A pair of children shoes comes in a box that measures 10.2 cm x 15 cm x 8 cm what is the volume of a shoebox A pair of children shoes comes in a box that measures 10.2 cm x 15 cm x 8 cm what is the volume of a shoebox

[tex]A) -27a^3 b^6 + 8a^9 b^{12}\\D) 27a^3 b^6 + 8a^9 b^{12}[/tex]

Step-by-step explanation:

Here, the given expressions are:

[tex]A) -27a^3 b^6 + 8a^9 b^{12}\\= (-3)^3(a^3)(b^2)^3 + (2)^3(a^3)3(b^4)^3\\= (-3ab^2)^3 +(2a^3b^4)^3[/tex]

So, the above expression is "sum of cubes".

[tex]B) -9a^3 b^6 + a^9 b^{10}\\[/tex]

But (-9) can not be expressed as a Perfect cube root.

So, the above expression is not "sum of cubes".

[tex]C) 9a^3 b^6 + 8a^9 b^{12}\\[/tex]

But (9) can not be expressed as a Perfect cube root.

So, the above expression is not "sum of cubes".

[tex]D) 27a^3 b^6 + 8a^9 b^{12}\\\\= (3)^3a^3(b^2)^3 + (2)^3(a^3)^3(b^4)^3\\= (3ab^2)^3+ (2a^3b^4)^3[/tex]

So, the above expression is  "sum of cubes".

Answer:

D

Step-by-step explanation:

Answer:

The required scatterplot is given in attached file.

Step-by-step explanation:

From the scatterplot we see that two study variables are not linearly related. There may be some non-linear relation between the two variables.

The question asks about the relationship between canary chick begging intensity and their relative growth rate. This can be determined by creating and interpreting a scatterplot of the provided data. The relationship would be considered linear if there's a consistent rate of change between begging intensity and growth rate, and non-linear if the rate of change varies.

The question is asking if the relationship between the relative growth rate of canary chicks and the difference in begging intensity is linear or not. By plotting the data on a scatterplot, we would visualize whether there is a consistent, straight-line relationship (linear) or not (non-linear) between these two variables.

Without the actual scatterplot, I cannot definitively say if the relationship is linear or not. However, linear relationships typically involve variables moving in the same direction at a constant rate, while non-linear relationships involve variables moving at different rates or directions. Therefore, if the increase in begging intensity is consistently associated with an increase in relative growth rate (and vice versa), the relationship could be considered linear. On the other hand, if increases or decreases in begging intensity inconsistently affect the relative growth rate, the relationship would likely be non-linear.

An important part of this research is the ability to interpret scatterplots and understand the concepts of linear and non-linear relationships in biological data. Interpreting such relationships is integral in the study of animal behavior and understanding how different factors, such as parental care and chick begging, affect survival and growth in bird species like canaries.

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Final answer:

To find the solutions to the system of equations, use the substitution method. The solutions are (1/2, -3/2) and (7, 5).

Explanation:

To find the solutions to the system of equations, we can use the substitution method. First, solve one of the equations for y in terms of x. Let's solve the second equation for y:

y = x - 2

Now substitute this expression for y into the first equation:

x - 2 = 2x^2 - 8x + 5

Now we have a quadratic equation. Rearrange it into standard form:

2x^2 - 9x + 7 = 0

Next, factor the quadratic equation:

(2x - 1)(x - 7) = 0

Set each factor equal to zero and solve for x:

2x - 1 = 0, x - 7 = 0

x = 1/2, x = 7

Now substitute these values of x back into either of the original equations to find the corresponding values of y:

For x = 1/2: y = 1/2 - 2 = -3/2

For x = 7: y = 7 - 2 = 5

So the solutions to the system of equations are (1/2, -3/2) and (7, 5).

The solution for the equation is [tex]p=\frac{9}{8}[/tex]

Explanation:

The given equation is [tex]p-\frac{1}{4}=\frac{7}{8}[/tex]

We need to solve the equation.

