Consider the differential equation y'(t)= t^2 - 5y^2 and the solution curve that passes through the point (2. - 1). What is the slope of the curve at (2. - 1)?

Answer:

Step-by-step explanation:.....

y = -t is  parametric equation of the line passing through a and parallel to b

Parametric Equation = Type of equation that employs an independent variable called a parameter (often denoted by t) and in which dependent variables are defined as continuous functions of the parameter and are not dependent on another existing variable.

⇒ parametric equation of line passing through a point (a₁, b₁, c₁)

  and parallel to a vector <a, b, c> is given by :

  x =a₁ + at ,   y = b₁ + bt   ,   z = c₁ + ct

now according to question:

given -

point, P(1, 0, -3)

line,   x = −1 + 2t , y = 2−t, and z = 3+3t.

so from the line the vector is=  <2, -1, 3>

now using above formula,

equation of line is =   x = 1 + 2t , y = −t, and z = -3+3t.

we have to solve for 'y' only,

⇒ y = -t    

read more on equations at

https://brainly.com/question/649785

#SPJ4

The particular equation of the outer ellipse is therefore: 1002x2 + 2402y2 - 2402 * 1002 = 0, the inner ellipse is therefore:502x2 + 2002y2 - 2002 * 502 = 0, Clearance= 343.29 yd and Approximate seats= 47124 seats

The plan for a new football stadium calls for the stands to be in a region defined by two concentric ellipses. The outer ellipse is to be 240 yd long and 200 yd wide. The inner ellipse is to be 200 yd long and 100 yd wide. A football field of standard dimensions, 120 yd by 160 ft, will be laid out in the center of the inner ellipse.

To find the particular equations of the two ellipses, we can make use of the following: standard form of equation of an ellipse:x2/a2 + y2/b2 = 1 The outer ellipse: Major axis = 240 yd = 480 yd/2a = 480/2 = 240 yd Minor axis = 200 yd/2b = 100 yd Now, substituting the values in the standard equation of an ellipse, we have:x2/2402 + y2/1002 = 1Multiplying both sides by 2402 * 1002, we have: 1002x2 + 2402y2 = 2402 * 1002

The particular equation of the outer ellipse is therefore: 1002x2 + 2402y2 - 2402 * 1002 = 0 The inner ellipse: Major axis = 200 yd = 400 yd/2a = 400/2 = 200 yd Minor axis = 100 yd/2b = 50 yd Now, substituting the values in the standard equation of an ellipse, we have: x2/2002 + y2/502 = 1 Multiplying both sides by 2002 * 502, we have:502x2 + 2002y2 = 2002 * 502The particular equation of the inner ellipse is therefore:502x2 + 2002y2 - 2002 * 502 = 0

The eccentricity of an ellipse is given by the formula: e = √(1 - b2/a2)For the outer ellipse: a = 240, b = 100∴ e = √(1 - 1002/2402) = √0.694 = 0.833 For the inner ellipse: a = 200, b = 50∴ e = √(1 - 502/2002) = √0.9375 = 0.968

To find the clearance between the corner of the field and the inner ellipse in the direction of the end line, we need to find the distance between the foci of the inner ellipse, which is given by: d = 2 * √(a2 - b2)where a = 200 yd and b = 50 yd. Substituting the values, we have: d = 2 * √(2002 - 502)≈ 387.29 yd The clearance between the corner of the field and the inner ellipse in the direction of the end line is approximately 387.29 - 120/2 = 343.29 yd.

The area of the stands is the area between the two ellipses. Using the formula given in the question, A = πab, where a = 240 yd/2 and b = 200 yd/2, the area is: A = π * 120 * 100 = 37,699.1 sq yd Approximate seating capacity of the stadium will be number of seats = 37,699.1 / 0.8 = 47123.88 ≈ 47124 seats.

A circle is a special case of an ellipse, where both radii are equal. In the case of a circle with radius r, the formula for the area of an ellipse is reduced to the formula for the area of a circle: A = πab = πr2 = area of a circle

To know more about ellipse, refer here:

https://brainly.com/question/19507943#

#SPJ11

(I) The proportion of times we get a red candy will approach 0.2 as the number of trials increases. (m) The probability of winning at a 'daily number' lottery game is 1/1000 implies that if we play the game repeatedly, the proportion of times we win will approach 1/1000 as the number of plays increases. (n) Saying there is a 30% chance of rain tomorrow indicates that if we observe the occurrence of rainy days over a long period. (o) If 70% of the adult American population wants to retain the penny, then randomly selecting. (p) If we take multiple random samples of 100 people from the adult American population.

