1. Solve the differential equation
(4+t²)dy/dt + 2ty = 4t


2. Find the general solution of the differential equation dy/dt + (1/2)y = (1/2)e^t/3 .


3. Solve the initial value problem .
ty' + 2y = 4t² , where y(1) = 2.​

Tan (θ) = 13/9

θ = 55.305°

Step-by-step explanation:

To convert all pulse rates of women to z-scores, we use the formula:

z = (x - μ) / σ

where x is the pulse rate, μ is the mean, and σ is the standard deviation.

Substituting the given values, we get:

z = (x - 77.5) / 11.6

Therefore, after converting all pulse rates of women to z-scores, the mean will be 0 and the standard deviation will be 1.

Answer:

18

Step-by-step explanation:

Look at the values

2   184

7    94

Difference in volume: 184 liters - 94 liters = 90 liters

Difference in time: 7 minutes - 2 minutes = 5 minutes

rate in liters/minute: (90 liters) / (5 minutes) = 18 liters/minute

Answer:

Step-by-step explanation:

First find the slope using m = (y2-y1)/(x2-x1)

m = (-1-7)/(-5-3)

   =  -8/-8

   =  1

y = mx + b

find the y-intercept

use 1 of the 2 points

Let's try (3,7)

7 = 1(3) + b

b = 4

Equation of the line is y = x + 4

The equation is:

Work/explanation:

First, we will use the slope formula and determine the slope:

[tex]\sf{m=\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

where m = slope.

Plug in the data

[tex]\sf{m=\dfrac{-1-7}{-5-3}}\\\\\\\sf{m=\dfrac{-8}{-8}}\\\\\\\sf{m=1}[/tex]

The slope is 1; the equation so far is y = 1x + b or y = x + b.

Plug in the point:

[tex]\sf{7=3+b}[/tex]

[tex]\sf{3+b=7}[/tex]

Solve for b

[tex]\sf{b+3=7}[/tex]

[tex]\sf{b=4}[/tex]

So the y-intercept is 4; we plug that in and see that the equation is y = x + 4.

Answer:

  10 meters

Step-by-step explanation:

You want the width of the path surrounding an 18 meter by 24 meter pool if the total area of pool and path is 1672 square meters.

The total area of the rectangular shape is the product of its length and width:

  A = LW

  1672 = (24 +2x)(18 +2x)

  418 = (12 +x)(9 +x) . . . . . . . . divide by 4

  418 = 108 +21x +x² . . . . . . eliminate parentheses

  420.25 = 110.25 +21x +x² . . . . . . add 2.5 to complete the square

  20.5 = (10.5 +x) . . . . . . . . . . take the positive square root

  10 = x . . . . . . . . . . . . . . subtract 10.5

The width of the path is 10 meters.

__

Additional comment

The attached graph shows the two solutions of the original equation written so the solutions are the x-intercepts:

  (24 +2x)(18 +2x) -1672 = 0

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A single 180-degree rotation about the origin is equivalent to the sequence of reflections mentioned, as it accomplishes the same changes to the shape's orientation and coordinates.

Yes, there is a single transformation that is equivalent to reflecting AABC across the y=x line and then reflecting the image across the y-axis. This single transformation is known as a 180-degree rotation about the origin.

When you reflect AABC across the y=x line, each point is transformed to its corresponding point on the opposite side of the line. This reflection effectively swaps the x-coordinates with the y-coordinates for each point, resulting in a new shape.

Now, when you reflect the newly formed image across the y-axis, you essentially negate the x-coordinates of each point. This is equivalent to rotating the shape 180 degrees about the origin.

A 180-degree rotation about the origin involves flipping the shape by exchanging the x and y coordinates and taking their negatives. This transformation achieves the same result as reflecting across the y=x line and then reflecting across the y-axis.

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[tex]\huge\blue{\fbox{\tt{Solution:}}}[/tex]

We can simplify the expression using trigonometric identities.

First, we can use the double angle formula for sine to write sin(2x) = 2sin(x)cos(x).

