Find the mass of the lamina that occupies the region D = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} with the density function ρ(x, y) = xye^(x+y)

Answer:

30 degrees

Step-by-step explanation:

u dot v=1*4+1*4+1*4+0*4=4+4+4+0=12

|u|=sqrt(1^2+1^2+1^2+0^2)=sqrt(3)

|v|=sqrt(4^2+4^2+4^2+4^2)=sqrt(4*4^2)=2*4=8

cos(theta)=u dot v/(|u||v|)

cos(theta)=12/(sqrt(3)*8)

cos(theta)=3/(sqrt(3)*2)

cos(theta)=sqrt(3)/2

theta=30 degrees

Answer:  0.3936

Step-by-step explanation:

Given: Mean : [tex]\mu =1387 \text{ grams}[/tex]

Standard deviation : [tex]\sigma = 161 \text{ grams}[/tex]

The formula to calculate z is given by :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x= 1,425 grams

[tex]z=\dfrac{1425-1387}{161}=0.23602484472\approx0.27[/tex]

The P Value =[tex]P(X>1425)=P(z>0.27)=1-0.6064198=0.3935802\approx0.3936[/tex]

Hence, the  probability that a randomly selected broiler weighs more than 1,425 grams =0.3936

Final Answer:

There is approximately a 40.66% chance that a randomly selected broiler weighs more than 1,425 grams.

Explanation:

To solve this problem, you will need to apply the properties of the normal distribution. We want to find out the probability that a broiler weighs more than 1,425 grams.
Given:
- Mean (μ) = 1387 grams
- Standard deviation (σ) = 161 grams
- X = 1425 grams (the value we're interested in)

Step 1: First, we compute the z-score for the weight of 1425 grams. The z-score is a measure of how many standard deviations an element is from the mean. It can be calculated using the formula:

[tex]\[ z = \frac{(X - \mu)}{\sigma} \][/tex]

where X is the value for which we're finding the probability, μ is the mean, and σ is the standard deviation.

Step 2: Insert the values into the formula to compute the z-score for 1425 grams:

[tex]\[ z = \frac{(1425 - 1387)}{161} \\\\\[ z = \frac{38}{161} \\\\\[ z \approx 0.236 \][/tex]
Step 3: Once we have the z-score, we can use the z-table (a standard normal distribution table) to find out the probability of a z-score being less than 0.236. However, since we want the probability that the broiler weighs more than 1425 grams, we are interested in the probability of a z-score being greater than 0.236.

Step 4: Look up the corresponding probability for z = 0.236 on the z-table. The z-table gives us the area under the normal curve to the left of the z-score.

Let's assume the z-table gives us a probability of P(Z < 0.236). The value would typically be around 0.5934, which means there is a 59.34% chance that a random broiler will weigh less than 1425 grams.

Step 5: To find the probability that a broiler weighs more than 1425 grams, we subtract the value found in the z-table from 1 because the total area under the curve equals 1 (which corresponds to the probability of all possible outcomes).

[tex]\[ P(Z > 0.236) = 1 - P(Z < 0.236) \\\\\[ P(Z > 0.236) = 1 - 0.5934 \\\\\[ P(Z > 0.236) \approx 0.4066 \][/tex]

The curl of the vector field F is 2i + 2j + 2k. The dot product of F and dr along the closed path C is -60√3.

To find the curl of vector field F, we need to compute the partial derivatives of its components with respect to x, y, and z. In this case, F = (z-y)i + (x-z)j + (y-x)k. Taking the partial derivatives, we get curlF = 2i + 2j + 2k.

The dot product of F and dr along the closed path C can be calculated using Stokes' Theorem. By evaluating the dot product and integrating over C, we find that F · dr = -60√3.

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Answer:

a^2bc

Step-by-step explanation:

The GCF of the expression a2b2c2 + a2bc2 - a2b2c is a2bc.

