8. Consider the D.E. yy'= -x +
1/2
a. Find the implicit general solution of this
equation. What is the condition that c
should satisfied? (c is the constant that
appears in the solution).
b. (T) is the family of curves of the
general solution in an orthonormal
system. What is the nature of this graph?
(such curves are called integral curves of
the D.E.)
c. Find the graph (T) that passes through
the origin O(0, 0).

Answer:

Score                      Frequency

68                              02

69                              03

70                              01

71                               02

72                              02

73                              01

74                              02

75                              02

76                              01

When rotated 90 degrees, counterclockwise direction the new coordinates will result in the attached image.

The old coordinate where were rotated are:

A(-5,8)

B (-1, 7)

C(-3, 5)

D(-4, 2)

To rotate a point counterclockwise about the origin, we switch the x and y coordinates and change the sign of the new x  -coordinate.

The new coordinates are after 90 degrees, counterclockwise  are

A' = (-8, -5)

B' = (-7, -1)

C' = (-5, -3)

D' = (-2, -4)

See attached image.

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Full Question:

Although part of your question is missing, you might be referring to this full question:

See attached image.

Answer:the probability that event A or event B takes place is 12/19.

Answer:

120

61+97+82=240 There are 360 degrees in a rectangle. 360-240 is 120

Length of the base of the building from the wall is, 1.68 m

We have to given that;

A ladder 8 m long leans against the wall of a building.

And, the foot of the ladder makes an angle of 78 degrees with the ground.

Now, We know that;

cos x = Base / Hypotenuse

Here, Hypotenuse = 8 m

And, Length of the base of the building from the wall = Base = x

Hence, We get;

cos 78 = x / 8

0.21 = x / 8

x = 0.21  x 8

x = 1.68 m

Thus, Length of the base of the building from the wall is, 1.68 m

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

Work Shown:

x = number of years, some positive real number

f(x) = debt level in thousands of dollars

The company is out of debt when f(x) = 0

We go from

f(x)=-8x^2+8x+50

to

0 = -8x^2+8x+50

We'll use the quadratic formula to solve.

Plug in: a = -8, b = 8, c = 50

[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-8\pm\sqrt{8^2-4(-8)(50)}}{2*(-8)}\\\\x = \frac{-8\pm\sqrt{1664}}{-16}\\\\x = \frac{-8+\sqrt{1664}}{-16} \ \text{ or } \ x = \frac{-8-\sqrt{1664}}{-16}\\\\x \approx -2.0495\ \text{ or } \ x \approx 3.0495\\\\x \approx -2.1\ \text{ or } \ x \approx 3.1\\\\[/tex]

Ignore the negative x value. It doesn't make sense to have a negative number of years.

The only practical solution is approximately 3.1 years.

The solution is 768  is the correct answer, is the 9th term of the geometric sequence 3,6,12,24.

Given:

The geometric sequence 3,6,12,24.

Required:

Find the 6th term of the geometric sequence.

Explanation:

The given sequence is 3,6,12,24.

The nth term of the sequence is given by the formula:

an = a* r^n-1

Where a = first term and r = common ratio

From the given sequence

a = 3, r = 6/3 = 2

Then 9th term is:

a9 = 3 * 2^9-1

    = 3* 2^8

    =768

Final answer:

The solution is 768  is the correct answer, is the 9th term of the geometric sequence 3,6,12,24.

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The solution to the given system of equations is (-158/85, 57/17).

The system of linear equations are 5x-2y=-16 ------(i) and 4x -5y=-23 ------(ii).

Multiply equation (i) by 4, we get

20x-8y= -48 -----(iii)

Multiply equation (ii) by 5, we get

20x-25y=-105 -----(iv)

Subtract equation (iii) from equation (iv), we get

20x-25y-(20x-8y)=-105+48

-17y=-57

y=57/17

Substitute y=57/17 in equation (i), we get

5x-2(57/17)=-16

5x-114/17=-16

5x=-16+114/17

5x=(-272+114)/17

x= -158/85

Therefore, the solution to the given system of equations is (-158/85, 57/17).

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Answer:

[tex]\textsf{1)} \quad f(x)=-\dfrac{1}{4}(x-2)^2+\dfrac{1}{2}[/tex]

[tex]\textsf{2)} \quad g(x)=-\dfrac{1}{4}(x+2)^2+\dfrac{1}{2}[/tex]

[tex]\textsf{3)} \quad h(x)=\dfrac{1}{4}(x-2)^2-\dfrac{1}{2}[/tex]

[tex]\textsf{4)} \quad p(x)=\dfrac{1}{4}(x+2)^2-\dfrac{1}{2}[/tex]

Step-by-step explanation:

From inspection of the given graph, function f(x) is a parabola that opens downwards. Its vertex is (2, 1/2) and its y-intercept is (0, -1/2).

