Triangle ABC has the following coordinates: A=(5,-5), B=(3,-3), C=(5,-3) What are the coordinates of triangle A'B'C' if it is created by dilating triangle ABC with the origin (0,0) as the center of dilation and with a scale factor of 3?

The probability that a randomly student at the university has a commuting time between ​​​​ 55 and 70 mins is about 0.56. Given information:The commuting time (the average number of hours spent commuting each week) for students at a particular university is normally distributed with a mean of 63 mins and standard deviation of 9.61.

Find: We are to determine the probability that a randomly student at the university has a commuting time between ​​​​ 55 and 70 mins.

Here,μ = 63 min σ = 9.61 min. We have to find the probability of a random student has commuting time between 55 and 70 min. That is P(55 ≤ X ≤ 70).First, we need to convert the given range to Standard Normal Distribution form.i.e., z-score for X = 55 and X = 70.Z-score formula:z = (X - μ) / σFor X = 55z = (55 - 63) / 9.61z = -0.83For X = 70z = (70 - 63) / 9.61z = 0.73. We need to find the probability of a random student has a z-score between -0.83 and 0.73.P(-0.83 < z < 0.73)

Using standard normal table or calculator, we can find the probability P(-0.83 < z < 0.73) = P(z < 0.73) - P(z < -0.83)= 0.7665 - 0.2033≈ 0.56

Thus, the probability that a randomly student at the university has a commuting time between ​​​​ 55 and 70 mins is about 0.56.

For more question on probability

https://brainly.com/question/30390037

#SPJ8

The limit of the expression lim (x² - √x⁴ + 3x²) as x approaches any value is indeterminate (∞ - ∞), except when x approaches zero, where the limit is 0.

To find the limit of the expression lim (x² - √x⁴ + 3x²) as x approaches a certain value, we can simplify the expression and evaluate the limit.

First, let's simplify the expression:

lim (x² - √x⁴ + 3x²)

= lim (4x² - x² - √x⁴)

= lim (3x² - √x⁴)

Now, let's consider the behavior of the expression as x approaches a value.

As x approaches any finite value, the term 3x² will approach a finite value.

For the term √x⁴, as x approaches a finite value, the square root of x⁴ will approach the absolute value of x².

Therefore, the limit becomes:

lim (3x² - √x⁴) = lim (3x² - |x²|)

Next, let's consider the different cases as x approaches positive infinity, negative infinity, and zero.

1. As x approaches positive infinity, the term 3x² will tend to positive infinity, and |x²| will also tend to positive infinity. Thus, the expression becomes:

lim (3x² - |x²|) = lim (∞ - ∞)

In this case, the limit is indeterminate (∞ - ∞).

2. As x approaches negative infinity, the term 3x² will tend to positive infinity, and |x²| will also tend to positive infinity. Thus, the expression becomes:

lim (3x² - |x²|) = lim (∞ - ∞)

Again, in this case, the limit is indeterminate (∞ - ∞).

3. As x approaches zero, the term 3x² will tend to zero, and |x²| will also tend to zero. Thus, the expression becomes:

lim (3x² - |x²|) = lim (0 - 0) = 0

Therefore, the limit of the expression lim (x² - √x⁴ + 3x²) as x approaches any value is indeterminate (∞ - ∞), except when x approaches zero, where the limit is 0.

learn more about limit: https://brainly.com/question/29412132

#SPJ1

The linear function that has the greatest y-intercept is [tex]y = 3x + 4[/tex].

In a linear equation, the y-intercept is where the line crosses the y-axis.

It is represented by the constant term in the equation.

So, to determine which linear function has the greatest y-intercept, we need to look at the constant term of each equation.

Let's consider each equation: [tex]y = 6x + 1[/tex]

The constant term in this equation is 1.

So, the y-intercept is 1.

On a coordinate plane, a line goes through points (0, 2) and (5, 0).

To find the equation of this line, we can use the point-slope form:

[tex]y - y1 = m(x - x1)[/tex]

where m is the slope and (x1, y1) is a point on the line.

Using the points (0, 2) and (5, 0), we get:

[tex]m = \frac{(0 - 2)}{(5 - 0)} =-\frac{2}{5}[/tex]

So, the equation of the line is:

[tex]y - 2 = (\frac{-2}{5} )(x - 0)[/tex]

[tex]y = (\frac{-2}{5} )x + 2[/tex]

The constant term in this equation is 2.

So, the y-intercept is 2.

On a coordinate plane, a line goes through points (1, 2) and (0, -3).