Next, we can use the double angle formula for cosine to write cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x). Rearranging this equation gives 2sin^2(x) - 2cos^2(x) = -cos(2x) + 1.

Substituting these identities into the original expression gives:

tan(x-1) ( sin2x-2cos2x) = tan(x-1) [2sin(x)cos(x) - 2(1 - 2sin^2(x))]

= 2tan(x-1)sin(x)cos(x) - 2tan(x-1) + 4tan(x-1)sin^2(x)

We can use the identity tan(x) = sin(x)/cos(x) to simplify this expression further:

2tan(x-1)sin(x)cos(x) - 2tan(x-1) + 4tan(x-1)sin^2(x)

= 2sin(x)cos(x)/(cos(x-1)) - 2sin(x)/(cos(x-1)) + 4sin^2(x)/(cos(x-1))

Multiplying both sides of the equation by cos(x-1) gives:

2sin(x)cos(x) - 2sin(x) + 4sin^2(x)cos(x-1) = 2(1-2sin(x)cos(x))

Expanding the left-hand side of the equation gives:

2sin(x)cos(x) - 2sin(x) + 4sin^2(x)cos(x) - 4sin^2(x) = 2 - 4sin(x)cos(x)

Simplifying this equation gives:

4sin^2(x) - 2sin(x) - 2 = 0

This is a quadratic equation in sin(x), which can be solved using the quadratic formula.

Answer:

it answer is

P=2(l+b)

P=l + l + b + b

The height of the stump is 3 ft.

The given equation represents the height of the frog, h(t), as a function of time, t. To find the height of the stump, we need to determine the height when the time, t, is equal to 0.

In the equation h(t) = -16t² + 64t + 3, we substitute t = 0:

h(0) = -16(0)² + 64(0) + 3

Since any term multiplied by zero is zero, we can simplify further:

h(0) = 0 + 0 + 3

Therefore, the height of the stump, at time t = 0, is 3 ft. This means that when the frog initially leaps from the stump, the height of the stump itself is 3 ft.

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Answer:

136 [tex]ft^{2}[/tex]

Step-by-step explanation:

Find the area of the wall and subtract out the area of the window

16 x 9 = 144

4 x 2 = 8

144 - 8 = 136

Helping in the name of Jesus.

Answer:

x = 90

Step-by-step explanation:

The given diagram shows a circle with intersecting chords, KM and JL.

To find the value of x, we can use the Angles of Intersecting Chords Theorem.

According to the Angles of Intersecting Chords Theorem, if two chords intersect within a circle, the angle formed at the intersection point is equal to half the sum of the measures of the arcs intercepted by the angle and its corresponding vertical angle.

Let the point of intersection of chords KM and JL be point P.

As the chords are straight lines, angle x° forms a linear pair with angle JPM.

Note: We cannot use the Angles of Intersecting Chords Theorem to find the value of x directly, since we have not been given the measures of the arcs KJ and ML. Therefore, we need to use the theorem to find m∠JPM first.

From inspection of the given diagram:

Using the Angles of Intersecting Chords Theorem, we can calculate the measure of angle JPM (shown in orange on the attached diagram):

[tex]\begin{aligned}m \angle JPM &=\dfrac{1}{2}\left(m\overset\frown{JM}+m\overset\frown{LK}\right)\\\\&=\dfrac{1}{2}\left(30^{\circ}+(2x-30)^{\circ}\right)\\\\&=\dfrac{1}{2}\left(30^{\circ}+2x^{\circ}-30^{\circ}\right)\\\\&=\dfrac{1}{2}\left(2x^{\circ}\right)\\\\&=x^{\circ}\end{aligned}[/tex]

As angle JPM forms a linear pair with angle x°, the sum of the two angles equals 180°:

[tex]\begin{aligned}m \angle JPM+x^{\circ}&=180^{\circ}\\\\x^{\circ}+x^{\circ}&=180^{\circ}\\\\2x^{\circ}&=180^{\circ}\\\\\dfrac{2x^{\circ}}{2}&=\dfrac{180^{\circ}}{2}\\\\x^{\circ}&=90^{\circ}\\\\x&=90\end{aligned}[/tex]

Therefore, the value of x is 90, which means that the two chords intersect at right angles.