The solution of the equation can be determined by finding the value for p.

Thus, from the equation, let us add both sides of the equation by [tex]\frac{1}{4}[/tex]

Hence, we have,

[tex]p-\frac{1}{4}+\frac{1}{4}=\frac{7}{8}+\frac{1}{4}[/tex]

Simplifying the equation, we get,

[tex]p=\frac{7}{8}+\frac{1}{4}[/tex]

Taking LCM for 4 and 8, we get,

[tex]p=\frac{7+2}{8}[/tex]

Adding the numerator, we have,

[tex]p=\frac{9}{8}[/tex]

Thus, the value of p is [tex]p=\frac{9}{8}[/tex]

Hence, the solution for the equation is [tex]p=\frac{9}{8}[/tex]

Answer:

8pi feet per second

Or, 25.1 feet per second (3 sf)

Step-by-step explanation:

C = 2pi×r

dC/dr = 2pi

dC/dt = dC/dr × dr/dt

= 2pi × 4 = 8pi feet per second

dC/dt = 25.1327412287

Answer:

40/13

The decimal form is going to be 3.076

Answer:

16 to the power of 1 over 4 equals 2 to the power of 4 to the power of 1 over 4 equals 2 to the power of 4 multiplied by 1 over 4 equals 2

Step-by-step explanation:

16 to the power of 1 over 4 equals 2 to the power of 4 to the power of 1 over 4 equals 2 to the power of 4 multiplied by 1 over 4 equals 2

(16)^1/4 = (2^4)^1/4

4 cancels 4

2^1 = 2

Answer:

Step-by-step explanation:

The answer is the first one.

[tex]16^{\frac{1}{4}}[/tex]  simplifies down to

[tex](2^4)^{\frac{1}{4}}[/tex]  The power to power rule is that you multiply the exponents together:

[tex]2^{\frac{4}{4}}[/tex]  which is [tex]2^1[/tex]  which is 2

I'm assuming that you are also working with radicals (since radicals and exponents are inverses of each other).  The way to write this is as a radical and simplify it is:

[tex]16^{\frac{1}{4}[/tex]  as a radical is

[tex]\sqrt[4]{16^1}[/tex]

To simplify, try to write the radicand (the number under the square root) so it's a number with a power that matches the index (the number in the "arm" of the radical sign.  Our index is a 4).  

16 is the same as 2⁴:

[tex]\sqrt[4]{2^4}[/tex]

The power on the 2 is a 4, which is the same as the index.  When the power matches the index, you pull out the base as a single number:

[tex]\sqrt[4]{2^4}=2[/tex]

Answer:

3.5x + 15 ≥ 55

Step-by-step explanation:

I think the question below contains the missing information.

Josh has a rewards card for a movie theater. - He receives 15 points for becoming a rewards card holder. - He earns 3.5 points for each visit to the movie theatre. - He needs at least 55 points to earn a free movie ticket. Which inequality can Josh use to determine x, the minimum number of visits he needs to earn his firs free movie ticket?

My answer:

Let x is the number of times he visits = 3.5x

Total points = Points received on becoming a member + Points received on x visits

So,

Total Points = 15 + 3.5x

We know the total points must be at least 55 for a free movie ticket.  This can be expressed as:

3.5x + 15 ≥ 55

Answer:

(1,-3)

Step-by-step explanation:

the x-axis for A is positive and the y-axis is negative. point A's X value is 1 because it is 1 point away from the origin and the value of the Y is 3 units away from the origin and it has to be negative.

23) x = [tex]\frac{-60}{9}[/tex] = -6.666.

24) x = [tex]\frac{-12}{7}[/tex] = -1.7142.

25) x = [tex]\frac{-37}{5}[/tex] = -7.4.

Step-by-step explanation:

Step 1; For [tex]\frac{x+6}{3}[/tex] = [tex]\frac{x+4}{12}[/tex], we cross multiply the denominators and get,

3 × (x + 4) = 12 × (x + 6),

3x + 12 = 12x + 72.

