Find the solution to the linear system of differential equations { -250 +42y -140 +24y satisfying the initial conditions x(0) = 5 and y(0) = 3. y' 2(t) = g(t) =

The formula for calculating the slope coefficient (b) in the simple linear regression model using the normal equations is b = Σ[(xᵢ - X)(yᵢ - Y)] / Σ[(xᵢ - X)²], representing the rate of change of y with respect to x.

A simple linear regression model describes the relationship between two continuous variables, denoted as x (explanatory variable) and y (response variable). The model equation is y = a + bx, where a represents the y-intercept, b represents the slope, and e represents the error term. The slope, b, quantifies the rate of change in y for a unit change in x.

To determine the line of best fit using the normal equations, we solve two simultaneous equations derived from the normal distribution of errors (e).

The first equation arises from the first-order condition for minimizing the sum of squared errors (SSE):

∂SSE/∂b = 0

Expanding SSE, we have:

SSE = Σ(yᵢ - a - bxᵢ)²

Differentiating SSE with respect to b and setting it equal to zero, we get:

Σ(xᵢyᵢ) - aΣ(xᵢ) - bΣ(xᵢ²) = 0

Rearranging the terms, we have:

Σ(xᵢyᵢ) - aΣ(xᵢ) = bΣ(xᵢ²)

To calculate the slope, b, we divide both sides by Σ(xᵢ²):

b = (Σ(xᵢyᵢ) - aΣ(xᵢ)) / Σ(xᵢ²)

To find the value of a, we substitute the sample means of x and y, denoted as X and Y respectively:

a = Y - bx

Thus, the solution for the slope, b, in the simple linear regression model, derived using the normal equations, is:

b = Σ(xᵢ - x)(yᵢ - y) / Σ(xᵢ - x)²

Whereas the solution for the y-intercept, a, is:

a = Y - b x

These equations enable the determination of the coefficients a and b, which yield the line of best fit that minimizes the sum of squared errors in the simple linear regression model.

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Answer:

-16/65

Step-by-step explanation:

Given sinα = 3/5 in quadrant 1;

Since  sinα = opp/hyp

opp = 3

hyp = 5

adj^2 = hyp^2 - opp^2

adj^2 = 5^2 = 3^2

adj^2 = 25-9

adj^2 = 16

adj = 4

Since all the trig identity are positive in Quadrant 1, hence;

cosα = adj/hyp = 4/5

Similarly, if tanβ = 5/12 in Quadrant III,

According to trig identity

tan theta = opp/adj

opp = 5

adj = 12

hyp^2 = opp^2+adj^2

hyp^2 = 5^2+12^2

hyp^2 = 25+144

hyp^2 = 169

hyp = 13

Since only tan is positive in Quadrant III, then;

sinβ = -5/13

cosβ = -12/13

Get the required expression;

sin(α - β) = sinαcosβ - cosαsinβ

Substitute the given values

sin(α - β) = 3/5(-12/13) - 4/5(-5/13)

sin(α - β)= -36/65 + 20/65

sin(α - β) = -16/65

Hence the value of sin(α - β) is -16/65

The answer is:

\(\large\textbf{They aren't equivalent.}}\)

In-depth explanation:

To determine the answer to this problem, we will use one of the exponent properties:

\(\sf{x^{-m}=\dfrac{1}{x^m}}\)

And

\(\sf{\dfrac{1}{x^{-m}}=x^m}\)

Now we apply this to the problem.

What is 4⁻³ equal to? Well according to the property, it's equal to:

\(\sf{4^{-3}=\dfrac{1}{4^3}}\)

And this question asks us if 4⁻³ is the same as 1/4⁻3.

Well according to the calculations performed above, they're not equivalent.

Answer:

$1,252.13 I believe  or $1,252

Step-by-step explanation:

Multiply the 1500 by .10 and you'll get 150

Now subtract that from 1500 that's 1350

.0725 multiplyed by 1350 is 97.875

Subtract that from 1350 and you'll get the answer!

The given expression √10 and √26  are irrational numbers.

The sum of √10 and √26 cannot be simplified because the two numbers are not like terms. However, we can estimate the value of the sum by using the fact that the square root of a number is between two consecutive perfect squares.

