x+y}{-x+3y}\right]=\dfrac{(-x+3y) \cdot \left(-1+1\dfrac{\text{d}y}{\text{d}x}\right)-(-x+y) \cdot \left(-1+3\dfrac{\text{d}y}{\text{d}x}\right)}{(-x+3y)^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{(3y-x)\left(\dfrac{\text{d}y}{\text{d}x}-1\right)-(y-x)\left(3\dfrac{\text{d}y}{\text{d}x}-1\right)}{(-x+3y)^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\left(3y\dfrac{\text{d}y}{\text{d}x}-3y-x\dfrac{\text{d}y}{\text{d}x}+x\right)-\left(3y\dfrac{\text{d}y}{\text{d}x}-y-3x\dfrac{\text{d}y}{\text{d}x}+x\right)}{(-x+3y)^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2y+2x\dfrac{\text{d}y}{\text{d}x}}{(-x+3y)^2}[/tex]

Substitute in dy/dx:

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2y+2x\left(\dfrac{-x+y}{-x+3y}\right)}{(-x+3y)^2}[/tex]

To find d²y/dx² at (2, 2), substitute x = 2 and y = 2 into the second derivative:

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2(2)+2(2)\left(\dfrac{-2+2}{-2+3(2)}\right)}{(-2+3(2))^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-4+4\left(\dfrac{0}{4}\right)}{(4)^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-4}{16}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=-\dfrac{1}{4}[/tex]

The situation representing independent events are ,

Drawing an ace and spade from a deck of 52 cards.

Choosing a destination by 3 randomly selected travelers.

Representing the dependent and the independent events,

Drawing an ace and then drawing a spade from a standard deck of 52 cards without replacement.

The probability of drawing a spade is not affected by whether or not an ace was drawn first,

This is independent event .

Training as a long-distance runner and winning a marathon.

The probability of winning a marathon depends on various factors such as training, skill, and the competition.

These factors are not independent.

so these events are dependent.

The destinations of 3 randomly selected travelers at an airport.

The choice of destination for each traveler is independent of the choices made by the other travelers.

so these events are independent.

Flipping a coin and getting heads twice in a row.

The probability of getting heads on the second flip depends on the outcome of the first flip.

If the first flip resulted in tails, then the probability of getting heads on the second flip is 1/2.

However, if the first flip resulted in heads, then the probability of getting heads on the second flip is still 1/2.

This events are dependent.

Therefore, the independent events are drawing an ace and spade from deck of cards and selecting destination of 3 randomly selected travelers.

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The value of x using Pythagoras Theorem on the triangle is: 27.741

Pythagoras Theorem is defined as the manner in which we find the missing lengths of a right angled triangle.

The triangle has three sides, namely the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).

Pythagoras theorem is in the form of;

a² + b² = c²

Using Pythagoras Theorem, we can find the smaller side of the triangle as:

y² = 12² - 10.5²

y = √(12² - 10.5²)

y = √33.75

y = 5.809

From the other triangle, we have:

x' = √(25² - 12²)

x' = √481

x' = 21.932

Thus:

x = 21.932 + 5.809

x = 27.741

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Answer:

Step-by-step explanation:

2 triangles on either side = 4*3/2 = 6 * 2 = 12

rectangle on the back = 4*8 = 32

rectangle on top = 5*8 = 40

rectangle on bottom = 3*8 = 24

12 + 32 + 40 + 24 = 108 cm

Hope this is correct

Tell me if im wrong

The values of the lengths of the sides and segments obtained using the equivalent ratios of the sides of similar triangles are;

1. AA

2. AA

3. 200 ft

4. 21 feet

5. 37.5 meters

6. 4.2 meters

7. 6 feet

Similar triangles are triangles that posses similar shape, but may have varied sizes.

1. Two angles in triangle ORS are congruent to two angles in triangle VUT, therefore, the triangles are similar by the Angle-Angle, AA similarity postulate

2. Whereby the angle ∠ABC is congruent to the angle ∠ADF, and the angle ∠ABC and ∠ADGF are corresponding angles, we get by the conversed of the corresponding angles theorem;

BC ║ DF

However, angle A is congruent to itself, by reflexive property, therefore;

Triangle ABC is similar to triangle ADF, according to the AA similarity postulate.

3. Using the proportional sides relationship between similar triangles or the definition of the tangent of an angle, we get;

50/12.5 = h/50

h = 50 × 50/12.5 = 200

The height of the building is 200 ft

4. Using similar triangles relationship, we get;

7/2 = h/6

h = (7/2) × 6 = 21

The height of the taller flagpole is 21 ft

5. Using the equivalent ratio of the sides of similar triangles, we get;

x/25 = 12/8

x = 25 × (12/8) = 37.5

The distance is 37.5 meters

6. The equivalent ratio of the sides of similar triangles indicates;

h/7 = 9/15

h = 7 × (9/15) = 4.2

The height of the brace is 4.2 meters

7. The ratios of the sides of similar triangles indicates that we get;

136/34 = h/(1 1/2)

h = 1 1/2 × (136/34) = 6

The height of the man is 6 feet

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so we have two triangles touching each other at a point on that vertical line, and the angles on either side of the vertical line are vertical angles and thus congruent, which means both triangles are similar by AA.

