Help me on #12 A&C, #13 a,b,&c plsss preferably step by step

To find the surface integral of the given function over the specified surface, we'll use the surface integral formula in Cartesian coordinates:

∫∫_S (2y^2 + 2^2) dS

where S is the part of the sphere x² + y² + z² = a² that is above the xy-plane.

First, let's parameterize the surface S in terms of spherical coordinates:

x = ρsinφcosθ

y = ρsinφsinθ

z = ρcosφ

where 0 ≤ φ ≤ π/2 (since we're considering the upper hemisphere) and 0 ≤ θ ≤ 2π.

Now, we need to find the expression for the surface element dS in terms of ρ, φ, and θ. The surface element is given by:

dS = |(∂r/∂φ) × (∂r/∂θ)| dφdθ

where r = (x, y, z) = (ρsinφcosθ, ρsinφsinθ, ρcosφ).

Let's calculate the partial derivatives:

∂r/∂φ = (cosφsinφcosθ, cosφsinφsinθ, -ρsinφ)

∂r/∂θ = (-ρsinφsinθ, ρsinφcosθ, 0)

Now, let's find the cross product:

(∂r/∂φ) × (∂r/∂θ) = (cosφsinφcosθ, cosφsinφsinθ, -ρsinφ) × (-ρsinφsinθ, ρsinφcosθ, 0)

= (-ρ^2sin^2φcosθ, -ρ^2sin^2φsinθ, ρcosφsinφ)

Taking the magnitude of the cross product:

|(∂r/∂φ) × (∂r/∂θ)| = √[(-ρ^2sin^2φcosθ)^2 + (-ρ^2sin^2φsinθ)^2 + (ρcosφsinφ)^2]

= √[ρ^4sin^4φ(cos^2θ + sin^2θ) + ρ^2cos^2φsin^2φ]

= √[ρ^4sin^4φ + ρ^2cos^2φsin^2φ]

= √[ρ^2sin^2φ(sin^2φ + cos^2φ)]

= ρsinφ

Now, we can rewrite the surface integral using spherical coordinates:

∫∫_S (2y^2 + 2^2) dS = ∫∫_S (2(ρsinφsinθ)^2 + 2^2) ρsinφ dφdθ

= ∫[0 to π/2]∫[0 to 2π] (2ρ^2sin^2φsin^2θ + 4) ρsinφ dφdθ

Simplifying the integrand:

∫[0 to π/2]∫[0 to 2π] (2ρ^2sin^2φsin^2θ + 4) ρsinφ dφdθ

= ∫[0 to π/2]∫[0 to 2π] (2ρ^2sin^3φsin^2θ + 4ρsinφ) dφdθ

Now, we can evaluate the double integral to find the surface integral value. However, without a specific value for 'a' in the sphere equation x² + y² + z² = a², we cannot provide a numerical result. The calculation involves solving the integral expression for a given value of a.

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On the cease of atrial systole, every ventricle contains the maximum amount of blood that it will hold. This volume of blood is referred to as the end- diastolic volume( EDV).

The end-diastolic volume (EDV) refers to the amount of blood present in each ventricle at the end of ventricular diastole, which is the relaxation phase of the cardiac cycle. During ventricular diastole, the ventricles fill with blood as the atria contract and push blood into the ventricles. This is known as atrial systole. At the end of this phase, the ventricles are maximally filled with blood, reaching their maximum capacity.

The volume of blood present in each ventricle at this point is the end-diastolic volume. It represents the preload, or the stretching of the ventricles, before they contract during ventricular systole to pump blood out of the heart. The end-diastolic volume is an important determinant of stroke volume, which is the amount of blood ejected by each ventricle with each heartbeat.

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The probability that the team wins the match is 1/9.

Here the odds against winning are given to be 8 to 1. It is also said that they lose more than they win.

The above statement indicates that out of every 8 + 1

= 9 matches, the team has a probability of losing 8 matches and winning one match

Probability

= no. of observations for the favorable event / Total number of observations

Here we need to find the probability of winning.

Let A be the event that the team wins the match.

As we can see the total number of observations is 9 matches.

Therefore, n = 9

n(A) = 1

Hence, P(A)

= n(A)/n

= 1/9

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Answer:

A. V= area of base x height 1/3

(20x15)x30x1/3= 3000cm

B.65cm

C.CTCN is the angle between

tan<TCN= TN/NC=12/5

=67.380

D. 10\sqrt(10)

e.tan<TMN=TN/MN=3

<TMN=71.57

1.  The average rate of change for P between 1992 and 2000 is  -1.025.

2.  For values of t = 1996 and t = 1992, the function has a negative average rate of change, and value of rate of change will be -2.4.

3.   For values of t = 2000 and t = 1996, the function has a positive average rate of change, and value of rate of change will be 0.35.

Rate is a mathematical term which we use, When two quantities are compared and expressed as a ratio and both have different units.