For √10, we know that 9 < 10 < 16, so √10 is between 3 and 4. For √26, we know that 25 < 26 < 36, so √26 is between 5 and 6. Therefore, we can estimate the sum of √10 and √26 to be between 8 and 10.

To get a more precise value, we can use a calculator to evaluate the sum of √10 and √26. Using a calculator, we find that the sum of √10 and √26 is approximately 7.051. Therefore, we can describe the sum of √10 and √26 as a decimal number that is approximately 7.051.

Alternatively, we can write the sum of √10 and √26 as a simplified radical expression. To do this, we need to combine the two radical terms into a single term using the distributive property of multiplication. We have:

√10 + √26 = √(10 × 1) + √(26 × 1) = √10 × √1 + √26 × √1 = √10 + √26

Therefore, we cannot simplify the sum of √10 and √26 any further, and we can describe it as the sum of two irrational numbers.

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The process the thickness of this part has a normal distribution with a mean 2.17 and a standard deviation of 0.018. 0.8675 of the manufactured parts will be accepted by the buyer. Answer: 0.8675

Given,Mean = μ = 2.17

Standard Deviation = σ = 0.018

Thickness = X

Specified value of thickness = 2.2 ± 0.05 cm

We know that Z score is given by `Z = (X-μ)/σ`

Now, for X = 2.15Z = (2.15 - 2.17) / 0.018

Z = -1.11

Now, for X = 2.25Z = (2.25 - 2.17) / 0.018Z = 4.44

Now, we have to calculate the proportion of parts falling within these Z scores.

Z score table gives the proportion of parts lying below the specified Z value.

Here, we have Z value on both sides and we need the proportion of parts lying in between them.

To find this, we can subtract the smaller value from the larger one.

Proportion of parts falling within the specified range = P( -1.11 < Z < 4.44 ) = P(Z < 4.44) - P(Z < -1.11)

Now, from Z score table, P(Z < -1.11) = 0.1325P(Z < 4.44) = 1.0 (As Z > 3.49)

Hence, P( -1.11 < Z < 4.44 ) = 1.0 - 0.1325 = 0.8675So, 0.8675 of the manufactured parts will be accepted by the buyer. Answer: 0.8675

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Answer:

Step-by-step explanation:

Answer:

3/5

Step-by-step explanation:

3 + 6 + 1 = 10

6 green, 6/10

3/5

Answer:

3/5

Step-by-step explanation:

We know that 68.26% of data is between 580 and 620 using a normal distribution table.

A data set must (when graphed) follow a bell-shaped, symmetrical curve that is centered around the mean in order to be regarded as having a normal distribution.

Additionally, it must follow the empirical rule that shows the proportion of the data set that lies within (plus or minus) 1, 2, and 3 standard deviations of the mean.

So, we know that:

μ = 600

σ = 20

Use the equation:
z - score = x-μ/σ

580 in the data's Z-score:

= 580-600/20

= -20/20

= -1

While the data's z-score in 620:

= 620-600/20

= 20/20

= 1

Using a table of the normal distribution:

P(580 < z) = 0.1587

Also, P(620 > z) = 0.8413

As a result, the percent of the data that falls between 580 and 620 is given by: P(620 > z) - P(580 z).

= 0.8413  - 0.1587

= 0.6826

= 68.26 %

Therefore, we know that 68.26% of the data is between 580 and 620 using a normal distribution table.

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There were 25 students in class yesterday.

To solve the given problem, let the number of students in class yesterday be x.

There were 30 students in class today.

This is an increase of 25% from yesterday.

To calculate how many students were in class yesterday, we'll use the formula for percent increase:

% increase = (new value - old value) / old value * 10025%

= (30 - x) / x * 100

We need to solve for x : 25x

= 30x - 750x

= 25.

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Answer:

25,45,5

Step-by-step explanation:

Answer:

m<AOB = 50°

Step-by-step explanation:

<ACB = 25° is an inscribed angle of the circle.

<AOB is a central angle of the circle.

Thus, based on the inscribed angle theorem which states that an inscribed angle is ½ the measure of the central angle, therefore:

m<ACB = ½(m<AOB)

Substitute

25° = ½(m<AOB)

Multiply both sides by 2

2*25 = m<AOB

m<AOB = 50°

The volume of the sphere figure is 697.19 cubic yards

From the question, we have the following parameters that can be used in our computation:

The sphere

The volume of a sphere can be calculated as

V = 4/3πr³

Where

r = Radius

In this case,

r = 11/2

Substitute the known values in the above equation, so, we have the following representation

V = 4/3 * 22/7 * (11/2)³

Evaluate

V = 697.19

Hence, the volume is 697.19 cubic yards

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Answer:

We have the proportion: 8/10 = 100/y

We can solve for y by cross-multiplying.