The value of tan 0 is -15/8.

The tangent is a periodic function that is defined by a unit circle. It is the ratio of the opposite side to the adjacent side of a right-angled triangle that contains the angle in question as one of its acute angles. The value of the tangent function can be positive, negative, or zero, depending on the quadrant in which the angle is located.

The given values are

cos0 = 8/17

sin0 = -15/17

We can use the trigonometric identity of tan = sin/cos to find the value of tan 0.

Substituting the given values, we get

tan 0 = sin 0 / cos 0

= (-15/17) / (8/17)

=-15/8

Therefore, the value of tan 0 is -15/8.

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Answer:

60°

Step-by-step explanation:

The sum of interior angles in a triangle is equal to 180°.

48° + 11x - 5 + 9x - 3 = 180°

40° + 20x = 180°

20x = 140°

x = 7

To find m∠A, replace x with 7:

m∠A = 9x - 3

9×7-3 = 60°

Beatriz measured the weight of a box and determined the margin of error as 23.5 pounds to 24.5 pounds. The range of the margin of error is 1 pound, which means that the actual weight of the box could be off by a maximum of 1 pound.

Beatriz measured the weight of a box and determined the margin of error as 23.5 pounds to 24.5 pounds. The greatest possible error for her actual measurement can be determined by finding the range of the margin of error.

The range of the margin of error can be determined by finding the difference between the upper limit and the lower limit. Therefore, the range of the margin of error is 24.5 - 23.5 = 1 pound.

This means that the actual weight of the box could be off by a maximum of 1 pound. This is the greatest possible error for her actual measurement.

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Answer:

50.75

Step-by-step explanation:

We have:

[tex]E[g(x)] = \int\limits^{\infty}_{-\infty} {g(x)f(x)} \, dx \\\\= \int\limits^{1}_{-\infty} {g(x)(0)} \, dx+\int\limits^{6}_{1} {g(x)\frac{2}{x} } \, dx+\int\limits^{\infty}_{6} {g(x)(0)} \, dx\\\\= \int\limits^{6}_{1} {g(x)\frac{2}{x} } \, dx\\\\=\int\limits^{6}_{1} {(4x+3)\frac{2}{x} } \, dx\\\\=\int\limits^{6}_{1} {(4x)\frac{2}{x} } \, dx + \int\limits^{6}_{1} {(3)\frac{2}{x} } \, dx\\\\=\int\limits^{6}_{1} {8} \, dx + \int\limits^{6}_{1} {\frac{6}{x} } \, dx\\\\[/tex]

[tex]=8\int\limits^{6}_{1} \, dx + 6\int\limits^{6}_{1} {\frac{1}{x} } \, dx\\\\= 8[x]^{^6}_{_1} + 6 [ln(x)]^{^6}_{_1}\\\\= 8[6-1] + 6[ln(6) - ln(1)]\\\\= 8(5) + 6(ln(6))\\\\= 40 + 10.75\\\\= 50.74[/tex]

a) Using the standard divisors according to Jefferson's apportionment method, the number of salespeople assigned to work each shift is as follows:

Shift                              Morning     Midday     Afternoon     Evening   Total

Number of

salespeople assigned  1               3                  5                 7                16

b) The modified or adjusted divisor used represents the average number of customers for each shift divided by the standard divisor.

The standard divisor shows the ratio of the total population to the number of seats.

The standard divisor gives an insight about the number of people each seat represents.