To find the equation of this line, we can use the point-slope form:

[tex]y - y1 = m(x - x1)[/tex]

where m is the slope and (x1, y1) is a point on the line.

Using the points (1, 2) and (0, -3), we get:

[tex]m = \frac{ (-3 - 2) }{(0 - 1)} = -5[/tex]

So, the equation of the line is:

[tex]y - 2 = (-5)(x - 1)y = -5x + 7[/tex]

The constant term in this equation is 7.

So, the y-intercept is 7.

[tex]y = 3x + 4[/tex]

The constant term in this equation is 4.

So, the y-intercept is 4.

Therefore, we can see that the linear function that has the greatest y-intercept is [tex]y = 3x + 4[/tex].

For such more questions on y-intercept

https://brainly.com/question/25722412

#SPJ8

Answer:

We don't need to worry about the displaystyle- {3} −3 anyway, because we dcided in the first step that displaystyle {x}ge- {2} x ≥ −2. So the domain for this case is displaystyle {x}ge- {2}, {x}ne {3} x≥ −2,x≠ 3, which we can write as displaystyle {left [- {2}, {3}right)}cup {left ({3},inftyright)} [−2,3)∪(3,∞).

Step-by-step explanation:

Step-by-step explanation:

a)  23.6 (1.08)^x        for 2016    x = 26   ('x' is the number of years past 1990)

        23.6 (1.08)^(26) = 174.6  billion

b)  109 = 23.6 ( 1.08)^x

    4.6186 = 1.08^x

          x =   log 4.6186 / log1.08   = 19.88 yrs  

                                         means  1990 + 19.88 yrs = year 2010

For accelerated motion, changes in distance compare with equal changes in velocity as follows:

B. Distance increases faster than time.

A reason why a person will not compute a slope for the accelerated motion graph is that it graphed as a straight line.

Accelerated motion is a form of motion in which the motion is not equal and the object moving does not complete the same distances in the same intervals of time.

The graph formed in the case of an accelerated motion is not a straight line. So, a reason why a person would not compute a slope for the accelerated motion is that it graphed as a straight line.

Learn more about accelerated motion here:

https://brainly.com/question/25876659

#SPJ1

a) Option a) 9 - 1/2 is equal to 17/2.

b) Option b) 27 - 2/3 is equal to 79/3.

a) The expression 9 - 1/2 can be simplified by finding a common denominator for the terms. The common denominator for 9 and 1/2 is 2.

Multiplying 9 by 2/2, we get:

9 * (2/2) = 18/2

So, the expression 9 - 1/2 can be simplified to:

18/2 - 1/2 = 17/2

Therefore, option a) 9 - 1/2 is equal to 17/2.

b) The expression 27 - 2/3 can be simplified in a similar manner by finding a common denominator for the terms. The common denominator for 27 and 2/3 is 3.

Multiplying 27 by 3/3, we get:

27 * (3/3) = 81/3

So, the expression 27 - 2/3 can be simplified to:

81/3 - 2/3 = 79/3

Therefore, option b) 27 - 2/3 is equal to 79/3.

for such more question on simplified

https://brainly.com/question/11680269

#SPJ8

Answer:

37.98

Step-by-step explanation:

Answer:

the interest rate. : 5%

[(1365-1300)/1300]*100 = 5%

Step-by-step explanation:

Answer:

∠ MHU = 54°

Step-by-step explanation:

the angle MHU between the tangent and the secant is half the measure of the intercepted arc HM , then

∠ MHU = [tex]\frac{1}{2}[/tex] × 108° = 54°

The correct statement about the mean is:

The mean equals the sum of all observations divided by the number of observations.

The mean is a commonly used measure of central tendency. It is calculated by summing up all the observations and then dividing the sum by the total number of observations. It provides an average value that represents the typical value of the data set.

To calculate the mean, it is not necessary to order the observations from least to most. The order of the observations does not affect the mean calculation.

The mean is not necessarily a whole number. It can be a decimal or a fraction, depending on the data set and the values of the observations. The mean represents the balance point of the data set and can take on any real number value.

for such more question on mean

https://brainly.com/question/14532771

#SPJ8

a. The  slope of the curve at the point (4,16)   is approximately 1.165, and at the point (16,4)  is approximately -0.496.

b. The   curve has a horizontal tangent line at the points(0,0) and (3,27).

c. The   curve has a vertical tangent lineat the points (0,0) and (65/2, (65/2)³).

a. To find the   slope of the curve given by the equation x³ + y³ - 65xy = 0 at the points (4,16) and (16,4),we can differentiate the equation implicitly with respect to x and solve for dy/dx.