The greatest common factor (GCF) of an algebraic expression is the largest polynomial that divides each of the terms without leaving a remainder. To find the GCF of the expression a2b2c2 + a2bc2 - a2b2c, first identify the common factors in each term.

Inspecting each term we see that a2 is a common factor for all of them, and the smallest power of b and c present in all terms is b and c, respectively. Therefore, the GCF is a2bc.

Answer:

16%

Step-by-step explanation:

Eight shots were missed. Take half of eight; 4. You now have 4\25, which is 160‰ [16%].

Answer:

%16

Step-by-step explanation:

Step 1:  Find the shots missed

25 - 17 = 8

Step 2:  Find half of the shots missed

8 / 2 = 4

Step 3:  Divide 4 by 25

4/25 = 0.16

Step 4:  Convert to Percent

0.16 * 100 = %16

Answer:  %16

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Answer:

Choice A

Step-by-step explanation:

a=-6           (1st term)

ar=             (2nd term)

ar^2=         (3rd term)

ar^3           (4th term)

ar^4=         (5th term)

ar^5=-1458 (6th term)

a=-6 so -6r^5=-1458

divide both sides by -6 giving r^5=243 so to obtain r you do the fifth root of 243 which is 3.

The common ratio is 3.

so ar=6(-3)=-18 (2nd term)

Only choice A fits this.

[tex]\bf \sqrt{xy}=y\implies \left( xy \right)^{\frac{1}{2}}=y\implies \stackrel{\textit{chain rule~\hfill }}{\cfrac{1}{2}(xy)^{-\frac{1}{2}}\stackrel{\textit{product rule}}{\left(y+x\cfrac{dy}{dx} \right)}}=\cfrac{dy}{dx} \\\\\\ \cfrac{1}{2\sqrt{xy}}\left(y+x\cfrac{dy}{dx} \right)=\cfrac{dy}{dx}\implies \cfrac{y}{2\sqrt{xy}}+\cfrac{x}{2\sqrt{xy}}\cdot \cfrac{dy}{dx}=\cfrac{dy}{dx}[/tex]

[tex]\bf \cfrac{x}{2\sqrt{xy}}\cdot \cfrac{dy}{dx}=\cfrac{dy}{dx}-\cfrac{y}{2\sqrt{xy}} \implies \cfrac{x}{2\sqrt{xy}}\cdot \cfrac{dy}{dx}-\cfrac{dy}{dx}=-\cfrac{y}{2\sqrt{xy}} \\\\\\ \stackrel{\textit{common factor}}{\cfrac{dy}{dx}\left( \cfrac{x}{2\sqrt{xy}}-1 \right)}=-\cfrac{y}{2\sqrt{xy}} \implies \cfrac{dy}{dx}=-\cfrac{y}{\left( \frac{x}{2\sqrt{xy}}-1 \right)2\sqrt{xy}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \cfrac{dy}{dx}=-\cfrac{y}{x-2\sqrt{xy}}~\hfill[/tex]

Answer:

= (x+5)² = x² + 10x + 25

= (x-2)² = x² - 4x + 4

= (x² + 10x + 25) - (x² - 4x + 4)

 = x² + 10x + 25 - x² + 4x - 4

 = 14x + 21  square units

The area of the shaded region is found by subtracting the area of the inner square, (x-2)², from the area of the outer square, (x+5)², resulting in the expression 14x + 21.

The area of the shaded region in this problem represents the difference between the area of the outer square and the inner square.

To find this, we calculate the area of each square individually and then subtract one from the other.

First, the area of the outer square is (x+5)² and the area of the inner square is (x-2)².

Now, we find the difference between these two areas to isolate the shaded region:

Area of shaded region = (x+5)² - (x-2)²

To expand this, we use the binomial expansion:

Now we subtract the smaller area from the larger area:

Shaded region = (x² + 10x + 25) - (x² - 4x + 4)

Shaded region = x² + 10x + 25 - x² + 4x - 4

Shaded region = 14x + 21

This expression represents the area of the shaded region in terms of x.