1.   we need to calculate the average rate of change in 1992 & 2000

    percentage of voters in 2000 = 34.5

    percentage of voters in  1996 =  33.1

    percentage of voters in 1992  = 42.7

it is known that,

Rate of change = change in percentage/change in years

Rate of change in between 1992 &1996 is,

R₁=   (33.1 - 42.7)/(1996-1992)

R₁=    -2.4  

Rate of change in between 1996 & 2000 is,

R₂=   (34.5-33.1)/(2000-1996)

R₂=    0.35

Average rate of change = (R₁ + R₂)/2

Average rate of change = (-2.4 + 0.35)/2

Average rate of change =  -1.025

2. One of the pair of values of t  for which rate of change will be negative are 1996 & 1992,

R=   (33.1 - 42.7)/(1996-1992)

R=   -2.4  

3. One of the pair of values of t  for which rate of change will be positive are 1996 & 2000,

R=   (34.5-33.1)/(2000-1996)

R=   0.35

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Answer:

The dependent variable is the time taken to run 100 metres

Step-by-step explanation:

A dependent variable is simply one that is being measured or sometimes tested in an experiment.

Now, in this case, what is being determined is the time each group of participants will take to run a 100-meter race.

Thus, the dependent variable is the time each group of participants will take to run a 100-meter race.

The best approximation of the cost for an additional square foot of living space for homes in this neighborhood is a decrease of $0.125 thousand, or $125, for every one additional square foot of living space.

Since we are looking for an approximation of the cost for an additional square foot of living space, we need to find the slope of the line of best fit for the given scatterplot. The slope of the line represents the change in the cost of the house for every one unit increase in the living space, which in this case is one square foot.

From the scatterplot, we can see that as the living space increases, the cost of the house also increases, indicating a positive linear relationship. We can also see that there is a line of best fit that closely approximates this relationship.

To find the slope of the line of best fit, we can choose any two points on the line and calculate the change in the y-coordinate (the cost) divided by the change in the x-coordinate (the living space). One way to do this is to choose the two points where the line intersects the y-axis and the x-axis. From the scatterplot, we can estimate these points to be approximately (0, 200) and (1600, 0), respectively.

Using these two points, the slope of the line of best fit is:

slope = (change in y) / (change in x)

slope = (200 - 0) / (0 - 1600)

slope = -0.125

Therefore, the best approximation of the cost for an additional square foot of living space for homes in this neighborhood is a decrease of $0.125 thousand, or $125, for every one additional square foot of living space.

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Answer:

False

Step-by-step explanation:

Equilateral triangles can only be acute.

Because a triangle can only have 180 degrees in angles, in an equilateral triangle all angles must be less than 90 degrees since they all measure the same (60 degrees)

I hope this helps!

-TheBusinessMan

Answer: False.

Step-by-step explanation:

All the angles of an equilateral triangle are 60° all less than 90° so only acute is correct.

The complete question is

"Find the general solution of the given differential equation

y''-y=0, y1(t)=e^t , y2(t)=cosht

The function \(y(t)=e^t\)  is the solution of the given differential equation.

The function y(t)=cosht is the solution of given differential equation.

The function is a type of relation, or rule, that maps one input to specific single output.

Given;

\(y_1(t) = e^t\)

Given differential equations are,

y''-y = 0

 So that,  

\(y' (t) = e^t, y'' (t) = e^t\)

Substitute values in the given differential equation.

\(e^t -e^t=0\)

Therefore, the function \(y(t)=e^t\)  is the solution of the given differential equation.

Another function;

\(y(t)=cosht\)  

So that,  

\(y"(t)=sinht\\\\y"(t)=cosht\)

Hence, function y(t)=cosht is solution of given differential equation.

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Answer: For 6 cups of gravy, Yolanda will have 36 servings of potatoes

Step-by-step explanation:

Step 1

Since  Yolanda has a  1/6-cup of ladle for one full serving

Then for every serving the 1/6- cup ladle  is used and can be expressed as

1 serving  =1/6-cup of ladle

Step 2--- Solving

Serving of potatoes for  1 cup is

1 serving  =1/6 cup of ladle

?? servings=1 cup of ladle ( cross multiply)

=  1 x 6/1 =  6 servings

Therefore  6 cups of gravy =  6 servings x 6 = 36 servings of potatoes

Answer:

-2 5/6

Step-by-step explanation:

First, you have to change the fraction 2/3 to have a common denominator with 1/6. 2/3 x 2/2 = 4/6.

Second, now your equation is -2 1/6 + ( -4/6 ).

Third, you have to add -2 1/6 and -4/6 which gets you -2 5/6

Hope this helps!

Answer:

4.00

Step-by-step explanation:

Given

x:   2  ,      5   ,     8     , 12

y: 8.50, 15.25, 22.00. 31.00

Required

Determine the y intercept

First, we calculate the slope:

\(m = \frac{y_2- y_1}{x_2 - x_1}\)

Where

\((x_1,y_1) = (2,8,50)\)

\((x_2,y_2) = (12,31.00)\)

\(m = \frac{31.00 - 8.50}{12 - 2}\)

\(m = \frac{22.5}{10}\)

\(m = 2.25\)

Next, we calculate the equation:

\(y - y_1 = m(x - x_1)\)

Where:

\((x_1,y_1) = (2,8,50)\)      

\(m = 2.25\)

The equation becomes:

\(y - 8.50 = 2.25(x - 2)\)

\(y - 8.50 = 2.25x -4.5\)

Make y the subject

\(y = 2.25x -4.5+8.50\)

\(y = 2.25x +4\)

Take \(x =0\); to get the y intercept

\(y = 2.25*0 +4\)

\(y = 4\)

Hence, the y intercept is 4.00

Answer:

The answer is B

Step-by-step explanation:

Check the picture below.