(I) The long-run relative frequency definition of probability states that if we repeatedly select M&M candies at random from a large bag, the proportion of times we get a red candy will approach 0.2 as the number of trials increases.

(m) The long-run relative frequency definition of probability states that if we play the 'daily number' lottery game repeatedly, the proportion of times we win will approach 1/1000 as the number of plays increases.

(n) The long-run relative frequency definition of probability states that if we observe the occurrence of rainy days over a long period of time, the proportion of days with rain will approach 30% as the number of days observed increases.

(o) The long-run relative frequency definition of probability states that if we randomly select individuals from the population of adult Americans repeatedly, the proportion of individuals who want to retain the penny will approach 0.70 as the number of selections increases.

(p) The long-run relative frequency definition of probability states that if we take multiple random samples of 100 people from the population of adult Americans, the proportion of samples in which the sample proportion exceeds 0.80 will approach 0.015 as the number of samples increases.

To know more about proportion:

https://brainly.com/question/31010676

#SPJ4

Answer:

See below ~

Step-by-step explanation:

The maximum Number of  scholars in each row is 12. This means that the  scholars can be arranged in rows with an equal number of  scholars, and each row can have a  outside of 12  scholars.

To find the maximum number of  scholars in each row, we need to determine the  topmost common divisor( GCD) of the total number of  scholars in each section. The GCD represents the largest number that divides all the given  figures unevenly.  

Given that there are 24, 36, and 60  scholars in the three sections, we can calculate the GCD as follows    Step 1 List the  high factors of each number  24 =  23 * 31  36 =  22 * 32  60 =  22 * 31 * 51    

Step 2 Identify the common  high factors among the three  figures  Common  high factors 22 * 31    Step 3 Multiply the common  high factors to find the GCD  GCD =  22 * 31 =  4 * 3 =  12  

 thus, the maximum number of  scholars in each row is 12. This means that the  scholars can be arranged in rows with an equal number of  scholars, and each row can have a  outside of 12  scholars.

For more questions on Number .

https://brainly.com/question/26460978

#SPJ8

Answer:

-1104

Step-by-step explanation:

(−23)(48)

= (−1104)

Give me Brainllest

Answer:

3

Step-by-step explanation:

First, we can start by multiplying the equation 2x - 3y = 12 by six to see where that gets us,

2x - 3y = 12 --> 12x - 18y = 72

12x - 18y = 72

5x + 6y = 18 (other equation)

We can notice that eliminating the y term would be the easiest to go about, so we want to find the number that we can multiply 6 by to get 18. That number is 3, so the factor is 3.

You can use the fact that non of 2 times 6 doesn't include 5 as factor.

There are two possible values of the factor by which if Yumiko multiplies the first equation and add the equations, a variable is eliminated.

Those values of the factor are \(3\)   (for elimination of y) or \(-\dfrac{12}{5}\) (for elimination of x)

Our aim in doing so is to make the coefficients of same variable equal and with opposite signs so that when we add the equations, that variable gets eliminated because of coefficient becoming zero.

For example, take the given equations and the factor by which Yumiko multiplied the second equation.

5x + 6y = 18

2x – 3y = 12 => 12x - 18y = 72   (since she multiplied this with 6)

Let she multiply the first equation with factor "p"

Then we will have the system as;

\(p \times 5x + p \times 6y = p \times 18\\12x - 18y = 72\)

Adding both the equations, we get:

\((5p + 12)x + (6p - 18)y = 18p + 72\)

Since there are two variables and since any one of them can be eliminated depending on the value of p, thus we have two cases:

Then we have coefficient of x = 0

or

\(5p + 12 = 0\\5p = -12\\p = -\dfrac{12}{5}\\\)

Then we have coefficient of y = 0

or

\(6p - 18 = 0\\6p = 18\\p = \dfrac{18}{6} = 3\)

Thus, there are two possible values of the factor by which if Yumiko multiplies the first equation and add the equations, a variable is eliminated.