We take all the x terms to the LHS and keep the constants on the RHS.

3x - 12x = 72 - 12,

-9x = 60, x = [tex]\frac{-60}{9}[/tex] = -6.6666.

Step 2; For [tex]\frac{-5}{x-4}[/tex] = [tex]\frac{9}{x+12}[/tex], we cross multiply the denominators and get,

-5 × (x + 12) = 9 × (x - 4),

-5x - 60 = 9x - 36.

We take all the x terms to the LHS and keep the constants on the RHS.

-5x - 9x = -36 + 60,

-14x = 24, x = [tex]\frac{-24}{14}[/tex] = -1.7142.

Step 3; For [tex]\frac{6}{11}[/tex] = [tex]\frac{x-1}{x-8}[/tex], we cross multiply the denominators and get,

6 × (x - 8) = 11 × (x - 1),

6x - 48 = 11x - 11.

We take all the x terms to the LHS and keep the constants on the RHS.

6x - 11x = -11 + 48,

-5x = 37, x = [tex]\frac{-37}{5}[/tex] = -7.4.

Answer:

Step-by-step explanation:

We would apply the formula for

exponential decay which is expressed as

A = P(1 - r)^t

Where

A represents the population after t years.

t represents the number of years.

P represents the initial population.

r represents rate of growth.

From the information given,

P = 1,450,000

r = 0.99% = 0.99/100 = 0.0099

t = x years

Therefore, an exponential equation to represent this situation after x years is

A = 1450000(1 - 0.0099)^t

A = 1450000(0.9901)^t

Answer:

Pressure would be approximately 4.66 pounds.

Step-by-step explanation:

Given:

Volume of gas (V) = 25 cubic cm

Pressure of the  gas (P) = 11 pounds

We need to find the pressure when volume is 59 cubic cm.

Solution:

Now Given:

[tex]V[/tex] ∝ [tex]\frac{1}{P}[/tex]

so we can say that;

[tex]V =\frac kP[/tex]

where k is a constant.

When V = 25 cubic cm, P =11 pounds.

[tex]25 = \frac{k}{11}\\\\k= 25\times 11 = 275\ cm^3.pounds[/tex]

So the equation becomes as.

[tex]V = \frac{275}{P}[/tex]

Now we need to find the pressure when Volume is 59 cubic cm.

[tex]59 =\frac{275}{P}\\\\P=\frac{275}{59}\\\\P\approx 4.66\ pounds[/tex]

Hence Pressure would be approximately 4.66 pounds.

Answer:

Explanation:

The question is incomplete. The complete question is:

A bag contains 6 red jelly beans, 4 green jelly beans, and 4 blue jelly beans.

If we choose a jelly bean, then another jelly bean without putting the first one back in the bag, what is the probability that the first jelly bean will be green and the second will be red?

The probability that the first jelly bean will be green is the number of green jelly beans divided by the total number of jelly beans:

After chosing the first green jelly bean, there will be 13 jelly beans, from which 6 are red. Thus, the probability that the second jelly bean will be red is:

The probability of the joint events is the product of the two consecutive events:

The probability that the first jelly bean will be green and the second will be red is 12/91.

We start by determining the total number of jelly beans in the bag, which is:

6 red + 4 green + 4 blue = 14 jelly beans.

Step 1: Probability of the first jelly bean being green

The probability of drawing a green jelly bean first is the number of green jelly beans divided by the total number of jelly beans:

P(Green first) = 4/14 = 2/7.

Step 2: Probability of the second jelly bean being red

Once the first green jelly bean is chosen, there are now 13 jelly beans left in the bag, with 6 being red:

P(Red second | Green first) = 6/13.

Step 3: Combined probability

The combined probability of both events happening (first green, then red) is given by multiplying their individual probabilities:

P(Green first and Red second) = (2/7) * (6/13) = 12/91.