Answer:

Step-by-step explanation:

1st one is x=26

2nd one is x=4

3rd is x=4

4th is x=-1

Hope that helps!

Answer:

so the answers are 26, 4, 4, and -1

Step-by-step explanation:

If you want me to solve all of them it is: Your getting x by itself

so do the opposite of each problem i'll do the first one

2x - 20 = 32

     + 20   +20

2x = 52 divide the 2

2       2

x  = 26

Hope my answer has helped you if not i'm sorry.

Answer:

b. (-3, -4) and (2, 6)

Step-by-step explanation:

By the transitive property of equality, if y equals thing 1 and y also equals thing 2, then thing1 and thing 2 are also equal.  So we will set them equal to each other and factor to solve for the 2 values of x:

[tex]2x+2=x^2+3x-4[/tex]

Get everything on one side of the equals sign, set the whole mess equal to 0, and combine like terms to get:

[tex]0=x^2+x-6[/tex]

Because this is a second degree polynomial, a quadratic to be precise, it has 2 solutions.  We need to find those 2 values of x and then use them in either one of the original equations to solve for the y values that go with each x.  

Factoring that polynomial above gives you the x values of x = -3 and 2.  Sub in -3 first:

y = 2(-3) + 2 and

y = -6 + 2 so

y = -4

Therefore, the coordinate is (-3, -4).

Onto the next x value of 2:

y = 2(2) + 2 and

y = 4 + 2 so

y = 6

Therefore, the coordinate is (2, 6)

To calculate the time required for an investment of $1000 at 3% interest compounded monthly to grow to $1500, use the compound interest formula. Solve for 't' using natural logarithms and rounding to the nearest month.

To determine how long it takes for $1000 invested at 3% interest compounded monthly to grow to $1500, we use the formula for compound interest:

Final Amount = Principal (1 + (Interest Rate / Number of Compounding Periods in a Year))^(Total Number of Compounding Periods)

Plugging in the values we have:

$1500 = $1000 (1 + 0.03/12)^(12t)

Where 't' is in years. To find 't', we need to isolate it in the equation:

1.5 = (1 + 0.03/12)^(12t)

Take the natural logarithm of both sides:

ln(1.5) = 12t * ln(1 + 0.03/12)

Then, solve for 't' by dividing both sides by 12 * ln(1 + 0.03/12), and round to the nearest month:

t = ln(1.5) / (12 * ln(1 + 0.03/12))

Question 1:

For this case we have the following expression:

[tex]\frac {\frac {y-1} {y ^ 2-36}} {\frac {1-7y} {y + 6}} =\\\frac {(7y-1) (y + 6)} {(y ^ 2-36) (1-7y)} =[/tex]

We have to:

[tex]y ^ 2-36 = (y + 6) (y-6)[/tex]

Rewriting:

[tex]\frac {(7y-1) (y + 6)} {(y + 6) (y-6) (1-7y)} =\\\frac {7y-1} {(y-6) (1-7y)} =[/tex]

We take common factor "-" in the denominator:

[tex]\frac {7y-1} {(y-6) * - (- 1 + 7y)} =\\\frac {7y-1} {- (y-6) * (7y-1)} =\\- \frac {1} {(y-6)}[/tex]

ANswer:

[tex]- \frac {1} {(y-6)}[/tex]

Question 2:

For this case we must simplify the following expression:

[tex](3n ^ 4 + 1) + (- 8n ^ 4 + 3) - (- 8n ^ 4 + 2) =[/tex]

We eliminate parentheses keeping in mind that:

[tex]+ * - = -\\- * - = +\\3n ^ 4 + 1-8n ^ 4 + 3 + 8n ^ 4-2 =[/tex]

We add similar terms:

[tex]3n ^ 4-8n ^ 4 + 8n ^ 4 + 1 + 3-2 =\\3n ^ 4 + 2[/tex]

Answer:

[tex]3n ^ 4 + 2[/tex]

F* you B*!!!!!! Your so S*! That's the easiest thing in the world!!