Those values of the factor are \(3\)  or \(-\dfrac{12}{5}\)

Learn more about system of linear equations here:

https://brainly.com/question/13722693

We can use Stokes' theorem to evaluate the line integral of the curl of a vector field F around a closed curve C, by integrating the dot product of the curl of F and the unit normal vector to the surface S that is bounded by the curve C.

Mathematically, this can be written as:

∫∫(curl F) · dS = ∫C F · dr

where dS is the differential surface element of S, and dr is the differential vector element of C.

In this problem, we are given the vector field F = (2y cos z, ex sin z, xey), and we need to evaluate the line integral of the curl of F around the hemisphere x^2 + y^2 + z^2 = 9, z ≥ 0, oriented upward.

First, we need to find the curl of F:

curl F = (∂Q/∂y - ∂P/∂z, ∂R/∂z - ∂Q/∂x, ∂P/∂x - ∂R/∂y)

where P = 2y cos z, Q = ex sin z, and R = xey. Taking partial derivatives with respect to x, y, and z, we get:

∂P/∂x = 0

∂Q/∂x = 0

∂R/∂x = ey

∂P/∂y = 2 cos z

∂Q/∂y = 0

∂R/∂y = x e^y

∂P/∂z = -2y sin z

∂Q/∂z = ex cos z

∂R/∂z = 0

Substituting these partial derivatives into the curl formula, we get:

curl F = (x e^y, 2 cos z, 2y sin z - ex cos z)

Next, we need to find the unit normal vector to the surface S that is bounded by the hemisphere x^2 + y^2 + z^2 = 9, z ≥ 0, oriented upward. Since S is a closed surface, its boundary curve C is the circle x^2 + y^2 = 9, z = 0, oriented counterclockwise when viewed from above. Therefore, the unit normal vector to S is:

n = (0, 0, 1)

Now we can apply Stokes' theorem:

∫∫(curl F) · dS = ∫C F · dr

The left-hand side is the surface integral of the curl of F over S. Since S is the hemisphere x^2 + y^2 + z^2 = 9, z ≥ 0, we can use spherical coordinates to parameterize S as:

x = 3 sin θ cos φ

y = 3 sin θ sin φ

z = 3 cos θ

0 ≤ θ ≤ π/2

0 ≤ φ ≤ 2π

The differential surface element dS is then:

dS = (∂x/∂θ x ∂x/∂φ, ∂y/∂θ x ∂y/∂φ, ∂z/∂θ x ∂z/∂φ) dθ dφ

= (9 sin θ cos φ, 9 sin θ sin φ, 9 cos θ) dθ dφ

Substituting the parameterization and the differential surface element into the surface integral, we get:

∫∫(curl F) · dS = ∫C F ·

To learn more about Stokes' theorem visit:

brainly.com/question/29751072

#SPJ11

0 x a3 the corresponding hexadecimal representation of the equation.

The correct option is D.

Numbers are represented using a radix of 16 in the positional numeral system known as hexadecimal.

Base-16 is the base of the hexadecimal number system. The use of fewer digits allows for the representation of huge quantities. A, B, C, D, E, and F are the first six alphabetic characters in this system, which consists of 16 symbols or possible digit values from 0 to 9.

What is  polynomial ?

Using just the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables, a polynomial is an expression made up of coefficients and indeterminates. The polynomial x2 4x + 7 is an illustration of one with a single indeterminate x.

To know more about hexadecimal visit:

https://brainly.com/question/13041189

#SPJ4

I understand that the question you are looking for is :

if p(x) = (x^7)+(x^5)+(x+1) what is the corresponding hexadecimal representation.

A. Oxe5

B.  Oxae

C. Oxe3

D.  Oxa3

(a) Using mathematical induction, we can prove that for any positive integer n, the sum of the cubes of the first n positive integers is equal to (n(n+1)/2)^2.(b) Similarly, by mathematical induction, we can prove that for any positive integer n, the sum of j*(2^j) for j = 1 to n is equal to (n - 1)2^n+1 + 2.(c) By applying mathematical induction, it can be shown that for any positive integer n, the sum of j*(j - 1) for j = 1 to n is equal to n(n^2 - 1)/3.

(a) To prove the statement using mathematical induction, we start by establishing the base case.