Thus, the combined probability is 12/91.

Complete question: A bag contains 6 red jelly beans, 4 green jelly beans, and 4 blue jelly beans. If we choose a jelly bean, then another jelly bean without putting the first one back in the bag, what is the probability that the first jelly bean will be green and the second will be red?

Answer:

x = 2

y = 2 +  t

z = -20 -20t

Step-by-step explanation:

First, we are going to find the equation for this parabola. We replace x = 2 in the equation of the paraboloid, thus:

[tex]z = 6-x-x^{2} -5y^{2}[/tex]

if x = 2, then

[tex]z = 6-(2)-2^{2}-5y^{2}[/tex]

[tex]z = -5y^{2}[/tex]

Now, we calculate the tangent line to this parabola at the point (2,2,-20)

The parametrization of the parabola is:

x = 2

y = t  

[tex]z = -5t^{2}[/tex]  since [tex]z = -5y^{2}[/tex]

We calculate the derivative

[tex]\frac{dx}{dt}= 0[/tex]

[tex]\frac{dy}{dt}= 1[/tex]

[tex]\frac{dz}{dt}= -10t[/tex]

we evaluate the derivative in t=2, since at the point (2,2,-20) y = 2 and y = t

Thus:

[tex]\frac{dx}{dt}= 0[/tex]

[tex]\frac{dy}{dt}= 1[/tex]

[tex]\frac{dz}{dt}= -10(2)= -20[/tex]

Then, the director vector for the tangent line is (0,1,-20)

and the parametric equation for this line is:

x = 2

y = 2 +  t

z = -20 -20t

The parametric equation of the tangent line is [tex]L(t)=(2,2+t,-20-20t)[/tex]

The equation of Paraboloid is,

                 [tex]z =6-x-x^{2} -5y^{2}[/tex]

Equation of parabola when [tex]x = 2[/tex] is,

       [tex]z=6-2-2^{2} -5y^{2} \\\\z=-5y^{2}[/tex]

The parametric equation of parabola will be,

     [tex]r(t)=(2,t,-5t^{2} )[/tex]

Now, we have to find Tangent vector to this parabola is,

    [tex]T(t)=\frac{dr(t)}{dt}=(0,1,-10t)[/tex]

We get, the point [tex](2, 2, -20)[/tex] when [tex]t=2[/tex]

The tangent vector will be,

 [tex]T(2)=(0,1,-20)[/tex]

The tangent line to this parabola at the point (2, 2, −20) will be,

     [tex]L(t)=(2,2,-20)+t(0,1,-20)\\\\L(t)=(2,2+t,-20-20t)[/tex]

Learn more about the Parametric equation here:

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Answer:

y = 3.6(sine( 6.2(x-4.2))+4.4

Step-by-step explanation:

(8.2-.6)/2 = altitude = 3.6

6.2 = Wavelength

(8.2+.6)/2 = 4.4 The "line" (the horizontal central line thingy whose name I forgot cuz it's 12:00)

4.2 = x shift

y = 3.6(sine( 6.2(x-4.2))+4.4

Answer:

12

Step-by-step explanation:

15000x = 9000x +72000

6000x = 72000

x = 12

Answer: the number of units that must be sold to break even is 12

Step-by-step explanation:

The cost function is expressed as

C(x)= 9000x +72000

The revenue function is expressed as

R(x) = 15000x

Profit = Revenue - cost

At the point of break even, the total revenue is equal to the total cost. This means that profit is zero. The expression becomes

Revenue - cost = 0

Revenue = cost

R(x) = C(x)

Therefore,

15000x = 9000x +72000

15000x - 9000x = 72000

6000x = 72000

x = 72000/6000

x = 12

Answer:

Total number of possibilities 2,095,681,645,538.

Step-by-step explanation:

The variables can be 1 to 8 characters long.

The first space can be filled by any of the 26 letters.

The remaining n places can be filled by any of the 26 letters or any of the 10 digits.