To determine the equation of f(x), we can use the vertex formula:

[tex]\boxed{\begin{minipage}{5.6 cm}\underline{Vertex form of a quadratic equation}\\\\$y=a(x-h)^2+k$\\\\where:\\ \phantom{ww}$\bullet$ $(h,k)$ is the vertex. \\ \phantom{ww}$\bullet$ $a$ is some constant.\\\end{minipage}}[/tex]

Substitute the vertex (2, 1/2) and the y-intercept (0, -1/2) into the formula to determine the value of a:

[tex]\begin{aligned}y&=a(x-h)^2+k\\\\\implies -\dfrac{1}{2}&=a(0-2)^2+\dfrac{1}{2}\\\\-1&=4a\\\\a&=-\dfrac{1}{4}\end{aligned}[/tex]

Substitute the found value of a and the vertex into the formula to create an equation for f(x):

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

[tex]\hrulefill[/tex]

To reflect a function in the y-axis, negate the x-value of each point, but leave the y-value the same:

Therefore, if g(x) is f(x) reflected in the y-axis, replace x with -x:

[tex]\begin{aligned}g(x)&=f(-x)\\\\&=-\dfrac{1}{4}(-x-2)^2+\dfrac{1}{2}\\\\&=-\dfrac{1}{4}(x+2)^2+\dfrac{1}{2}\end{aligned}[/tex]

The vertex of g(x) is (-2, 1/2) and its y-intercept is (0, -1/2).

[tex]\hrulefill[/tex]

To reflect a function in the x-axis, negate the y-value of each point, but leave the x-value the same:

Therefore, if h(x) is f(x) reflected in the x-axis, then:

[tex]\begin{aligned}h(x)&=-f(x)\\\\&=-\left(-\dfrac{1}{4}(x-2)^2+\dfrac{1}{2}\right)\\\\&=\dfrac{1}{4}(x-2)^2-\dfrac{1}{2}\end{aligned}[/tex]

The leading coefficient is positive, so the parabola opens upwards.

The vertex of h(x) is (2, -1/2) and its y-intercept is (0, 1/2).

[tex]\hrulefill[/tex]

If p(x) is g(x) reflected in the y-axis, then:

[tex]\begin{aligned}p(x)&=-g(x)\\\\&=-\left(-\dfrac{1}{4}(-x-2)^2+\dfrac{1}{2}\right)\\\\&=\dfrac{1}{4}(-x-2)^2-\dfrac{1}{2}\\\\&=\dfrac{1}{4}(x+2)^2-\dfrac{1}{2}\end{aligned}[/tex]

The leading coefficient is positive, so the parabola opens upwards.

The vertex of p(x) is (-2, -1/2) and its y-intercept is (0, 1/2).

The area of the given rectangular frame is: 352 in²

The formula for the area of a rectangle is expressed as:

A = L * W

Where:

A is Area

L is Length

W is width

Now, we are told that the parameters are:

Width of frame = 16 inches

Length of frame = 22 inches

Thus:

Area of frame = 22 * 16

Area = 352 in²

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Harvey bought a frame in which he put his family's picture. Frame is 16 inches wide and 22 inches length. Picture area is 12 inches wide and 18 inches length. What is the area of the frame

Answer:

The 10th term of the sequence is 60.

Step-by-step explanation:

The given sequence is arithmetic because each term is obtained by adding 6 to the previous term. To find the 10th term, we use the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n-1)d

where a_1 is the first term, d is the common difference, and n is the term number we want to find.

For this sequence, a_1 = 6 and d = 6, so we have:

a_10 = 6 + (10-1)6

a_10 = 6 + 54

a_10 = 60

Therefore, the 10th term of the sequence is 60.

The arc length of arc AB is approximately 8.3076π units.

To find the arc length of arc AB, we need to know the angle subtended by the arc and the radius of the circle. In this case, we are given that angle AB is 115 degrees and the radius BC is 13 units.

The formula for the arc length (L) of a circle is given by L = (θ/360) * 2πr, where θ is the central angle in degrees and r is the radius of the circle.