Answer:

V = x³ + 20x² + 124x + 240

Step-by-step explanation:

Volume of a rectangular prism is width times length times height.

V = wlh

Given w = x+4, l = x+10, and h = x + 6:

V = (x + 4)(x + 10)(x + 6)

V = (x + 4)(x² + 16x + 60)

V = x²(x + 4) + 16x(x + 4) + 60(x + 4)

V = x³ + 4x² + 16x² + 64x + 60x + 240

V = x³ + 20x² + 124x + 240

Final answer:

The volume of the rectangular aquarium is given by the function V = x²t + 6x² + 14xt + 84x + 40t + 240, representing the product of its length, width, and height with given dimensions.

Explanation:

To determine a simplified function that represents the volume of the aquarium with given dimensions, we need to use the formula for the volume of a rectangular prism, which is length × width × height. The problem provides expressions for these dimensions: length is (x + 10), width is (x + 4), and height is (t + 6).

Therefore, the volume V of the aquarium can be calculated as follows:

V = (x + 10) × (x + 4) × (t + 6)

To simplify this, we multiply the expressions:

V = (x² + 14x + 40)(t + 6)

Expanding this, we get:

V = x²t + 6x² + 14xt + 84x + 40t + 240

This is the simplified function for the volume of the aquarium in terms of x and t.

A) 3x²-5x-28. The area of the rectangle  with length 3x+7 and width x-4 can be represented as 3x²-5x-28.

The equation to find the area of ​​the rectangle is simply A = l * w. This means that the area of ​​a rectangle is equal to the product of its length (l) by its width (w), or of its length by its width.

A = w*l

A = (3x + 7)(x -4) = (3x)(x) + (3x)(-4) + (7)(x) + (7)(-4)

A = 3x² - 12x + 7x - 28

A = 3x² -5x - 28

Answer:

200

Step-by-step explanation:

Goal 57600 heaters per year

160 hr per 1 month

so 160(12)hr per 1 year

that is 1920 hr per 1 year

We also have that .15 heaters are produced every 1 hour

so multiply 1920 by .15 and you have your answer

160(12)(.15)=288 heaters are produced per one person per year

so we need to figure how many people we need by dividing year goal by what one person can do

57600/288=200 people needed

200 laborers are employed at the plant.

First find out the number of hours each worker will have to work in a year:

= Number of hours per month x 12 months

= 160 * 12

= 1,920 hours

Find out the number of units each worker will produce in those hours:

= Annual number of hours x Units per hour

= 1,920 * 0.15

= 288 heaters

The number of laborers employed is:

= Yearly demand of heaters / Number of heaters produced per worker

= 57,600 / 288

= 200 laborers

The plant employs 200 laborers.

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Answer:  The required result is 15.

Step-by-step explanation:  We are given to evaluate the following :

"6 choose 4".

Since we are to choose 4 from 6, so we have to use the combination of 6 different things chosen 4 at a time.

We know that

the formula for the combination of n different things chosen r at a time is given by

[tex]^nC_r=\dfrac{n!}{r!(n-r)!}.[/tex]

For the given situation, n = 6  and  r = 4.

Therefore, we get

[tex]^6C_4=\dfrac{6!}{4!(6-4)!}=\dfrac{6!}{4!2!}=\dfrac{6\times5\times4!}{4!\times2\times1}=15.[/tex]

Thus, the required result is 15.

Answer:

Step-by-step explanation:

Let n represent the number of 42-cent stamps Sabrina bought. Then 52-n is the number of 22-cent stamps she bought. Her total expense was ...