For a single character variable the number of ways to select a variable name is,

n (1 character) = 26

For two character variable the number of ways to select a variable name is,

n (2 character) = 26 × 36 = 936

For three character variable the number of ways to select a variable name is,

n (3 character) = 26 × 36 × 36 = 26 × 36² = 33,696

For four character variable the number of ways to select a variable name is,

n (4 character) = 26 × 36 × 36 × 36 = 26 × 36³ = 1,213,056

And so on.

Similarly for the eight character variable the number of ways to select a variable name is,

n (8 character) = 26 × 36 × 36... × 36 = 26 × 36⁷ = 2,037,468,266,496

Total number of possibilities 2,095,681,645,538.

Answer:

105 miles

Step-by-step explanation:

The question seeks to know the number of miles traveled by Tiera given that she received a certain amount of money in payment.

The total amount of money she received is $147. She receives $7 for every 5 miles traveled. The number of 5 miles traveled is calculated as 147/7 = 21

This means she traveled 5 miles 21 times.

Thus, the total number of miles she had traveled would be 21 * 5 = 105 miles in total

Answer:

P =  x³ − 9x² + 36x − 54

Step-by-step explanation:

Complex roots come in conjugate pairs.  So if 3−3i is a zero, then 3+3i is also a zero.

P = (x − 3) (x − (3−3i)) (x − (3+3i))

P = (x − 3) (x − 3 + 3i) (x − 3 − 3i)

P = (x − 3) ((x − 3)² − (3i)²)

P = (x − 3) ((x − 3)² + 9)

P =  (x − 3)³ + 9 (x − 3)

P =  x³ − 9x² + 27x − 27 + 9x − 27

P =  x³ − 9x² + 36x − 54

Answer: kg*m^2 / s^3

​

Answer:

Answer: kg*m^2 / s^3

Step-by-step explanation:

Answer:

A. I Only.

Step-by-step explanation:

To begin, we must first be clear that it is the median and that it is the arithmetic mean:

Median is the middle value of a sequence of ordered numbers, for example:

{4,4,4,4,4}, the median is 4 despite being the same numbers.

Now the arithmetic mean is the average value of the samples and is independent of the amplitudes of the intervals.

Then let's analyze each of our options:

I. At least one of the homes was sold for more than $ 165,000.

We know through the flushed:

X1 + X2 +. . . + X7 + (X8 = $130,000) + X9 +. . . + X15 = 15 ∗ 150,000 = $ 2,250,000

Now we will assume the lowest possible value from X1 to X8 = $ 130,000 and from X9 to X15 = X, which is what we want to calculate. That is to say:

X1 = X2 = X3 = X4 = X5 = X6 = X7 = X8 = 130 and X9 = X10 = X11 = X12 = X13 = X14 = X15 = X,

knowing that the total value must be the average of 15, which is equal to $ 2250000 , we have the following equation:

8 ∗ $ 130,000 + 7X = $ 2,250,000

Rearranging:

X = ($ 2,250,000 $ - $ 1,040,000) / 7

X = $ 172,857

Therefore the first statement is true, because at least one house was sold at $ 172,857 which is more than $ 165,000

Evaluating the second option

II. At least one of the homes was sold for more than $ 130,0000 and less than $ 150,000

As the example of the median in the previous case you could have 8 houses that were sold for $ 130,000 or less, therefore here it loses validity, statement II is false.

Evaluating the third option

III. At least one of the homes was sold for less than $ 130,000.

We know that the eighth house sold for $ 130,000, but houses 1 to 7 may also have been sold for that same price. The statement III is false.

Therefore the answer is A. I Only.