That is, 8y = 100 * 10 Simplifying the right-hand side, 8y = 1000

Dividing both sides by 8 to solve for y, y = 125

Hence, the value of y is 125.

Finally, we have the expression: y = √80We can simplify this expression by factoring 80 into its prime factors:80 = 2 * 2 * 2 * 2 * 5

Taking the square root of 2 * 2 * 2 * 2, we have:√(2 * 2 * 2 * 2) = 2 * 2 = 4

Therefore, y = 4√5The value of y is 4√5.

Step-by-step explanation:

Hope this helps!! Have a good day/night!!

Answer:

W=x/7+ 8z/7 + 4/7

Step-by-step explanation:

The value for the expression 6 + 9w - 5s is 59.

You must substitute a number for each variable and carry out the arithmetic procedures to evaluate an algebraic expression. Since 6 + 6 equals 12, the variable x in the case above equals 6. If we are aware of the values of our variables, we can replace the variables with those values before evaluating the expression.

When the variables and constants of an expression are given values, the outcome of the computation it describes yields the expression's value. The quantity that the function assumes for these argument values is the value of the function, given the value(s) assigned to its argument(s).

The expression 6 + 9w - 5s when w = 7 and s = 2.

                          6 +(9*7 - 5*2) = 6+63 - 10 = 59.

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Answer:

128

Step-by-step explanation:

Sue's average for 9 games of bowling is 108. Therefore, the total scores for the 9 games is:

\(= 108 \times 9\)

\(= 972\)

To have an average of 110 in 10 games, her total score will be:

\(= 110 \times 10\)

\(= 1100\)

Therefore, the lowest score that she can get will be the difference between the values calculated above and this will be:

\(= 1100 - 972\)

\(\bold{= 128}\)

Answer:

60° and 30°

Step-by-step explanation:

Complementary angles sum to 90°

sum the parts of the ratio, 6 + 3 = 9 parts

Divide 90° by 9 to find the value of one part of the ratio.

90° ÷ 9 = 10° ← value of 1 part of the ratio, then

6 parts = 6 × 10° = 60°

3 parts = 3 × 10° = 30°

The measure of the angles is 60° and 30°

The volume of the region bounded by the two paraboloids is 32π/5 cubic units.

To find the volume of the region bounded by the two paraboloids, we need to determine the limits of integration for each variable.

Since the two paraboloids intersect in a curve, we can use this curve as a boundary to split the region into two parts.

First, let's find the curve of intersection by setting the two equations equal to each other:

\(12 - x^2 - y^2 = 2x^2 + 2y^2\\10x^2 + 10y^2 = 12\\x^2 + y^2 = 6/5\)

This is the equation of a circle with center at the origin and radius \(\sqrt{(6/5)\)

So we can use cylindrical coordinates to integrate over this region.

The limits for z are from the lower paraboloid to the upper paraboloid:

\(2x^2 + 2y^2 \leq z\leq 12 - x^2 - y^2\)

In cylindrical coordinates, we have:

\(0 \leq r \leq \sqrt{(6/5)}\\0 \leq \theta \leq 2\pi \\2r^2 \leq z \leq 12 - r^2\)

So the volume of the region is given by the triple integral:

V = ∫∫∫ dz r dr dθ

where the limits of integration are as described above. Therefore, we have:

\(V = \int\limits^{2\pi }_0 {\int\limits^{\sqrt{6/5}}_0 {\int\limits^{12-r^2}_{2r^2} \, dz}\ r \, dr } \, d\theta\)

Evaluating the integral, we get:

V = 32π/5

Therefore, the volume of the region bounded by the two paraboloids is 32π/5 cubic units.

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Answer:

y = 4x - 6

Step-by-step explanation:

linear equation:

y = mx + b

1. find the slope m

first select two point

im gonna take point 1 (-1,-7) and point 2 (3,9)

m = (y2-y1)/(x2-x1)

m = (9-(-7))/(3-(-1)) = 16/4 = 4

2. find b

first select a point

im gonna reuse point 2 (3,9)

y = mx + b

9 = 5(3) + b

9 = 15 + b

b = 9 - 15 = -6

3. keep x y, substitute m,b to linear equation

y = 4x - 6

The height of the 10th bounce of the ball will be 0.6 feet.