Answer: B.146 feet

Step-by-step explanation:

The coordinates of the points after dilation is D'(-5/3,1),  E'(1,-2/3), F'(5/3,-1), and C'(-1,-4/3)

Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size.

Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size.

Given that the scale factor is 1/3. The coordinates of the points after dilation is,

D(-5,3) = D'(-5/3,1)

E(3,-2) = E'(1,-2/3)

F(5,-3) = F'(5/3,-1)

C(-3,-4) = C'(-1,-4/3)

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Answer:

uuuuuuuuu

Step-by-step explauuuuuuuuuuu

The required number of plants from the function is 509 when there are 27 rows.

Given that,

The function g(x) = 18x+23 represents the total number of plants at the gardening store if there are x rows of plants in the field. What is the value of g(27) is to be determined.

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

Here,
Number of plants,
g(x) = 18x+23
Since the number of rows is 27
Now, the number of plants,
g(27) = 27 * 18 + 23
g(27) = 486 + 23
g(27) = 509

Thus, The required number of plants from the function is 509 when there are 27 rows.

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Answer:

\(x=5/2\)

Step-by-step explanation:

When \(f(x)=30\), we have \(3x+9x=30\).

Combining like terms on the right side, we have \(12x=30\).

Finally, dividing by \(12\) we get \(x=30/12\).

We note that \(30=6*5\) and \(12=6*2\), so \(30/12=5/2\).

So, \(\boxed{x=5/2}\) and we're done!

Answer:

85.5

Step-by-step explanation:

85 • 9 is 765

If you add 90 to 765 and then divide the sum by 10, you get 85.5.

Answer:

you have it correct sir

Step-by-step explanation:

Answer:

8 squares

Step-by-step explanation:

i think 8 squres iscorrect answer

Answer:

8 square units

Step-by-step explanation:

count the number of squares the rectangle is taking

or length times width

2x4

============================================

Explanation:

x = length of the red and purple lunch box

y = length of the blue lunch box

volume = length*width*height

The volume expression for the red and purple lunch box is x*7.5*4 = 30x

Set this equal to the stated volume 300 cubic inches, and solve for x.

30x = 300

x = 300/30

x = 10

The red and purple lunch box has length of 10 inches.

-----------------

The blue lunch box has volume of

length*width*height = y*8*3 = 24y

Set this equal to 300 and solve for y

24y = 300

y = 300/24

y = 12.5

The blue lunch box has length of 12.5 inches

Therefore the blue lunch box is longer and is longer by 12.5-10 = 2.5 inches.

Answer:

11.25 months

Step-by-step explanation:

7000-2500

=4500

4500/400

=11.25

plz mark me as brainliest.

Option E (A and C only) is the correct answer. Subject variables refer to qualities or characteristics of the participants in a study that cannot be manipulated by the experimenter, such as age, gender, personality traits, etc.

These variables are traditionally used to group participants based on those qualities or traits, and they can have an impact on the outcome of the study. However, subject variables are not considered to be independent variables, as they are not manipulated by the experimenter. Independent variables are manipulated in an experiment to observe their effect on the dependent variable. It is important for researchers to control for subject variables by either stratifying or randomizing participants to ensure that any observed differences between groups are not due to differences in the subject variables. Therefore, option A is true because subject variables cannot be manipulated by the experimenter, and option C is true because subject variables refer to qualities or characteristics of the participants themselves.

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Using concepts of Errors and Measurement, we got 252cm³ will be  the maximum error in volume of sphere.

We know that volume(V) of sphere is given by  =\(\frac{4.\pi .r^{3} }{3}\),where r is the radius of the volume.

If the error in the measured value of r is denoted by then the corresponding error in the calculated value of Volume is , which can be approximated by the differentiating the volume with respect to radius.

When r = 20 and dr = 0.05, this becomes:

d(V)/dr = 4×π×r²

=>d(V)=(4×π×r²)×dr

On putting the values,

=>d(V)=4×π×(20²)×(0.05)

=>d(V)=4×3.14×400×0.05

=>d(V)=252

Hence, the maximum error is 252cm³ in calculating the volume of sphere if we use the given radius.

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Applying algebraic equations, the value of the number is: 12.

We can solve word problems like the one given above by translating the statement into an algebraic expression by assuming the unknown number is a variable. Then solve for the variable to determine its value by isolating it to one side of the equation.

Let the unknown number in the problem above be represented as x.

"5 more than twice the number" is translated as 2x + 5.

"3 times the number decreased by 7" would also be translated as 3x - 7.

The algebraic equation would be:

2x + 5 = 3x - 7

Combine like terms

2x - 3x = -5 - 7

-x = -12

Divide both sides by -1

-x/-1 = -12/-1

x = 12

The number is 12.

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