Answer:

(a) [tex]P(X < 12)=0.26[/tex]

(b) [tex]P(X=2)=0.15[/tex]

Step-by-step explanation:

Question a

This is a Poisson distribution. The average/mean, μ = 14

So, probability that fewer than 12 field mice are found on a given acre is:

[tex]P(X < 12) = e^{-14}(\frac{14^{0}}{0!} +\frac{14^{1}}{1!} + \frac{14^{2}}{2!} + \frac{14^{3}}{3!} +\frac{14^{4}}{4!} + \frac{14^{5}}{5!} +\frac{14^{6}}{6!}+\frac{14^{7}}{7!}+\frac{14^{8}}{8!} +\frac{14^{9}}{9!}+\frac{14^{10}}{10!}+\frac{14^{11}}{11!})\\ \\P(X < 12) = e^{-14}(1+14+98+457.33+1600.67+4481.87+10457.69+20915.38+36601.91+56936.31+79710.83+101450.15)\\\\P(X < 12) = 8.315*10^{-7}(312725.1248)=0.26 \\\\P(X < 12)=0.26[/tex]

Question b

This is a Binomial distribution with:

Probability of success, p = 0.26

n = 3

[tex]P(X=2)= (3C2)p^{2}(1-p)=\frac{3!}{2!(3-2)!}*(0.26^{2})*(1-0.26)\\ \\P(X=2)=3(0.0676)(0.74)=0.15\\\\P(X=2)=0.15[/tex]

Final answer:

To find the probability that fewer than 12 field mice are found on a given acre and on 2 of the next 3 acres inspected, use the cumulative distribution function (CDF) of the Poisson distribution and the binomial distribution.

Explanation:

To find the probability that fewer than 12 field mice are found on a given acre, we need to use the cumulative distribution function (CDF) of the Poisson distribution. The average number of field mice per acre is 14, so the parameter of the Poisson distribution is also 14.

(a) To find the probability that fewer than 12 field mice are found on a given acre, we calculate P(X < 12) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 11), where X is the number of field mice found on a given acre

(b) To find the probability that fewer than 12 field mice are found on 2 of the next 3 acres inspected, we calculate P(X < 12) for each acre and use the binomial distribution to determine the probability of 2 successes out of 3 trials.

Answer: $406

Step-by-step explanation:

Answer: she earned $406 last week.

Step-by-step explanation:

Last week, she worked 2 hours on Monday, 10 hours on Tuesday, and 9 hours on Wednesday. This means that the number of hours that she worked for the first three days is

2 + 10 + 9 = 21 hours

She had Thursday off, and then she worked 8 hours on Friday. Therefore, the total number of hours that she worked for the week is 21 + 8 = 29 hours.

If Nanette earns $14 per hour, then the total amount of money that Nanette earned in all last week is

29 × 14 = $406

Answer:It depends

Step-by-step explanation:

In a cyclic quadrilateral ( a quadrilateral that is inscribed in a circle),

opposite angles add up to 180 degrees. So you can form an equation and solve for x, and thus angle E.

Therefore:

(-2 + 6x) + (7x - 13) = 180

13x - 15 = 180

13x = 195

x = 15

So angle E = 5x

                 = 5 (15)

                 = 75 degrees

Answer:

Fruit smoothie: $3.4

Protein bar: $2.5

Step-by-step explanation:

Let x represent cost of fruit smoothie and y represent cost of protein bar.

We have been given that Jake buys a fruit smoothie and a protein bar for​ $5.90. We can represent this information in an equation as:

[tex]x+y=5.90...(1)[/tex]

[tex]x=5.90-y...(1)[/tex]

We are also told that Kobe buys 2 fruit smoothies and 4 protein bars. He pays​ $16.80. We can represent this information in an equation as:

[tex]2x+4y=16.80...(2)[/tex]

Upon substituting equation (1) in equation (2), we will get:

[tex]2(5.90-y)+4y=16.80[/tex]

[tex]11.80-2y+4y=16.80[/tex]

[tex]2y=16.80-11.80[/tex]

[tex]2y=5[/tex]

[tex]y=\frac{5}{2}=2.5[/tex]

Therefore, each protein bar costs $2.5.