  0.42n +0.22(52 -n) = 16.24 . . . . total price of stamps

  0.20n + 11.44 = 16.24 . . . . . . . . . simplify

  0.20n = 4.80 . . . . . . . . . . . . . . . . subtract 11.44

  n = 24 . . . . . . . . . . . . . . . . . . . . . . divide by the coefficient of n

  52-n = 28 . . . . . . . . . . . . . . . . . . . find the number of 22-cent stamps

She bought 24 42-cent stamps and 28 22-cent stamps.

She bought 24-42 cent stamps

And, 28-22 cent stamps.

Here we assume  n be the number of 42-cent stamps

The equation should be

0.42n +0.22(52 -n) = 16.24

0.20n + 11.44 = 16.24

0.20n = 4.80

n = 24

Now

= 52 - n

= 52 - 24

= 28

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Step-by-step explanation:

[tex]3n-7\\\\\text{The difference between three times the number n and seven.}[/tex]

An expression is a set of numbers, variables, and mathematical operations. The Variable Expression 3n -7 into Verbal Expression​ can be written as expression 7 less than 3 times a number 'n'.

In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.

The expression that is given to us is 3n -7, this expression can be written as a verbal expression 7 less than 3 times a number 'n' or 7 subtracted from  3 times of number 'n'.

Hence, the Variable Expression 3n -7 into Verbal Expression​ can be written as expression 7 less than 3 times a number 'n'.

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Answer: 2.71

Step-by-step explanation:

We know that the formula to calculate the standard error is given by :-

[tex]S.E.=\dfrac{\sigma}{\sqrt{n}}[/tex], where [tex]\sigma[/tex] is the standard deviation and 'n' is the sample size.

Given : Standard deviation : [tex]\sigma=\$9[/tex]

Sample size : [tex]n=11[/tex]

Then , the standard error of the mean is given by :-

[tex]S.E.=\dfrac{9}{\sqrt{11}}=2.7136021012\approx2.71[/tex]

Hence, the standard error of the mean = 2.71

Final answer:

The standard error of the mean for the size of individual customer orders with a standard deviation of $9 and a sample size of 11 is approximately $2.71.

Explanation:

The Ransin Sports Company is looking to calculate the standard error of the mean for the size of individual customer orders. The standard error of the mean (SEM) is found by dividing the standard deviation by the square root of the sample size. Given a standard deviation of $9 and a sample size of 11 players (the soccer team), the standard error of the mean can be calculated using the formula SEM = σ / √n, where σ is the standard deviation and n is the sample size.

SEM = $9 / √11
SEM = $9 / 3.316...
SEM = approximately $2.71.

Therefore, the standard error of the mean is $2.71.

click on picture, sorry if it's hard to read, but my phone messed up the typing

The phrase 'Nine times the difference of a number and 8' is translated into the algebraic expression 9(n - 8) and simplified to 9n - 72.

The phrase 'Nine times the difference of a number and 8' translates to an algebraic expression by following specific mathematical operations. To represent an unknown number, we use a variable, such as 'n', and the phrase 'the difference of a number and 8' would be written as 'n - 8'. To adhere to the phrase 'nine times', we multiply the difference by 9, leading to the expression 9(n - 8).

When we simplify the expression, we need to distribute the 9 to both terms within the parentheses: 9 × n and 9 × (-8), which gives us 9n - 72. Thus, the simplified algebraic expression for the phrase 'Nine times the difference of a number and 8' is 9n - 72.

Final answer:

The parameter in this question refers to the population proportion. To compute a 95% confidence interval for the proportion, you can use the formula: p ± z × √(p × (1-p) / n). The sample proportion is 0.53 and the sample size is 98. By plugging these values into the formula, you can calculate the confidence interval.

Explanation:

The parameter in this question refers to the population proportion. In statistics, a parameter is a measure that describes a characteristic of a population. In this case, the parameter is the proportion of all adults living in the US who have been active in a veteran's group. To compute a 95% confidence interval for this proportion, you can use the formula:  p ± z × √(p × (1-p) / n), where p is the sample proportion, z is the z-score corresponding to the desired confidence level, and n is the sample size.