For n = 1, the left-hand side (LHS) is 1^3 = 1, and the right-hand side (RHS) is \((1(1+1)/2)^2 = (1/2)^2 = 1/4\). Since LHS = RHS, the statement holds true for n = 1.

Next, we assume that the statement is true for some positive integer k, i.e., \(sigma_j=1^k j^3 = k(k+1)/2^2\). We need to show that it holds for n = k + 1.

Using the assumption, \(sigma_j=1^k j^3 = k(k+1)/2^2\). Adding \((k+1)^3\) to both sides gives \(sigma_j=1^{(k+1)} j^3 = k(k+1)/2^2 + (k+1)^3\). Simplifying the RHS, we get \((k^3 + 3k^2 + 2k + 2) / 4\).

Rearranging the terms and factoring, the RHS becomes\(((k+1)(k+2)/2)^2\). Therefore, we have established that the statement holds for n = k + 1.

By mathematical induction, we conclude that the statement \(sigma_j=1^m j^3 = (n(n+1)/2)^2\)holds for any positive integer n.

The proofs for parts (b) and (c) are similar and can be done by following the same steps of base case verification and the induction assumption, and then deriving the result for n = k + 1.

To know more about mathematical induction refer here:

https://brainly.com/question/30577446#

#SPJ11

The estimated number of employees in 7 years, considering an 8% decrease rate annually, is approximately 361 employees.

To find the number of employees in 7 years, we need to apply the 8% decrease rate annually to the current number of employees.

Let's calculate the number of employees each year:

Year 1: 610 - 8% of 610

Year 2: (610 - 8% of 610) - 8% of (610 - 8% of 610)

Year 3: ((610 - 8% of 610) - 8% of (610 - 8% of 610)) - 8% of ((610 - 8% of 610) - 8% of (610 - 8% of 610))

Year 4: ...

Year 5: ...

Year 6: ...

Year 7: ...

Since the process of calculating each year's employee count becomes quite complex, let's use a simplified formula to find the number of employees in 7 years. The formula is:

Number of employees in 7 years = Current number of employees × \((1- Decrease rate)^{Number of years}\)

Let's plug in the values:

Number of employees in 7 years = 610 × (1 - 0.08)⁷

Now we can calculate it:

Number of employees in 7 years ≈ 610× (0.92)⁷

≈ 610× 0.5929

≈ 361.456

Therefore, the estimated number of employees in 7 years, considering an 8% decrease rate annually, is approximately 361 employees.

Learn more about approximation here:

https://brainly.com/question/29669607

#SPJ11

Answer:

A.) True

B.) True

C.) False

D.)True

Step-by-step explanation:

Answer: 208 points in total

Step-by-step explanation:

multiply 52(the amount of games) by the rate of change (which is 4)

The triple integral of the given solid is equal to (2/3) pi.

To evaluate the triple integral of a solid bounded by the cylinder and the planes in the first octant, we need to break down the integral into three separate integrals for each variable x, y, and z.

Firstly, we can determine the limits of integration for each variable by looking at the boundaries of the solid. The cylinder is defined by the equation \(x^{2}\) + \(y^{2}\) = 4, while the planes are defined by z = 0, x = 0, and y = 0.

In the first octant, we can set the limits of integration to be from 0 to 2 for x and from 0 to sqrt(4 - \(x^{2}\)) for y, as we are only considering the first quadrant of the cylinder. For z, the limits of integration are from 0 to 1, as we are only considering the solid bounded by the planes.

We can then set up the triple integral as follows:

∫∫∫ (1) dz dy dx

where (1) represents the constant function 1, as we are not given any specific function to integrate.

Using the limits of integration we determined earlier, we can simplify the integral to:

∫\(0^{2}\) ∫\(0^{\sqrt{4-x^{2} } }\) ∫\(0^{1}\) (1) dz dy dx

Evaluating this integral yields:

∫\(0^{2}\)∫\(0^{\sqrt{4-x^{2} } }\)(z)|\(0^{1}\)dy dx

which further simplifies to:

∫\(0^{2}\) (1/2)(\({\sqrt{4-x^{2} } }\)) dx

Using the substitution u = x/2, we can simplify the integral further to:

(1/4) ∫\(0^{4\sqrt{4-u^{2} } }\) du

This integral can be evaluated using the trigonometric substitution u = 2 sin(theta), which yields:

(1/2) ∫\(0^{\pi /2}\)(2 cos(θ)\()^{2}\) d(θ)

Simplifying this integral further yields:

(1/2) ∫0^pi/2 (4 \(cos^{2}\)(tθ) - 2) d(θ)

Evaluating the integral gives us:

(1/2) [(4/3) \(sin^{3}\)(θ) - 2 θ]|\(0^{\pi }\)

Finally, plugging in the limits of integration yields:

(2/3) \(\pi\)

Therefore, the triple integral of the given solid is equal to (2/3) \(\pi\).