Upon substituting [tex]y=2.5[/tex] in equation (1), we will get:

[tex]x=5.90-2.5=3.4[/tex]

Therefore, each fruit smoothie costs $3.4.

Each fruit smoothie costs [tex]3.40\ dollars[/tex], and each protein bar costs [tex]2.50\ dollars[/tex].

To solve for the cost of each fruit smoothie [tex](\( x \))[/tex] and each protein bar [tex](\( y \))[/tex], we'll use the given system of equations:

1. [tex]\( x + y = 5.90 \)[/tex]

2. [tex]\( 2x + 4y = 16.80 \)[/tex]

Let's solve this step by step.

Step 1: Solve the first equation for [tex]\( x \)[/tex]

[tex]\[ x + y = 5.90 \][/tex]

[tex]\[ x = 5.90 - y \][/tex]

Step 2: Substitute [tex]\( x = 5.90 - y \)[/tex] into the second equation:

[tex]\[ 2(5.90 - y) + 4y = 16.80 \][/tex]

[tex]\[ 11.80 - 2y + 4y = 16.80 \][/tex]

[tex]\[ 2y = 16.80 - 11.80 \][/tex]

[tex]\[ 2y = 5 \][/tex]

[tex]\[ y = \frac{5}{2} \][/tex]

[tex]\[ y = 2.50 \][/tex]

Step 3: Substitute [tex]\( y = 2.50 \)[/tex] back into [tex]\( x = 5.90 - y \)[/tex]

[tex]\[ x = 5.90 - 2.50 \][/tex]

[tex]\[ x = 3.40 \][/tex]

Answer:

The answer to the question is

The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is  (-∞, 4)

Step-by-step explanation:

To apply look for the interval, we divide the ordinary differential equation by (t-4) to

y'' + [tex]\frac{3t}{t-4}[/tex] y' + [tex]\frac{4}{t-4}[/tex]y = [tex]\frac{2}{t-4}[/tex]

Using theorem 3.2.1 we have p(t) =  [tex]\frac{3t}{t-4}[/tex], q(t) =  [tex]\frac{4}{t-4}[/tex], g(t) = [tex]\frac{2}{t-4}[/tex]

Which are undefined at 4. Therefore the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution, that is where p, q and g are continuous and defined is (-∞, 4) whereby theorem 3.2.1 guarantees unique solution satisfying the initial value problem in this interval.

The existence and uniqueness theorems for ODEs determine that the longest interval where the initial value problem has a unique and twice-differentiable solution is (0, 4), avoiding discontinuities at t=0 and t=4.

The initial value problem provided is a second-order linear ordinary differential equation (ODE) of the form:

t(t-4)y"+3ty'+4y=2, with initial conditions y(3)=0 and y'(3)=-1.

To determine the longest interval in which the solution is guaranteed to be unique and twice-differentiable, we need to consider the existence and uniqueness theorems for ODE's, which are predicated on the functions of the equation being continuous over the interval considered. Here, the coefficients of y" and y' are t(t-4) and 3t respectively. The problematic points occur where the coefficient of y" is zero because it will make the equation not well-defined, which occurs at t=0 and t=4. Therefore, the longest interval around the initial condition t=3 that avoids these points is (0, 4). Within this interval, the coefficients are continuous, and hence, the conditions for the existence and uniqueness of the solution are satisfied.

Answer:

P ( E_1*E_2*E_3*E_4 ) = 0.1055

Step-by-step explanation:

Given:

- 52 cards are dealt in 1 , 2 , 3 , 4 hands.

- Events:

             E_1       Hand 1 has exactly 1 ace

             E_2       Hand 2 has exactly 1 ace

             E_3       Hand 3 has exactly 1 ace

             E_4       Hand 4 has exactly 1 ace

Find:

p =P ( E_1*E_2*E_3*E_4 )

Solution:

Multiplication rule.

