How to change 2/3 to eighteenths

Final answer:

To find the probability of the sum of two dice being greater than 4 given the second die is a 3, first identify the sample space for the first die as {1, 2, 3, 4, 5, 6}. Then, calculate the favorable outcomes where the first die, added to 3, results in a number greater than 4, which are {2, 3, 4, 5, 6}. The probability is 5/6, rounded to approximately 0.8333.

Explanation:

Sample Space and Probability Calculation

When two dice are rolled one after another, and the second die results in a 3, we consider the outcomes of the first die only. As the first die is also a fair, six-sided die with faces numbered from 1 to 6, the sample space for the first die is S = {1, 2, 3, 4, 5, 6}.

The question asks for the probability of the sum being greater than 4 given that the second die is a 3. This means we are looking for the sum to be 5 or more. We can calculate the possible outcomes where the first die, when added to 3, results in a total greater than 4.

Therefore, the outcomes in the sample space that result in a sum greater than 4 are {2, 3, 4, 5, 6}. The probability of this event, given the second die is a 3, is the number of favorable outcomes divided by the total number of possible outcomes of the first die. There are 5 favorable outcomes and 6 possible outcomes, so the probability is 5/6 or approximately 0.8333 when rounded to four decimal places.

To construct the sample space, we need to consider all possible outcomes of rolling two dice. The probability of the sum of the dots on the dice being greater than 4 given that the second die is a 3 is 5/36.

To construct the sample space, we need to consider all possible outcomes of rolling two dice. Since each die has six sides numbered 1 to 6, the sample space will consist of 36 outcomes. We can represent the outcomes as pairs of numbers, where the first number represents the result of the first die and the second number represents the result of the second die. For example, (1, 1) represents both dice landing on 1, (1, 2) represents the first die landing on 1 and the second die landing on 2, and so on.

To determine the probability of the sum of the dots on the dice being greater than 4 given that the second die is a 3, we need to identify the outcomes where the second die is 3 and the sum is greater than 4. These outcomes are (2, 3), (3, 3), (4, 3), (5, 3), and (6, 3). There are a total of 5 outcomes that satisfy these conditions. Since the sample space has 36 outcomes, the probability is 5/36. To find the probability that the sum of the dots on two dice is greater than 4 given the second die is a 3, we list the possible outcomes for the first die as {1, 2, 3, 4, 5, 6}. The favorable outcomes are those that, when added to 3, result in a number greater than 4: {2, 3, 4, 5, 6}. This results in a probability of 5/6.

Answer:

[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx = \frac{1}{2}\sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n + 1)^2(2n)!}} + C[/tex]

General Formulas and Concepts:

Calculus

Integration

Sequences

Series

Taylor Polynomials

Power Series

Integration of Power Series:

Step-by-step explanation:

*Note:  

You could derive the Taylor Series for sin(x) using Taylor polynomials differentiation but usually you have to memorize it.

We are given the integral and are trying to find the infinite series of it:

[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx[/tex]

We know that the power series for sin(x) is:

[tex]\displaystyle sin(x) = \sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n + 1)!}[/tex]

To find the power series for  [tex]\displaystyle \frac{sin(x)}{2x}[/tex], divide the power series by 2x:

[tex]\displaystyle \frac{sin(x)}{2x} = \sum^{\infty}_{n = 0} \bigg[ \frac{(-1)^nx^{2n + 1}}{(2n + 1)!} \cdot \frac{1}{2x} \bigg][/tex]

Simplifying it, we have:

[tex]\displaystyle \frac{sin(x)}{2x} = \sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n}}{2(2n + 1)!}[/tex]

Rewrite the original integral:

[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx = \int {\sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n}}{2(2n + 1)!}} \, dx[/tex]

Integrate the power series:

[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx = \sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{2(2n + 1)(2n + 1)!}} + C[/tex]

Simplify the result:

[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx = \frac{1}{2}\sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n + 1)^2(2n)!}} + C[/tex]

And we have our final answer.

Topic: AP Calculus BC (Calculus I + II)  

Unit: Power Series

The equivalent value of the expression k = ( 2F/p )

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

F = ( 1/2 ) kp

On simplifying , we get

Multiply by 2 on both sides , we get

2F = kp

Divide by p on both sides , we get

k = 2F/p

Hence , the expression is k = 2F/p

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Using the concept of similar triangles, we found that Dixon's height is 6 feet, assuming that the light source causing the shadows is consistent.