Substituting the given values into the formula, we have:

L = (115/360) × 2π(13)

Simplifying this expression:

L = (0.3194) × 2π(13)

L ≈ 0.3194 × 26π

L ≈ 8.3076π

So, the arc length of arc AB is approximately 8.3076π units.

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The logarithmic eqaution log₂ 4 + log₂ (c-9) = 5 has its solution to be 17

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

log₂ 4 + log₂ (c-9) = 5

Apply the product rule of logarithm

So, we have

log₂(4 * (c-9)) = 5

This can then be expressed as

(4 * (c-9)) = 2⁵

Divide both sides by 4

c - 9 = 2³

So, we have

c - 9 = 8

When evaluated, we have

c = 17

Hence, the logarithmic eqaution has its solution to be 17

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The comparison of each expression is completed as shown below

7/5 x 11 / 10 > 7 / 5

7/5 x 3 / 3 = 7 / 5

7/5 x 4 / 9 < 7 / 5

In mathematics, inequality signs are symbols used to compare the relative size or value of two numbers or expressions. The most common inequality signs are:

Greater than: ">"

This symbol indicates that the number or expression on the left side is greater than the number or expression on the right side. For example, 5 > 3 means "5 is greater than 3."

Less than: "<"

This symbol indicates that the number or expression on the left side is less than the number or expression on the right side. For example, 2 < 7 means "2 is less than 7."

Greater than or equal to: "≥"

This symbol indicates that the number or expression on the left side is greater than or equal to the number or expression on the right side. For example, 4 ≥ 4 means "4 is greater than or equal to 4."

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The value of angle QRU is 102° and the value of angle VUW is 72°.

As we can see, the line QRS and the Line TUv are parallel to each other,

So, the angle ∠wuv and ∠urs are the same side interior angle,

So ∠wuv =∠urs

Also from the properties of the supplementary angles,

∠urs and ∠QRu are supplementary angles,

Thus,

∠urs +∠QRu = 180

4x + 6x = 180

10x = 180

x = 18 degrees

Therefore, the value of angle QRU = 18 * 6 = 102 degrees

and the value of angle VUW = 4 *18 = 72 degrees.

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The first quartile is 19, the second quartile (median) is 25, and the third quartile is 39 for the given data set A.

We have,

To find the first quartile, second quartile (which is also the median), and third quartile of a data set, we need to arrange the data in ascending order first.

Arranging the data set A = {15, 19, 23, 25, 37, 39, 43} in ascending order:

15, 19, 23, 25, 37, 39, 43

First, let's find the second quartile (median):

Since there are 7 data points, the median will be the value in the middle, which is the fourth value in the sorted data set:

Median = 25

Next, let's find the first quartile:

To find the first quartile, we need to find the median of the lower half of the data set. In this case, the lower half of the data set is {15, 19, 23}:

Since there are 3 data points in the lower half, the median will be the second value in the lower half:

First Quartile = 19

Finally, let's find the third quartile:

To find the third quartile, we need to find the median of the upper half of the data set. In this case, the upper half of the data set is {37, 39, 43}:

Since there are 3 data points in the upper half, the median will be the second value in the upper half:

Third Quartile = 39

Therefore,

The first quartile is 19, the second quartile (median) is 25, and the third quartile is 39 for the given data set A.

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The probability that a randomly chosen battery will last more than 3.664 years is 0.9673.

(a) The probability that a randomly chosen battery will last fewer than 5 years is 0.8413. This is calculated by subtracting the cumulative probability of 4.2 years (the mean life of the battery) from the cumulative probability of 5 years. We can calculate the cumulative probability using the z-score formula: z = (x - μ) / σ, where μ is the mean and σ is the standard deviation. We can also use the standard normal table to find the cumulative probability.

Z = (5 - 4.2) / 0.8 = 0.75

Cumulative probability of 5 years = 0.7734 (from the standard normal table)

Cumulative probability of 4.2 years = 0.7257 (from the standard normal table)

Therefore, the probability that a randomly chosen battery will last fewer than 5 years = 0.7734 - 0.7257 = 0.8413

(b) The probability that a randomly chosen battery will last between 1.8 and 6.6 years is 0.9545. This is calculated by subtracting the cumulative probability of 1.8 years from the cumulative probability of 6.6 years.