Using the provided information, the sample proportion is 52/98 = 0.53. To find the z-score for a 95% confidence level, you can use a standard normal distribution table or a calculator with the function invNorm(0.975). The z-score for a 95% confidence level is approximately 1.96. The sample size is 98. Plugging these values into the formula, you can calculate the confidence interval for the population proportion.

Confidence interval = 0.53 ± 1.96 × √(0.53 × (1-0.53) / 98) = 0.53 ± 0.0907

The parameter p is the true proportion of adults in the US who have ever been active in a veteran's group, and the 95% confidence interval for this parameter is (0.4317, 0.6295).

The formula for a 95% confidence interval for a proportion is given by:

[tex]\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \][/tex]

where z is the z-score corresponding to the desired confidence level. For a 95% confidence interval, the z-score is approximately 1.96.

Let's calculate the confidence interval:

 1. Calculate the sample proportion [tex]\( \hat{p} \)[/tex]:

[tex]\[ \hat{p} = \frac{52}{98} \approx 0.5306 \][/tex]

2. Calculate the standard error of the proportion:

[tex]\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.5306(1 - 0.5306)}{98}} \approx \sqrt{\frac{0.2503}{98}} \approx \sqrt{0.002554} \approx 0.0505 \][/tex]

3. Find the z-score for a 95% confidence interval, which is approximately 1.96.

4. Calculate the margin of error:

[tex]\[ ME = z \times SE \approx 1.96 \times 0.0505 \approx 0.0989 \][/tex]

5. Calculate the confidence interval:

[tex]\[ \text{Lower bound} = \hat{p} - ME \approx 0.5306 - 0.0989 \approx 0.4317 \] \[ \text{Upper bound} = \hat{p} + ME \approx 0.5306 + 0.0989 \approx 0.6295 \][/tex]

Therefore, the 95% confidence interval for the proportion p of all adults living in the US who have ever been active in a veteran's group is approximately (0.4317, 0.6295).

Answer:

2nd answer.

Step-by-step explanation:

see attached.

Answer with Step-by-step explanation:

We have to find the solution of the equation:

[tex]x^4+6x^2+5=0[/tex]

Let u=x²

Then, above equation is transformed to:

[tex]u^2+6u+5=0[/tex]

it could also be written as:

[tex]u^2+5u+u+5=0[/tex]

u(u+5)+1(u+5)=0

(u+1)(u+5)=0

either  u+1=0 or u+5=0

either u= -1 or u= -5

Putting u=x²

x² = -1 or x² = -5

On taking square root both sides

x= ± i  or  x= ± i√5

Hence, roots of the equation [tex]x^4+6x^2+5=0[/tex] are:

i , -i , i√5 and -i√5

Answer:

[tex] log_{a}(x) + log_{a}(y) = log_{a}(xy) [/tex]

In this high school level mathematics problem, the Product Rule of Logarithms is applied to rewrite the given expression using the appropriate property.

The property used to rewrite the expression is the Product Rule of Logarithms. According to this property, when adding two logarithms with the same base, it is equivalent to multiplying the values inside the logarithms.

So, log base 4 of 7 + log base 4 of 2 can be rewritten as log base 4 of (7*2), which simplifies to log base 4 of 14.

Answer:

The face value would be $75,000

Step-by-step explanation:

Maturity value = $76,386.99

Time = 90 days

Rate of interest = 7.5%

Let face value be 'x'

By using the formula [tex]A=P(1+\frac{RT}{100})[/tex]

                      $76,386.99 = [tex]x(1+\frac{7.5\times \frac{90}{365}}{100})[/tex]

Time in years = [tex]\frac{90}{365}[/tex]

⇒ $76,386.99 = x( 1 + 0.01849315 )

⇒ x = [tex]\frac{76,386.99}{1.01849315}[/tex]

x = $75,000

The face value would be $75,000

The alternative hypothesis would state that the proportion of Republicans who favor a bill to make gun ownership easier is not equal to the proportion of Democrats who favor the same.