This question is about the concept of similar triangles in Mathematics. If Ariadne's shadow is 15 feet long and she is 5 feet tall, it means the ratio of her height to her shadow length is 5:15 or 1:3. If Dixon's shadow is 18 feet long, and we assume the light source creating the shadows is the same, then the same ratio can apply to him, since their shadows will be proportional to their heights. Therefore, if the ratio of Ariadne's height to her shadow length is equal to the ratio of Dixon's height to his shadow length, we can form the following equation and solve for Dixon's height: 5/15 = x/18 where 'x' is Dixon's height. Solving this equation, we find that Dixon's height is 6 feet.

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To find tan x/2, given that tan x=3 and x terminates in π < x < (3π/2), we can use the half-angle formula for tangent. The value of tan (x/2) is ±1/√2.

To find tan x/2, given that tan x=3 and x terminates in π < x < (3π/2), we can use the half-angle formula for tangent. The half-angle formula for tangent is tan(x/2) = ±√((1-cosx) / (1+cosx)). Since tan x=3, we need to find the value of cos x first.

Given that tan x = 3, we can use the fact that tan x = sin x / cos x to find the value of cos x. Rearranging the equation, we have cos x = sin x / tan x = 1 / 3. Now, we can substitute this value of cos x into the half-angle formula to find tan (x/2).

tan (x/2) = ±√((1-cos x) / (1+cos x))
tan (x/2) = ±√((1-1/3) / (1+1/3))
tan (x/2) = ±√((2/3) / (4/3))
tan (x/2) = ±√(2/4)
tan (x/2) = ±√(1/2)
tan (x/2) = ±1/√2

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Answer: Amount she earn on the sale is $42.9.

Step-by-step explanation:

Since we have given that

A furniture salesperson sells a couch for $1560.

Percentage of commission she receive on the sale of the couch = 2.75%

So, Amount she earn on the sale is given by

[tex]2.75\%\ of\ 1560\\\\=\frac{2.75}{100}\times 1560\\\\=0.0275\times 1560\\\\=\$42.9[/tex]

Hence, amount she earn on the sale is $42.9.

Final answer:

By applying the given linear equation, we can calculate the average cinema admission price for specific years, find out in which year the price was approximately $7.88, and predict when it might reach $10.04.

Explanation:

The question involves solving a linear equation to complete a table, find a specific year based on the ticket price, and predict when the ticket price will reach a certain amount. To complete these steps, we apply the equation y=0.28x+5.92, where x represents the number of years after 2003, and y gives the price in dollars.

For a, plug in the values of x (2, 5, 8) into the equation to find y.

For b, set y=7.88 and solve for x (years after 2003) by rearranging the equation.

For c, with a target price of $10.04, use the equation again to solve for x.

When x=2, y=6.48.

When x=5, y=7.32.

When x=8, y=8.16.

For a ticket price of $7.88, solve for x: x =  (7.88 - 5.92) / 0.28 = 7 years after 2003, which is 2010.

To predict when the ticket price reaches $10.04, solve for x: x =  (10.04 - 5.92) / 0.28 = 14.71, rounding to 15 years after 2003, which is 2018.

There are 1001 different choices of 10 questions.

d. Since the student can choose any 10 questions out of the 14, the number of different choices of 10 questions is given by the combination formula, which is C(14,10). Using the formula for combinations, we have:

C(14,10) = 14! / (10!(14-10)!)

= 14! / (10!4!)

= (14*13*12*11) / (4*3*2*1) = 1001

Therefore, there are 1001 different choices of 10 questions.

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To complete the partial worksheet for Bill's Company, you need to record the appropriate adjusting entries using the given data. Once the adjustments are recorded, you can transfer the balances to the adjusted trial balance column.

To complete the partial worksheet for Bill's Company, we need to record the appropriate adjusting entries using the given data. Let's go step by step:

Once you complete these steps, you will have the adjusted trial balance with the appropriate balances extended from the adjustments.

the area of the region between the curves [tex]\(y = x\) and \(y = x^2\) for \(x\) in \([-2, 1]\) is \( \frac{29}{6} \)[/tex] square units.

To find the area of the region between the curves [tex]\(y = x\) and \(y = x^2\)[/tex]for x in the interval [-2, 1], we need to set up the integral and integrate with respect to x.

First, let's graph the curves [tex]\(y = x\) and \(y = x^2\) over the interval \([-2, 1]\)[/tex] to visualize the region.

Now, let's find the points of intersection between the curves [tex]\(y = x\) and \(y = x^2\).[/tex]

Setting [tex]\(y = x\) equal to \(y = x^2\)[/tex], we get:

[tex]\[ x = x^2 \][/tex]

[tex]\[ x - x^2 = 0 \][/tex]

[tex]\[ x(1 - x) = 0 \][/tex]

This equation gives us two solutions: x = 0 and x = 1. So, the curves intersect at x = 0 and x = 1.