- For n ε N and events E_1 , E_2 , ... , E_n:

P ( E_1*E_2*......*E_n ) = P (E_1)*P(E_2|E_1)*P(E_3|E_2*E_1)*......*(E_n|E_1*E_2...E_n-1 )

- So for these events calculate 4 probabilities:-

-  For E_1, is to choose an ace multiplied by the number of ways to choose remaining 12 cards out of 48 non-aces:

                               P ( E_1 ) = 4C1 * 48C12 / 52C13

- For E_2 | E_1 , one ace and 12 other cards have already been chosen. there are 39C13 equally likely hands. The number of different one ace hand 2 is the number of ways to choose an ace from 3 remaining multiplied by the number of ways to choose the remaining 12 from 36, we have:

                               P ( E_2 | E_1  ) = 3C1 * 36C12 / 39C13

                               P ( E_3| E_2*E_1  ) = 2C1 * 24C12 / 26C13

                               P ( E_4 | E_3*E_2*E_1  ) = 1C1*12C12 / 13C13 = 1

- So the multiplication rule for n = 4 is as follows:

     P ( E_1*E_2*E_3*E_4 ) = P (E_1)*P(E_2|E_1)*P(E_3|E_2*E_1)*P ( E_4 | E_3*E_2*E_1  ) = [ 4C1 * 48C12 / 52C13 ] * [ 3C1 * 36C12 / 39C13 ] * [ 2C1 * 24C12 / 26C13 ]

     P ( E_1*E_2*E_3*E_4 ) = [ 4!*48! / (12!)^4 ] / [ 52! / (13!)^4 ]

     P ( E_1*E_2*E_3*E_4 ) = [ 4!*13^4 / (52*51*50*49) ]

    P ( E_1*E_2*E_3*E_4 ) = 0.1055

The probability that each hand in a deck of 52 cards gets exactly one ace is approximately 10.5%.

To determine the probability that each hand in a randomly divided deck of 52 cards has exactly one ace, we use the concept of conditional probability.
Let's find it step by step

Step 1 : consider the event E1 that the first hand has exactly one ace:

There are 4 aces and 52 total cards. The probabilities for drawing an ace for the first hand are affected by the decreasing number of both aces and cards.

The probability of the first hand receiving one ace is calculated as:

P(E1) = (4/52) * (48/51) * (47/50) * ... * (36/39)

Step 2 : consider the event E2 that the second hand receives exactly one ace, given that the first hand already has one:

With one ace already given to the first hand, there are 3 aces remaining and 39 cards left for the second hand.

The probability is calculated as:

P(E2|E1) = (3/39) * (35/38) * ... * (25/26)

Step 3 : Proceed similarly for the third and fourth hands:

P(E3|E1E2) = (2/26) * ... * (12/13)

P(E4|E1E2E3) = 1 (since only one ace remains for the last hand)

Step 4 : Using the multiplication rule, the overall probability P(E1E2E3E4) is calculated by multiplying the individual probabilities:

P(E1E2E3E4) = P(E1) * P(E2|E1) * P(E3|E1E2) * P(E4|E1E2E3)

Step 5 : After performing the calculations, we find:

The combined probability P(E1E2E3E4) = (4/52)*(3/39)*(2/26)(1/13) after simplifying is approximately 0.105 or 10.5%.

What is the volume of a cylinder, in cubic m, with a height of 5m and a base diameter of 20m? Round to the nearest tenths place.

Answer: 1570.8

The volume of a cylinder with a height of 5m and a base diameter of 20m is approximately 1,570.8 cubic meters when rounded to the nearest tenths place.

To find the volume of a cylinder with a height of 5m and a base diameter of 20m, we will use the formula for the volume of a cylinder: V = πr²h , where V is volume, r is the radius of the base, and h is the height of the cylinder. The radius is half of the diameter, so for a diameter of 20m, the radius is 10m. Substituting these values into the formula gives us V = (π × 10² × 5), which we can calculate as V = 3.1416 × 100 × 5 = 1,570.8 cubic meters, rounded to the nearest tenths place.