The question was regarding how to construct an alternative hypothesis for a study on political beliefs and opinions on firearm ownership. In this case, the alternative hypothesis statement goes against the null hypothesis. The null hypothesis would be that there's no significant difference between the proportions of Republicans and Democrats that favor a bill making gun ownership easier. So, the alternative hypothesis can be written as: 'The proportion of Republicans (Group A) who favor a bill making it easier for someone to own a firearm is not equal to the proportion of Democrats (Group B) who favor the same.'

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The alternative hypothesis can be written as: H_A: The proportion of Republicans who favor a bill to make it easier for someone to own a firearm differs from the proportion of Democrats who favor the same.

The alternative hypothesis can be written as:

HA: The proportion of Republicans who favor a bill to make it easier for someone to own a firearm differs from the proportion of Democrats who favor the same.

Alternatively, it can be written as:

HA: pA ≠ pB

where pA is the proportion of Republicans who favor the bill and pB is the proportion of Democrats who favor the bill.

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Answer: 5\times10

Step-by-step explanation:

We know that the scientific notation is a representation of a very large or a very small number in the product of a decimal form of number (commonly between 1 and 10) and powers of ten.

Given : There are red blood cells contained in 50 cubic millimeters of blood .

The representation of 50 cubic millimeters in scientific notation is given by :-

[tex]5\times10\ \text{cubic millimeters }[/tex]

Answer: Hence, the distance covered in a straight line from the starting point is 13 miles.

Step-by-step explanation:

Since we have given that

Distance between AB = 5 miles

Distance between BC = 12 miles

We need to find the distance covered from the starting point.

We will use "Pythagorean Theorem":

[tex]H^2=P^2+B^2\\\\AC^2=AB^2+BC^2\\\\AC^2=5^2+12^2\\\\AC^2=25+144\\\\AC^2=169\\\\AC=\sqrt{169}\\\\AC=13\ miles[/tex]

Hence, the distance covered in a straight line from the starting point is 13 miles.

Triangles with height [tex]h[/tex] and base [tex]b[/tex], with [tex]b=h[/tex] have area [tex]\dfrac{b^2}2[/tex].

Such cross sections with the base of the triangle in the disk [tex]x^2+y^2\le25[/tex] (a disk with radius 5) have base with length

[tex]b(x)=\sqrt{25-x^2}-\left(-\sqrt{25-x^2}\right)=2\sqrt{25-x^2}[/tex]

i.e. the vertical (in the [tex]x,y[/tex] plane) distance between the top and bottom curves describing the circle [tex]x^2+y^2=25[/tex].

So when [tex]x=4[/tex], the cross section at that point has base

[tex]2\sqrt{25-16}=6[/tex]

so that the area of the cross section would be 6^2/2 = 18.

In case it's relevant, the entire solid would have volume given by the integral

[tex]\displaystyle\int_{-5}^5\frac{b(x)^2}2\,\mathrm dx=4\int_0^5(25-x^2)\,\mathrm dx=\frac{1000}3[/tex]

The question is about finding the volume of a solid with a circular base and equilateral triangular cross-sections, and the area of a cross section at x = 4. The base is defined by the circle equation x2 + y2 = 25 and the height and base of triangles are equal.

The question relates to the calculation of the volume of a solid object and the area of its cross section. The base of the solid is a circle defined by x2 + y2 = 25, which is a circle of radius 5. As the cross sections perpendicular to the x-axis are equal in height and base, they form equilateral triangles.

So the area A of the triangle at x = 4 is given by A = 1/2 * Base * Height. But in an equilateral triangle, the base and height are equal, so A = 1/2 * b2. From the equation of circle, the value of 'b' at x = 4 can be calculated as √(25 - 42) = 3. To get the volume we integrate the area A over the x domain of [-5,5].

https://brainly.com/question/33370732

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