Now, to find the area of the region between the curves, we integrate the difference of the curves from [tex]\(x = -2\) to \(x = 0\), and from \(x = 0\) to \(x = 1\)[/tex], and then add the absolute value of these results:

[tex]\[ \text{Area} = \int_{-2}^{0} (x - x^2) \, dx + \int_{0}^{1} (x^2 - x) \, dx \][/tex]

Let's solve these integrals separately:

1. [tex]\[ \int_{-2}^{0} (x - x^2) \, dx \][/tex]

[tex]\[ = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_{-2}^{0} \][/tex]

[tex]\[ = \left[ \left(\frac{0^2}{2} - \frac{0^3}{3}\right) - \left(\frac{(-2)^2}{2} - \frac{(-2)^3}{3}\right) \right] \][/tex]

[tex]\[ = \left[ 0 - \left(\frac{4}{2} - \frac{-8}{3}\right) \right] \][/tex]

[tex]\[ = \left[ 0 - \left(2 + \frac{8}{3}\right) \right] \][/tex]

[tex]\[ = -2 - \frac{8}{3} \][/tex]

[tex]\[ = -\frac{6}{3} - \frac{8}{3} \][/tex]

[tex]\[ = -\frac{14}{3} \][/tex]

2. [tex]\[ \int_{0}^{1} (x^2 - x) \, dx \][/tex]

[tex]\[ = \left[ \frac{x^3}{3} - \frac{x^2}{2} \right]_{0}^{1} \][/tex]

[tex]\[ = \left[ \left(\frac{1^3}{3} - \frac{1^2}{2}\right) - \left(\frac{0^3}{3} - \frac{0^2}{2}\right) \right] \][/tex]

[tex]\[ = \left[ \left(\frac{1}{3} - \frac{1}{2}\right) - (0 - 0) \right] \][/tex]

[tex]\[ = \left( \frac{1}{3} - \frac{1}{2} \right) \][/tex]

[tex]\[ = \frac{1}{3} - \frac{1}{2} \][/tex]

[tex]\[ = \frac{2}{6} - \frac{3}{6} \][/tex]

[tex]\[ = -\frac{1}{6} \][/tex]

Now, we add the absolute values of these results:

[tex]\[ \text{Area} = \left| -\frac{14}{3} \right| + \left| -\frac{1}{6} \right| \]\\[/tex]

[tex]\[ \text{Area} = \frac{14}{3} + \frac{1}{6} \]\\[/tex]

[tex]\[ \text{Area} = \frac{28}{6} + \frac{1}{6} \]\\[/tex]

[tex]\[ \text{Area} = \frac{29}{6} \][/tex]

Therefore, the area of the region between the curves [tex]\(y = x\) and \(y = x^2\) for \(x\) in \([-2, 1]\) is \( \frac{29}{6} \)[/tex] square units.

The probable question maybe:

What is the area of the region between the curves [tex]\(y = x\) and \(y = x^2\)[/tex]for x in the interval [-2, 1]?

Answer:

5%

Step-by-step explanation:

to work out what percent 17 is out of 340 we can formulate an equation

so 340(x%)=17

when we solve for x we get 5

Given a Poisson distribution with 0.05 flaws/sq ft and 10 sq ft panels, each car has a 60.65% chance of no flaws.

The probability of at least 1 car with flaws is 99.35%.

Considering only 1 car with flaws, the final probability of at most 1 car with flaws is 90.2%.

Probability of at most 1 car with flaws: 90.2%

Here's how to calculate the probability that at most 1 car out of 10 has any surface flaws, given the Poisson distribution parameters:

1. Define parameters:

Mean flaws per square foot (λ) = 0.05

Area of plastic panel per car (A) = 10 square feet

Number of cars (N) = 10

2. Calculate probability of no flaws:

Probability of no flaws in one car (P(X=0)) = e^(-λA) = e^(-0.0510) ≈ 0.6065

3. Calculate probability of 1 car with flaws (complementary probability):

Probability of at least 1 car with flaws (1 - P(no flaws in all cars))

P(X ≥ 1) = 1 - (P(X=0))^N = 1 - (0.6065)^10 ≈ 0.9935

Probability of exactly 1 car with flaws (P(X=1)) = N * P(X=0) * (1-P(X=0))^N-1

≈ 10 * 0.6065 * (1 - 0.6065)^9 ≈ 0.3869

4. Final probability:

Probability of at most 1 car with flaws (P(X ≤ 1)) = P(X=0) + P(X=1) ≈ 0.6065 + 0.3869 ≈ 0.9934

The probability that at most 1 car out of 10 has any surface flaws is approximately 90.2%.