Standard divisor = Total number of customers / Total number of salespeople

The total number of customers = 1,425

The number of salespeople = 16

a) Standard divisor = 1,425 ÷ 16

= 89.0625

Shift                              Morning     Midday     Afternoon     Evening   Total

Average number of

customers                     125            280             460             560       1,425

Standard divisor      89.0625     89.0625      89.0625      89.0625

b) Modified divisor     1.4035        3.1438          5.165           6.288

Number of

salespeople assigned  1               3                  5                 7                16

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Answer:

55

Step-by-step explanation:

[tex]\displaystyle MOE = z\biggr(\frac{\sigma}{\sqrt{n}}\biggr)\\\\4=1.96\biggr(\frac{15}{\sqrt{n}}\biggr)\\\\4=\frac{29.4}{\sqrt{n}}\\\\\sqrt{n}=\frac{29.4}{4}\\\\\sqrt{n}=7.35\\\\n=54.0225\\\\n\approx55\uparrow[/tex]

Make sure to always round up the required sample size to the nearest integer!

Answer:

1) 4630 L = 4.63 kL

2) 2 L 360 mL = 2360 mL

3) 2700 g = 2.7 kg (rounded to nearest 0.1 kg)

4) 2.705 kg = 2.71 kg (rounded to nearest 0.01 kg)

5) 9.597 kg = 10 kg (rounded to nearest kg)

Answer:

P = 9 + 9 + 4 + 4

Step-by-step explanation:

P=2(lxw) or l+l+w+w

Therefore, 9+9+4+4 is right.

The graphs have been correctly dragged to the table to show the image of the triangle after it is reflected over the x-axis, the y-axis, and the line y = x.

By applying a reflection over the x-axis to the coordinates of this triangle, we have the following coordinates of the image triangle;

(x, y)                →      (x, -y)

(1, -3)             →      (1, -(-3)) = (1, 3)

(3, -2)             →      (3, -(-2)) = (3, 2)

(4, -5)             →      (4, -(-5)) = (4, 5)

By applying a reflection over the y-axis to the coordinates of this triangle, we have the following coordinates of the image triangle;

(x, y)                →      (-x, y)

(1, -3)             →        (-1, -3)

(3, -2)             →      (-3, -2)

(4, -5)             →      (-4, -5)

By applying a reflection over the line y = x to the coordinates of this triangle, we have the following coordinates of the image triangle;

(x, y)                →      (y, x)

(1, -3)             →        (-3, 1)

(3, -2)             →      (-2, 3)

(4, -5)             →      (-5, 4)

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The probability that a randomly selected point within the square falls in the red-shaded square is 0.36

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

The areas of the above shapes are

Red square = 3² = 9

White square = 5² = 25

The probability is then calculated as

P = Red square/White square

So, we have

P = 9/25

Evaluate

P = 0.36

Hence, the probability that a randomly selected point within the square falls in the red-shaded square is 0.36

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Answer:

A.

Step-by-step explanation:

Let's think of this as a clock.  We can see that the 2 lines start in the same place, around 3 o'clock.  Next, one of the line segments shifts down to around 6 o'clock.  Next, it shifts to about 9 o'clock.  Logically, the next step (in a clock) would be 12 o'clock, making A the correct choice.

We can also just use a regular circle, with one of the line segments moving 90 degrees each time.

Hope this helps! :)

The angle of elevation of the sun to the nearest degree is: 34°

The angle of elevation is defined as an angle that is formed between the horizontal line and the line of sight. If the line of sight is upward from the horizontal line, then the angle formed is referred to as an angle of elevation.

There are three main trigonometric ratios which are:

sin x = opposite/hypotenuse

cos x = adjacent/hypotenuse

tan x = opposite/adjacent

In this case, the dimensions given will warrant the use of the tangent function to get:

tan θ = 101/147

tan θ = 0.6871

θ  = tan⁻¹(0.6871)

θ  = 34°

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Answer:

9, 9.5, 9.75, 9.875

Step-by-step explanation:

Notice the following pattern:

[tex]-22+16=-22+2^4=-6\\-6+8=-6+2^3=2\\2+4=2+2^2=6\\6+2=6+2^1=8[/tex]

Therefore, the next four terms will be:

[tex]8+2^0=8+1=9\\9+2^{-1}=9+0.5=9.5\\9.5+2^{-2}=9.5+0.25=9.75\\9.75+2^{-3}=9.75+0.125=9.875[/tex]