Z = (1.8 - 4.2) / 0.8 = -1.5

Cumulative probability of 1.8 years = 0.0668 (from the standard normal table)

Z = (6.6 - 4.2) / 0.8 = 1.75

Cumulative probability of 6.6 years = 0.9619 (from the standard normal table)

Therefore, the probability that a randomly chosen battery will last between 1.8 and 6.6 years = 0.9619 - 0.0668 = 0.9545

(c) The probability that a randomly chosen battery will last more than 3.664 years is 0.9673. This is calculated by subtracting the cumulative probability of 3.664 years from the cumulative probability of 1.

Z = (3.664 - 4.2) / 0.8 = -0.55

Cumulative probability of 3.664 years = 0.7118 (from the standard normal table)

Cumulative probability of 1 = 0.8413 (from the standard normal table)

Thus, 0.8413 - 0.7118 = 0.9673

Therefore, the probability that a randomly chosen battery will last more than 3.664 years is 0.9673.

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Answer:

8750

Step-by-step explanation:

The solutions to the equation 8x^3 - 1 = 0 are x = 1/2, x = (-2 + 2√2i)/8, and x = (-2 - 2√2i)/8.None of the given Values is NOT a solution of the equation.

The solution(s) of the equation 8x^3 - 1 = 0, we need to determine the values of x that satisfy the equation. We can solve this equation by setting it equal to zero and factoring:

8x^3 - 1 = 0

(2x)^3 - 1^3 = 0

(2x - 1)(4x^2 + 2x + 1) = 0

Now we can find the values of x that make each factor equal to zero:

2x - 1 = 0

x = 1/2

4x^2 + 2x + 1 = 0

Using the quadratic formula, we can solve for x and find two additional solutions:

x = (-2 ± √(-8))/8

x = (-2 ± 2√2i)/8

Therefore, the solutions to the equation 8x^3 - 1 = 0 are x = 1/2, x = (-2 + 2√2i)/8, and x = (-2 - 2√2i)/8.None of the given values is NOT a solution of the equation.

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Amanda will catch up to Ben after approximately 36 minutes.

To determine how long it will take for Amanda to catch up to Ben, we can set up an equation based on their respective speeds and the time difference.

the distance traveled by Ben is given by:

Distance = Speed * Time

Distance = 2 mi/h * (t + 1.5) h

the distance traveled by Amanda is given by:

Distance = Speed * Time

Distance = 7 mi/h * t h

equating both distances

2(t + 1.5) = 7t

2t + 3 = 7t

3 = 5t

t = 3/5

Therefore, it will take Amanda 3/5 of an hour (or 36 minutes) to catch up to Ben.

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a) The fixed cost for ABC Company in the year 2003 was Br. 600,000.

b) The variable expense at the break-even point for 2003 was Br. 375,000.

The break-even point refers to the sales units or revenue at which the total sales revenue equals the total costs (variable and fixed).

At the break-even point, ABC Company does not generate any profit or incur a loss.

Sales price per unit = Br. 13

Variable cost per unit = Br. 5

Contribution margin per unit = Br. 8 (Br. 13 - Br. 5)

Contribution margin ratio = 0.6154 (Br. 8 ÷ Br. 13)

Sales revenue for 2003 = Br. 1,040,000

Sales units for 2003 = 80,000 units (Br. 1,040,000 ÷ Br. 13)

Break-even units for 2003 = 75,000 units (80,000 - 5,000)

Break-even sales revenue = Br. 975,000 (Br. 13 x 75,000)

Variable expense at the break-even point = Br. 375,000 (Br. 5 x 75,000)

Fixed expenses = Br. 600,000 (Br. 975,000 - Br. 375,000)

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It  had started at (0,120) and ran for 40 minutes, Coordinates would be approximately (60.00, 103.92).

a) To find the coordinates after 20 minutes of running counter-clockwise starting at (120,0), we need to determine the distance covered in that time period.

Since you are running at a speed of 3 meters per second, in 20 minutes (or 20 * 60 = 1200 seconds), you would have covered a distance of 3 * 1200 = 3600 meters.

Given that the equation of the circular track is x^2 + y^2 = 14400, which represents a circle with a radius of 120 units (since the radius squared is 14400), we can determine your new position on the track.

Starting from (120,0) and moving counter-clockwise, you would have traveled a distance of 3600 meters along the circumference of the circle. This corresponds to a fraction of 3600 / (2 * π * 120) = 15 / π of the circle's circumference.

Using this fraction, we can calculate the angle in radians by multiplying it by 2π. Therefore, the angle is 2π * (15 / π) = 30 radians.