Know more about   triple integral   here:

https://brainly.com/question/31315543

#SPJ11

Stephen makes the mistake in the expression as he uses the 4 in the root and the 3 in the power and the expression actually is: ∛(16)/e

We start with the expression:

exp( (4/3)*ln(2) - 1)

Here we can use that:

exp(ln(x)) = x.

and e^(a + b) = e^a*e^b.

the first step here is:

e^((4/3)*ln(2) - 1) = e^((4/3)*ln(2)*e^(-1)

So the first step of Stephen is correct, but the first step of Helen is not, you can not simplify the expression in that way.

now, we have that:

a*ln(x) = ln(x^a)

then we can write:

(4/3)*ln(2) = ln(2^(4/3))

and e^(ln(2^(4/3)) = 2^(4/3)

then we have:

e^((4/3)*ln(2)*e^(-1) = 2^(4/3)/e

now we can write this as:

∛(2^4)/e

Here is where Stephen makes the mistake, he uses the 4 in the root and the 3 in the power.

The expression actually is: ∛(16)/e

Learn more about expressions on:

https://brainly.com/question/723406

#SPJ1

Answer:

24 degrees

Step-by-step explanation:

90-66=24

Answer:

−0.0005 m/sec.

Step-by-step explanation:

Let us denote the volume of the can as V

let us denote the radius of the circular disc in lake as r

Let us denote the thickness of the circular disc as h

But we know the volume of the can as πr^2h then we can calculate How fast it is decreasing when the radius is 2 m

CHECK THE ATTACHMENT FOR DETAILED EXPLANATION

Answer:

-45

Step-by-step explanation:

Sum of two integers = -30

one of them = 15

let another number be 'x'

x + 15 = -30

x = -30 - 15

x = -45

so the other number is -45

Answer:

Step-by-step explanation:

7.056

7 is units place = 7 *1

0 is tenth place = 0 *(1/10)

5 is hundredth place = 5 *(1/100)

6 is thousandth place = 6 * (1/1000)

\(7.056 = 7*1+ 0*\frac{1}{10}+5*\frac{1}{100}+6*\frac{1}{1000}\)

Seven and fifty six thousandths

18.3

Eighteen and one third

\(18.3 = 1*10 + 8*1 + 3*\frac{1}{10}\)

918.0201

Nine hundred eighteen and two hundred  one ten thousandths

\(918.0201=9*100 + 1*10 + 8*1 +0 +2*\frac{1}{100}+0+1*\frac{1}{10000}\)

Based on the given information, the amount of capital this year (K1) could be something less than 25 (option c).

To determine the amount of capital this year based on the given information, we can use the investment and depreciation rates.

Let's denote the amount of capital this year as K1.

According to the information provided:

People invest 55% of income, but we don't have any information about income. Therefore, we cannot determine the exact investment amount.

10% of capital depreciates. Based on this, the capital at the beginning of this year (K1) can be calculated as follows:

K1 = K - 0.1K

= 0.9K

Since we know that the capital last year was equal to 25, we substitute K = 25 into the equation above:

K1 = 0.9 * 25

= 22.5

Therefore, based on the given information, the amount of capital this year (K1) could be something less than 25 (option c).

Learn more about information from

https://brainly.com/question/27894163

#SPJ11

The Pearson correlation coefficient for the given data set is 0.5 (option a).

To calculate the Pearson correlation coefficient,

1: Calculate the mean of x and y.

Mean of x: (2 + 1 + 3) / 3 = 2

Mean of y: (6 + 2 + 4) / 3 = 4

2: Calculate the deviation of each value from the mean for both x and y.

x deviation: (2 - 2), (1 - 2), (3 - 2) = 0, -1, 1

y deviation: (6 - 4), (2 - 4), (4 - 4) = 2, -2, 0

3: Calculate the product of the deviations for each pair of values.