A geometric sequence is a sequence in which each term is found by multiplying the preceding term by the same value.

The nth term of the geometric sequence is given by

\(\sf T_n=ar^{n-1}\)

Where,

According to the given question.

During the first bounce, height of the ball from the ground, a = 4.5 feet

And, the each successive bounce is 73% of the previous bounce's height.

So,

During the second bounce, the height of ball from the ground

\(\sf = 73\% \ of \ 10\)

\(=\dfrac{73}{100}(10)\)

\(\sf = 0.73 \times 10\)

\(\sf = 7.3 \ feet\)

During the third bounce, the height of ball from the ground

\(\sf = 73\% \ of \ 7.3\)

\(=\dfrac{73}{100}(7.3)\)

\(\sf = 5.33 \ feet\)

Like this we will obtain a geometric sequence 7.3, 5.33, 3.11, 2.23,...

And the common ratio of the geometric sequence is 0.73

Therefore,

The sixth term of the geometric sequence is given by

\(\sf T_{10}=10(0.73)^{10-1\)

\(\sf T_{10}=10(0.73)^{9\)

\(\sf T_{10}=10(0.059)\)

\(\sf T_{10}=0.59\thickapprox0.6 \ feet\)

Hence, the height of the 10th bounce of the ball will be 0.6 feet.

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Angles Formulas at the center of a circle can be expressed as:

Central angle, θ = (Arc length × 360º)/(2πr) degrees

Sum of Interior angles=180°(n-2)

The angles formulas are used to find the measures of the angles. An angle is formed by two intersecting rays, called the arms of the angle, sharing a common endpoint.

The corner point of the angle is known as the vertex of the angle. The angle is defined as the measure of the turn between the two lines.

There are various types of formulas for finding an angle; some of them are the central angle formula, double-angle formula, etc...

We use the central angle formula to determine the angle of a segment made in a circle.

We use the sum of the interior angles formula to determine the missing angle in a polygon.

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Answer:

A: 71°

Explanation:

The interior angles of a triangle must always equal 180°.

We have two of the angles, and if we add them up and subtract it from 180°, we can get the missing angle.

49+60+x=180

Add 49 and 60

109+x=180

Isolate the x, so subtract 109 from 180

x=71°

Answer:second account

Step-by-step explanation:

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Answer:

The result can be shown in multiple forms.

Exact Form:

− 3/2

Decimal Form:

− 1.5

Mixed Number Form:

−1 1/2

Step-by-step explanation:

Answer:

\(\displaystyle -\frac{3}{2}\)

Step-by-step explanation:

\(\displaystyle \biggr(\frac{7}{4}-\frac{8}{3}\biggr)-\biggr|\frac{1}{6}-\frac{3}{4}\biggr|\\ \\=\biggr(\frac{7*3}{4*3}-\frac{8*4}{3*4}\biggr)-\biggr|\frac{1*2}{6*2}-\frac{3*3}{4*3}\biggr|\\ \\=\biggr(\frac{21}{12}-\frac{32}{12}\biggr)-\biggr|\frac{2}{12}-\frac{9}{12}\biggr|\\ \\=-\frac{11}{12}-\biggr|-\frac{7}{12}\biggr|\\\\=-\frac{11}{12}-\frac{7}{12}\\\\=-\frac{18}{12}\\ \\=-\frac{3}{2}\)

Answer:

3+3 and 2(3)

Step-by-step explanation:

3+3 and 2(3)

Answer:

yes

Step-by-step explanation:

x+3 = 6 if X is 3

2x = 6

2(3) = 6

so the statement is correct

Answer:

73 - 2x = 86 - 3x

So 3x - 2x =86 - 73 so x=13

Step-by-step explanation:

Then m<1 = 73-2(13) = 47

M<2=82-3(13)=47

So both angles are having same value

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To construct a frequency distribution with a class width of 10,000, we'll divide the income ranges into appropriate intervals and count the frequencies within each interval. Here's the frequency distribution:

Income Range Frequency

35,000 - 44,999 6

45,000 - 54,999 9

55,000 - 64,999 8

65,000 - 74,999 2

Now, let's construct the relative frequency distribution. To calculate the relative frequency, we divide the frequency of each interval by the total number of data points. In this case, the total number of data points is the sum of the frequencies.