[tex]\cfrac{4}{2}=\cfrac{x}{3.5}\implies 2=\cfrac{x}{3.5}\implies 7=x[/tex]

If Fred and Rhoda together had 97 more apples than Tom, then the number of apples that Fred has is 105 apples.

Let the number of apples Fred had be denoted by variable "F".

We know that, Rhoda had 17 apples and Tom had 25 apples. We also know that Fred and Rhoda had 97 more apples than Tom.

We can write this information in equation as :

⇒ F + 17 = 97 + 25,

Simplifying this equation:

We get,

⇒ F + 17 = 122

Subtracting 17 from both sides:

⇒ F = 105

Therefore, Fred had a total of 105 apples.

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The given question is incomplete, the complete question is

Tom, Fred, and Rhoda combined their apples for a fruit stand. Fred and Rhoda together had 97 more apples than Tom. Rhoda had 17 apples. Tom had 25 apples. How many apples did Fred have?

Answer:

8 miles i think

Step-by-step explanation:

i hope this helps

Hence, after 12 minutes, there are 936 liters of water in the pond overall as the rate of water addition is constant at 28 liters per minute.

To answer mathematical issues involving proportional relationships between two or more quantities, one uses the unitary method. It is a technique for determining the value of one unit depending on the value of another, in other words. In several disciplines, including mathematics, physics, economics, and engineering, the unitary technique is frequently utilized. Finding the value of a single unit of a given quantity and using that value to calculate the value of a desired number of units is what this process entails.

given

The initial amount of water plus the water that has been added over time equals the total amount of water in the pond at any one time. We can apply the following formula because the rate of water addition is constant at 28 liters per minute:

W = 600 + 28T

where W is the pond's total volume of water (measured in liters), and T is the passing of time (in minutes).

T = 12 can be used in the equation to determine how much water was in the pond overall after 12 minutes:

W = 600 + 28T

W = 600 + 28(12) (12)

W = 600 + 336

W = 936

Hence, after 12 minutes, there are 936 liters of water in the pond overall as the rate of water addition is constant at 28 liters per minute.

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The complete question is:-   Writing and evaluating a function that models a real-world...

Owners of a recreation area are filling a small pond with water. They are adding water at a rate of 28 liters per minute. There are 600 liters in the pond to start. Let W represent the total amount of water in the pond (in liters), and let 7' represent the total number of minutes that water has been added. Write an equation relating W to T. Then use this equation to find the total amount of water after 12 minutes.

For the number of small aircraft that arrive during a 75-min period, the expected value if 10 and standard deviation is 3.17.

The expected value of the number of small aircraft that arrive during a 75-minute period is ⇒ E[X] = μ = 8t.

We need to convert the time period to hours,

So, t = 75/60 = 1.25 hours.

So, the expected number of small aircraft arrivals during a 75-minute period is ⇒ E[X] = μ = 8(1.25) = 10,

To find the standard deviation, we know that for a Poisson distribution, the standard deviation(σₓ) is the square root of mean.

So, σₓ = √μ = √8t,

Substituting t = 1.25,

We get,

⇒ σₓ = √8(1.25) = √10 ≈ 3.17,

Therefore, the expected value for number of aircraft is 10, and standard deviation is 3.17.

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The given question is incomplete, the complete question is

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α =8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t.

What are the expected value and standard deviation of the number of small aircraft that arrive during a 75-min period?

The surface area of the storage container is 1536 square feet. To find the surface area of a storage container with dimensions 24 feet long, 18 feet wide, and 8 feet high, we need to calculate the area of each of its six faces and then add them together.

The front and back face each have an area of 24 feet x 8 feet = 192 square feet.

The top and bottom faces each have an area of 18 feet x 24 feet = 432 square feet.

The left and right faces each have an area of 8 feet x 18 feet = 144 square feet.

Adding these areas together, we get:

Surface area of the container = 2(192 sq. ft.) + 2(432 sq. ft.) + 2(144 sq. ft.)

The surface area of the container = 384 sq. ft. + 864 sq. ft. + 288 sq. ft.

The surface area of the container = 1536 square feet

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For the given equation of function, the production is equal when x = 1/3 and x = 3.

A relationship between a group of inputs and one output each is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function. Typically, f(x), where x is the input, is used to represent a function. y = f is how functions are typically represented (x).

The equation of the functions are:

A(x) = 3x²

B(x) = 8x + 3

The production when the functions are equal is given by:

3x² = 8x + 3

3x² - 8x - 3 = 0

3x² - 9x + x - 3 = 0

3x (x - 3) + 1 (x - 3) = 0

(3x + 1) (x - 3) = 0

x = 1/3 and x = 3

Thus, the production is equal when x = 1/3 and x = 3.