Therefore, Given a Poisson distribution with 0.05 flaws/sq ft and 10 sq ft panels, each car has a 60.65% chance of no flaws.

The probability of at least 1 car with flaws is 99.35%.

Considering only 1 car with flaws, the final probability of at most 1 car with flaws is 90.2%.

The probable question may be: The number of surface flaws in plastic panels used in the interior of automobile has a Poisson distribution with a man of 0.05 flaws per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel. If 10 cars are sold to a rental company, what is the probability that at most 1 car has any surface flaws?

a. The probability of no surface flaws in an auto's interior is approximately 0.7408.

b. The probability of none of the 10 cars having any surface flaws is approximately 0.0498.

c. The probability of at most 1 car having any surface flaws is approximately 0.9631.

a. Probability of no surface flaws:

Calculate the lambda parameter: The lambda parameter for the Poisson distribution represents the expected number of flaws, which is calculated as the mean flaws per square foot multiplied by the total area.

In this case, [tex]\lambda[/tex] = 0.03 flaws/sq ft * 10 sq ft = 0.3 flaws.

Use the Poisson probability formula: The probability of no flaws (x = 0) in a Poisson distribution is given by [tex]e^{(-\lambda)[/tex].

Plugging in lambda = 0.3, we get

P(x = 0) = [tex]e^{(-0.3)[/tex] ≈ 0.7408.

b. Probability of none of the 10 cars having flaws:

Treat each car as an independent event: Since the flaws are random and independent for each car, we can treat each car as a separate event with the same probability of no flaws (0.7408) calculated in part (a).

Calculate the combined probability: To get the probability of none of the 10 cars having flaws, we simply multiply the individual probabilities.

P(no flaws in all 10 cars) = [tex](0.7408)^{10[/tex] ≈ 0.0498.

c. Probability of at most 1 car having flaws:

Calculate probabilities for 0 and 1 flaws: We need the probabilities of 0 flaws (already calculated in part (a)) and 1 flaw (x = 1) to determine the probability of at most 1 flaw.

Probability of 1 flaw: Using the Poisson formula again,

P(x = 1) = [tex]\lambda[/tex] * [tex]e^{(-\lambda)[/tex] = 0.3 * [tex]e^{(-0.3)[/tex] ≈ 0.2222.

Probability of at most 1 flaw: This includes both scenarios with 0 and 1 flaws. P(at most 1 flaw) = P(0 flaws) + P(1 flaw) = 0.7408 + 0.2222 ≈ 0.9631.

Question:-

The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.03 flaws per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel.

a. What is the probability that there are no surface flaws in an auto's interior?

b. If 10 cars are sold to a rental company, what is the probability that none of the 10 cars has any surface flaws?

c. If 10 cars are sold to a rental company, what is the probability that at most 1 car has any surface flaws? Round your answers to four decimal places (e.g. 98.7654).

Barbara Mack owes approximately $2543.78 at the end of 4 years with 6% interest compounded continuously.

To find the total amount Barbara Mack owes after 4 years with a 6% continuously compounded interest rate, we use the formula:

A = P[tex]e^{rt}[/tex]

with P = $2000,

r = 0.06, and

t = 4

A = 2000 x [tex]e^{0.06 * 4}[/tex].

When we calculate e(0.06*4), we'll get a certain number that you then multiply by 2000 to find the total amount owed.

A= 2000 x [tex]2.71828 ^{0.24}[/tex]

A= 2543.78 ( approx)

So, Barbara Mack owes approximately $2543.78 at the end of 4 years with 6% interest compounded continuously.

Answer:

497.2 miles

Step-by-step explanation:

Great question, it is always good to ask away and get rid of any doubts that you may be having.

To begin solving this problem we first need to calculate how much gas Tyler has in his jeep after stopping at the gas station. We calculate this by multiplying the total bill by the price per gallon of gas, and then we add the amount that was left in the tank.

[tex](22/1.25)+5 = 22.6gallons[/tex]

After stopping at the gas station Tyler has 22.6 gallons of gas in his jeep. Since he gets 22 miles per gallon we multiply this by the amount of gallons in his car to calculate the distance they can travel.

[tex]\frac{22.miles}{gallon} * 22.6gallons = 497.2miles[/tex]

Tyler and his friends can travel 497.2 miles with the amount of gas they have.

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