To find the new coordinates, we can use the polar coordinates formula:

x = r * cos(θ)

y = r * sin(θ)

Plugging in the values:

x = 120 * cos(30) ≈ 103.92

y = 120 * sin(30) ≈ 60.00

So, after 20 minutes of running counter-clockwise, your coordinates would be approximately (103.92, 60.00).

b) If you had started at (0,120) and ran for 40 minutes, the process is similar. Since you are running at a speed of 3 meters per second, in 40 minutes (or 40 * 60 = 2400 seconds), you would have covered a distance of 3 * 2400 = 7200 meters.

Using the same circle equation x^2 + y^2 = 14400, we determine the new position on the track.

Starting from (0,120) and moving counter-clockwise, you would have traveled a distance of 7200 meters along the circumference of the circle. This corresponds to a fraction of 7200 / (2 * π * 120) = 30 / π of the circle's circumference.

The angle in radians is given by 2π * (30 / π) = 60 radians.

Using the polar coordinates formula:

x = r * cos(θ)

y = r * sin(θ)

Plugging in the values:

x = 120 * cos(60) ≈ 60.00

y = 120 * sin(60) ≈ 103.92

So, if you had started at (0,120) and ran for 40 minutes, your coordinates would be approximately (60.00, 103.92).

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Answer:

Function f is translated to the right 2 units, and then up 1 unit to obtain function g.

To see this, note that g(x) = (x+1)³ - 2 = (x-(-1))³ - 2. Comparing this to f(x) = (x-2)³, we see that replacing x with x+3 in f(x) gives g(x) = (x+3-2)³ = (x+1)³, so this transformation moves the graph of f(x) three units to the left. Then, adding the constant -2 to f(x) translates the graph down 2 units. Finally, adding the constant 2 to the result of the previous step translates the graph up 2 units, so the graph of g(x) is obtained by first translating the graph of f(x) to the right 2 units, and then translating it up 1 unit. Therefore, the correct answers are:

Function f is translated to the right 2 units.

Function f is translated up 1 unit.

Function f is translated to the right 2 units, and then up 1 unit to obtain function g.

To see this, note that g(x) = √x - 2 + 1 = f(x-2) + 1. This means that the graph of g(x) is obtained by translating the graph of f(x) to the right 2 units, and then translating it up 1 unit. Therefore, the correct answers are:

Function f is translated to the right 2 units.

Function f is translated up 1 unit.

Step-by-step explanation:

1/25 is the probability that a randomly selected male student plays both basketball and football

Let A be the  Male students who play football

B be Male students who play basketball

We are looking for the probability of P(A ∩ B), which represents the probability that a male student plays both basketball and football.

Using the formula for the probability of the intersection of two events:

P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

First, we need to find the probability of P(A ∪ B), which represents the probability that a male student plays either basketball or football.

Since we know the total number of male students who play either sport (90), we can calculate it as follows:

P(A ∪ B) = Total number of male students who play either basketball or football / Total number of male students

P(A ∪ B) = 90 / 200

P(A ∪ B) = 9/20

Next, we can calculate the probability of P(A) and P(B) individually:

P(A) = Number of male students who play football / Total number of male students

P(A) = 58 / 200

P(A) = 29/100

P(B) = Number of male students who play basketball / Total number of male students

P(B) = 40 / 200

P(B) = 1/5

Now, we can substitute the values into the formula to find P(A ∩ B):

P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

P(A ∩ B) = 29/100 + 1/5 - 9/20

P(A ∩ B) = 29/100 + 20/100 - 45/100

P(A ∩ B) = 4/100

P(A ∩ B) = 1/25

Therefore, the probability that a randomly selected male student plays both basketball and football is 1/25.

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The values of the trigonometric ratios are: a) - 7/5, b) -75 c ) -1/5  d ) -25/12

Trigonometric ratios are measurements of the lengths of two sides of a triangle. There are three basic trigonometric ratios: sine, cosine, and tangent.  Sine ratios are the ratios of the length of the side opposite the angle they represent over the hypotenuse

the give parameters are:

tanα = -4/3 where π/2 <α<π and sinβ = √3/2, 0<β<π/2

a)  sin(α+β)

Using the trig ratios tan = opp/adj

but from pyth. rule, 4² + 3² = h²

16+9 = h²

h = √25 = 5

Now from trig Sin(α+β)

4/5 + 3/5

=- 7/5

b) cos(α+β)

cos = ad/hypo

4/5 + 3/5

= -7/5

c) Sin(α-β)

4/5 - 3/5

-1/5

d) tan(α-β)

= -4/3 - 3/4

= (-16 - 9)/ 12

-25/12

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