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Answer:

The answer would be

Step-by-step explanation:

A. develop a point estimate of the population mean number of units sold per month.

I must say, I do hope this helps so good luck dude

The equation representing the least amount of money needed is 23x + 15.

What is an equation?

An equation is a statement stating the equality of two expressions containing variables and/or numbers. The attempts to logically resolve equations, which are essentially questions, have been the basic driving forces behind the development of mathematics.

Let the number of people be x.

Cost of tickets = $12x

Cost of food = $11x

Cost of large kettle corn = $15

Now, on adding all the values, we get the equation as

M = 12x + 11x + 15

M = 23x + 15

Hence, the equation representing the least amount of money needed is 23x + 15.

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[tex]\text{Yesterday} = \$9.80[/tex]

[tex]\text{Today} = \$9.71[/tex]

[tex]\% \ \text{Increase} = ((9.71 - 9.80) \div 9.80) \times 100\%[/tex]

                    [tex]= (-0.09\div9.80) \times 100%[/tex]

                    [tex]\thickapprox -0.0092 \times 100%[/tex]

                   [tex]\thickapprox -0.92\%[/tex]

                  [tex]\thickapprox \bold{-0.9 \%}[/tex]   [tex]\bold{to \ the \ nearest \ tenth.}[/tex]

Answer:

The temperature will be the same in two days

Step-by-step explanation:

We can set up two equations to represent the temperatures in Boston (B) and San Francisco (SF) after a certain number of days (d):

B = -1 + 3d

SF = 6 - (d/2)

To find the number of days when the two temperatures will be the same, we can set the equations equal to each other and solve for d:

-1 + 3d = 6 - (d/2)

Multiplying both sides by 2 to eliminate the fraction:

-2 + 6d = 12 - d

Combining like terms:

7d = 14

Dividing both sides by 7:

d = 2

Therefore, the two temperatures will be the same after 2 days.

Answer:

it is typically 22 to 7

Step-by-step explanation:

if you divide 22 by 7 you get 3.142857143...

Answer:

  22/7

Step-by-step explanation:

You want to know a ratio commonly used as an alternative to 3.14 for an approximation to pi.

The value 3.14 as an approximation of pi is obtained by simply truncating the value to 2 decimal places. That makes the approximation lower than the actual value by about 0.0507%.

Rational approximations can be found from the expression of pi as a continued fraction. Truncating that fraction at various points gives the approximations ...

  22/7 . . . . high by about 0.0402%

  333/106 . . . . low by about 0.0026%

  355/113 . . . . . high by about 0.0000085%

22/7 is the most commonly used rational approximation of pi instead of 3.14.

__

Additional comment

The continued fraction is non-repeating. It starts ...

  [tex]3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\dots}}}}}[/tex]

The graph of the exponential function f(x) = [tex]5^{x}[/tex] is an upward-sloping curve that starts at (0, 1) and gets steeper and steeper as x increases. It never touches or crosses the x-axis, but it gets arbitrarily close to it as x approaches negative infinity.

As we may raise 5 to any power, all real numbers (-∞,∞ ) are the function's domain (positive, negative, or zero).

Range: The range of the function is all positive real numbers (0, ∞), since the exponential function always gives a positive result.

Asymptote: The graph of the function approaches the x-axis as x approaches negative infinity. Therefore, the x-axis is the horizontal asymptote.

Y-intercept: The y-intercept of the function is (0, 1), since[tex]5^{0}[/tex] = 1.

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Surface area of the cylinder in terms of pi, given a diameter of 3.4 inches and a height of 3 inches, is [tex]28.36\pi[/tex] square inches.

To find the surface area of a cylinder in terms of pi, you need to add the areas of the top and bottom circles (which are identical) and the area of the curved lateral surface.

The formula for the area of a circle is [tex]A = \pi r^2[/tex], where r is the radius of the circle. Since the diameter of the circle is given as 3.4 inches, we can find the radius by dividing the diameter by 2:

r = 3.4 / 2 = 1.7 inches

Using this radius, the area of one circle is:

A = [tex]\pi (1.7)^2[/tex] = [tex]9.08\pi[/tex] square inches

Since the cylinder has two identical circles, the total area of both circles is:

2A = 2([tex]9.08\pi[/tex]) = [tex]18.16\pi[/tex] square inches

The lateral surface area of a cylinder can be found by multiplying the circumference of the base circle by the height of the cylinder. The formula for the circumference of a circle is C = 2πr, so the circumference of our circle is:

C =[tex]2\pi (1.7)[/tex]= [tex]3.4\pi[/tex] inches

The height of the cylinder is given as 3 inches, so the lateral surface area is:

A = Ch = ([tex]3.4\pi[/tex])(3) = [tex]10.2\pi[/tex] square inches

To find the total surface area of the cylinder, we add the areas of the two circles and the lateral surface:

Total surface area = 2A + A = [tex]18.16\pi + 10.2\pi[/tex] = [tex]28.36\pi[/tex] square inches

Therefore, the surface area of the cylinder in terms of pi, given a diameter of 3.4 inches and a height of 3 inches, is [tex]28.36\pi[/tex] square inches.

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