Product of deviations: (0 * 2), (-1 * -2), (1 * 0) = 0, 2, 0

4: Calculate the squared deviation for each value of x.

x squared deviation: \((0^2), (-1^2), (1^2)\) = 0, 1, 1

5: Calculate the squared deviation for each value of y.

y squared deviation: \((2^2), (-2^2), (0^2)\) = 4, 4, 0

6: Sum up the products of the deviations and divide by the square root of the product of the squared deviations of x and y.

Pearson correlation coefficient: (0 + 2 + 0) / sqrt((0 + 1 + 1) * (4 + 4 + 0))

Pearson correlation coefficient: 2 / sqrt(2 * 8)

Pearson correlation coefficient: 2 / sqrt(16)

Pearson correlation coefficient: 2 / 4

Pearson correlation coefficient: 0.5

Therefore, the Pearson correlation coefficient for the given data set is 0.5, which corresponds to option a.

For more such questions on coefficient, click on:

https://brainly.com/question/17291621

#SPJ8

To rotate a point 180° clockwise about the origin, we essentially need to flip the point across the x-axis and then across the y-axis. So the x-coordinate of point A' is -3.

This means that the x-coordinate of the point will become its opposite (negation) and the y-coordinate of the point will also become its opposite.

So, in this problem, we have the point A with coordinates (3, 2). To rotate this point 180° clockwise about the origin, we will negate both the x and y coordinates of the point:

The negation of 3 is -3, so the new x-coordinate of the point will be -3.

The negation of 2 is -2, so the new y-coordinate of the point will be -2.

Putting these together, we get the new coordinate of the point A' as (-3, -2).

So the x-coordinate of point A' is -3.

To learn more about origin please click on below link.

https://brainly.com/question/28528569

#SPJ4

The estimated monthly average daily radiation on a horizontal surface in June in Amman is approximately 7.35 kWh/m(^2)/day.

To estimate the monthly average daily radiation on a horizontal surface H in June in Amman, we can use the following equation:

\([H = S \times H_s \times \frac{\sin(\phi)\sin(\delta)+\cos(\phi)\cos(\delta)\cos(H_a)}{\pi}]\)

where:

S is the solar constant, which is approximately equal to 1367 W/m(^2);

\(H(_s)\) is the average number of sunshine hours per day in Amman in June, which is given as 10 hours;

(\(\phi\)) is the latitude of the location, which for Amman is approximately 31.9 degrees North;

(\(\delta\)) is the solar declination angle, which is a function of the day of the year and can be calculated using various methods such as the one given in the answer to Q1;

\(H(_a)\) is the hour angle, which is the difference between the local solar time and solar noon, and can also be calculated using various methods such as the one given in the answer to Q1.

Substituting the given values, we get:

\([H = 1367 \times 10 \times \frac{\sin(31.9)\sin(\delta)+\cos(31.9)\cos(\delta)\cos(H_a)}{\pi}]\)

Since we are only interested in the monthly average daily radiation, we can assume an average value for the solar declination angle and the hour angle over the month of June. For simplicity, we can assume that the solar declination angle (\(\delta\)) is constant at the value it has on June 21, which is approximately 23.5 degrees North. We can also assume that the hour angle \(H(_a)\) varies linearly from -15 degrees at sunrise to +15 degrees at sunset, with an average value of 0 degrees over the day.

Substituting these values, we get:

\([H = 1367 \times 10 \times \frac{\sin(31.9)\sin(23.5)+\cos(31.9)\cos(23.5)\cos(0)}{\pi}]\)

Simplifying the equation, we get:

\([H \approx 7.35 \text{ kWh/m}^2\text{/day}]\)

Therefore, the estimated monthly average daily radiation on a horizontal surface in June in Amman is approximately 7.35 kWh/m(^2)/day.

Learn more about "average daily radiation " : https://brainly.com/question/32911494

#SPJ11

Answer:

416 cm2

Step-by-step explanation:

The formula for surface area of prism.

2Ab+ Pb*h

2(1/2)(12)(8)+ 32*10

= 96+320

= 416 cm2

if my answer helps please mark as brainliest.