Total number of data points = 6 + 9 + 8 + 2 = 25

Income Range Frequency Relative Frequency

35,000 - 44,999 6 6/25 = 0.24

45,000 - 54,999 9 9/25 = 0.36

55,000 - 64,999 8 8/25 = 0.32

65,000 - 74,999 2 2/25 = 0.08

The relative frequency distribution is as follows:

Income Range Relative Frequency

35,000 - 44,999 0.24

45,000 - 54,999 0.36

55,000 - 64,999 0.32

65,000 - 74,999 0.08

Note: The relative frequencies are expressed as decimals, not rounded to the nearest decimal place.

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The two possible functions is F(x)= \frac{x^4}{12}+3x^2 and F(x)= \frac{x^4}{12}+3x^2+x.

The derivative of the first derivative of the given function is known as the second order derivative. We can learn about the slope of the tangent at a particular position or the instantaneous rate of change of a function at that point from the first-order derivative at that point.

F”(x)= x^2+6

F’(x)= \int f"(x) dx

       = \int( x^2+6) dx

       = \frac{x^3}{3}+6x+c_1                                  c_1 is the integral constant.

F(x)= \int f\prime(x) dx

      = \int f (\frac{x^3}{3}+6x+c_1) dx

      = \frac{x^4}{12}+3x^2+c_1x+c_2

(i) if c_1=0 , and c_2=0

   F(x)= \frac{x^4}{12}+3x^2

(ii) if c_1=1, and c_2=0

        F(x)= \frac{x^4}{12}+3x^2+x

Therefore the two possible functions is F(x)= \frac{x^4}{12}+3x^2 and F(x)= \frac{x^4}{12}+3x^2+x

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The solution of the system of differential equations satisfying the initial conditions x(0)=5 and y(0)=3 is given by:

\($x(t) = e^{at} (5 \cos(bt) + 3 \sin(bt))$\)

\($y(t) = e^{ct} (5 \sin(bt) - 3 \cos(bt))$\)

Let x(t) and y(t) be the solution of the linear system of differential equations:

\($\frac{dx}{dt} = ax + by$\)

\($\frac{dy}{dt} = cx + dy$\)

with initial conditions x(0)=5 and y(0)=3.

The general solution of the system of differential equations can be written as:

\($x(t) = e^{at} (X0 \cos(bt) + Y0 \sin(bt))$\)

\($y(t) = e^{ct} (X0 \sin(bt) - Y0 \cos(bt))$\)

where X0 = 5 and Y0 = 3.

Therefore, the solution of the system of differential equations satisfying the initial conditions x(0)=5 and y(0)=3 is given by:

\($x(t) = e^{at} (5 \cos(bt) + 3 \sin(bt))$\)

\($y(t) = e^{ct} (5 \sin(bt) - 3 \cos(bt))$\)

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The average rate of change of the function f(x) = x² + 7x + 6 over the interval -4 ≤ x ≤ -1 is -8/3 or about -2.67.

To determine the average rate of change of a function over a specific interval, we use the following formula:

\($$ \frac{f(b) - f(a)}{b - a} $$\)

where a and b are the endpoints of the interval.

In this case, we have the function f(x) = x² + 7x + 6 and the interval -4 ≤ x ≤ -1. To find the average rate of change of the function over this interval, we need to evaluate the function at the endpoints of the interval and substitute these values into the formula.

Therefore:

\($$ \text{Average rate of change} = \frac{f(-1) - f(-4)}{-1 - (-4)} $$\)

We start by evaluating the function at the endpoints of the interval: \($$ f(-1) = (-1)^2+ 7(-1) + 6 = -2 $$\)

\($$ f(-4) = (-4)^2 + 7(-4) + 6 = 6 $$\)

Substituting these values into the formula, we get: \($$ \text{Average rate of change} = \frac{-2 - 6}{-1 - (-4)} = \frac{-8}{3} $$\)

Therefore, the average rate of change of the function f(x) = x² + 7x + 6 over the interval -4 ≤ x ≤ -1 is -8/3 or about -2.67.

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