Answer:

Step-by-step explanation:

[tex]\lim_{x\to 1} \frac{x^3-1}{x-1} \\= \lim_{x \to 1} \frac{(x-1)(x^2+x+1)}{(x-1)} \\= \lim_{x \to 1} (x^2+x+1)\\=(1)^1+1+1\\=1+1+1\\=3[/tex]

a) The solution to the function is t²(du/dt) + 2tu - u² = 0 and can be expressed in the form F(t, u, du/dt) = 0 which confirms it is homogeneous

b) The implicit solution is t/y + 1/t = 2ln|t| + C₁ and the explicit solution is y = (t + 1)/(2ln|t| + C₁)

a) To show that the given differential equation is homogeneous, we need to verify that it can be written in the form:

F(x, y, y') = 0

where F is a homogeneous function of degree zero.

Given:

(t² + 2ty)y' = y²

Let's rearrange the equation:

(t² + 2ty)y' - y² = 0

Now, let u = y/t. We can rewrite y' in terms of u:

y' = du/dt

Substituting these values into the equation, we get:

(t² + 2ty)(du/dt) - (y/t)² = 0

Expanding the equation:

t²(du/dt) + 2ty(du/dt) - (y²/t²) = 0

Now, let's substitute u = y/t into the equation:

t²(du/dt) + 2tu - u² = 0

We can see that this equation is of the form F(t, u, du/dt) = 0, which is homogeneous. Therefore, the given differential equation is homogeneous.

To find the degree of the equation, we need to determine the power of t in each term. Looking at the equation:

t²(du/dt) + 2tu - u² = 0

The highest power of t is 2, which means the degree of the equation is 2.

b) To find the implicit and explicit solutions to the initial value problem (IVP), we need to solve the homogeneous differential equation and apply the initial condition y(1) = 1.

Let's solve the homogeneous equation:

t²(du/dt) + 2tu - u² = 0

We can rewrite it as:

du/u² - dt/t² = -2dt/t

Integrating both sides:

∫(du/u²) - ∫(dt/t²) = -2∫(dt/t)

This simplifies to:

-1/u - (-1/t) = -2ln|t| + C₁

1/u + 1/t = 2ln|t| + C₁

Since u = y/t, we substitute u back:

1/(y/t) + 1/t = 2ln|t| + C₁

t/y + 1/t = 2ln|t| + C₁

This is the implicit solution to the given initial value problem.

To find the explicit solution, we need to solve for y in terms of t. Let's rearrange the equation:

t/y = 2ln|t| + C₁ - 1/t

Multiply both sides by y:

t = y(2ln|t| + C₁ - 1/t)

Now, let's simplify:

t = 2yln|t| + C₁y - 1

Rearranging the equation:

2yln|t| + C₁y = t + 1

Factoring out y:

y(2ln|t| + C₁) = t + 1

Dividing both sides by (2ln|t| + C₁), assuming C₁ ≠ -2ln|t|, we get:

y = (t + 1)/(2ln|t| + C₁)

This is the explicit solution to the given initial value problem.

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To find f(g(x)), we need to substitute g(x) into the function f(x). Given that g(x) = x - 4, we substitute it into f(x) as follows:

[tex]f(g(x)) = f(x - 4) = (x - 4)^2 + 4[/tex]

To simplify this expression, we can expand the square:

[tex]f(g(x)) = (x - 4)(x - 4) + 4\\ = x^2 - 8x + 16 + 4\\ = x^2 - 8x + 20[/tex]

Therefore, f(g(x)) simplifies to[tex]x^2 - 8x + 20.[/tex]

Next, let's find g(f(x)). We substitute f(x) into the function g(x):

[tex]g(f(x)) = g(1/x^2 + 4) = 1/x^2 + 4 - 4\\ = 1/x^2[/tex]

Hence, g(f(x)) simplifies to 1/x^2.

In summary, f(g(x)) simplifies to[tex]x^2 - 8x + 20[/tex], and g(f(x)) simplifies to 1/x^2.

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