Differentiating the equation with respect to x, we have  -

3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0

To find the slope at a specific point, substitute the x and y coordinates into the equation and solve for dy/dx.

For the point (4,16)  -

3(4)² + 3(16)²(dy/dx) - 65(16) - 65(4)(dy/dx) = 0

48 + 768(dy/dx) - 1040 - 260(dy/dx) = 0

508(dy/dx) = 592

(dy/dx) = 592/508

(dy/dx) ≈ 1.165

For the point (16,4)  -

3(16)² + 3(4)²(dy/dx) - 65(4) - 65(16)(dy/dx) = 0

768 + 48(dy/dx) - 260 - 1040(dy/dx) = 0

(-992)(dy/dx) = 492

(dy/dx) = 492/(-992)

(dy/dx) ≈ -0.496

Thus, the slope of the   curve at the point (4,16) isapproximately 1.165, and at the point (16,4) is approximately -0.496.

b. To find the point where the curve has   a horizontal tangent line, we need to find the x-coordinate(s)where dy/dx equals zero.

This means   the slope is zero and the tangent line is horizontal.

From the derivative we obtained earlier  -

3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0

Setting dy/dx equal to zero  -

3x² - 65y = 0

Substituting y = x³/65 into the equation  -

3x² - 65(x³/65) = 0

3x² - x³ = 0

Factoring out an x²  -

x²(3 - x) = 0

This equation has two solutions  -  x = 0 and x = 3.

hence, the curve has a horizontal   tangent line at the points(0,0) and (3,27).

c. To find the point where the curve has a vertical tangent line, we need to find the x-coordinate(s)   where the derivative is undefinedor approaches infinity.

From the derivative  -

3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0

To find the vertical tangent line, dy/dx should be undefined or infinite. This occurs when the denominator of dy/dx is zero.

Setting the denominator equal to zero:  -

65x = 65y

x = y

Substituting this condition back into the original equation  -

x³ + x³ - 65x² = 0

2x³ - 65x² = 0

x²(2x - 65) = 0

This equation has two solutions  - x = 0 and x = 65/2.

Therefore, the curve has a vertical tangent line   at the points (0,0)

and(65/2, (65/2)³).

Learn more about slope:
https://brainly.com/question/3493733
#SPJ1

The points where the rate of change of f(x) equal to zero are A, C and E

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

The graph

The point where the rates is 0 are the points where movement is at a constant

using the above as a guide, we have the following:

The points are A, C and E

Hence, the point where the rates is 0 are A, C and E

Read more about rate of change at

https://brainly.com/question/31965106

#SPJ1

Answer:

99

Step-by-step explanation:

since the height is 9 and the base is 11 we use the formula BH=V

substitute 9x11 and get 99

The measure of the arc DGF which substends the angle DEF at the circumference of the circle is equal to 190°

The angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference.

Given that the arc DGF = (62x + 4)°

62x + 4 = 2(30x + 5)

62x + 4 = 60x + 10

62x - 60x = 10 - 4 {collect like terms}

2x = 6

x = 6/2

x = 3

arc DGF = 62(3) + 4 = 190°

Therefore, the measure of the arc DGF which substends the angle DEF at the circumference of the circle is equal to 190°

Read more about angle here:https://brainly.com/question/24423151

#SPJ1

Answer:

To write the polynomial function m(x) = x + x^2 - 1 in standard form, we rearrange the terms in descending order of degree:

m(x) = x^2 + x - 1

Polynomial name: Quadratic polynomial

Degree: 2 (the highest exponent is 2)

Leading coefficient: 1 (the coefficient of the highest-degree term)

Constant term: -1 (the term without any variable)

Answer:

180 meters

Step-by-step explanation:

To find the perimeter of the actual swimming pool, you need to first find the length and width of the actual swimming pool by multiplying the length and width of the pictured swimming pool by the scale factor of 1200 cm.

Now that we know that the perimeter of the actual pool is 18000 centimeters, we need to convert that to meters! Keep in mind that:

Now we can divide 18000 by 100:

18000 cm ÷ 100 = 180 m

Therefore, the perimeter of the actual swimming pool is 180 m.

Trent will need approximately 2.86 bottles of waterproof spray to cover his tent.

To calculate the number of bottles of waterproof spray Trent will need to cover his tent, we first need to find the surface area of the tent.

The surface area of a square-based pyramid is given by the formula:

Surface Area = Base Area + (0.5 x Perimeter of Base x Slant Height)

The base of the pyramid is a square with a side length of 14 feet, so the base area is:

Base Area = (Side Length)^2 = 14^2 = 196 square feet

To find the slant height of the pyramid, we can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by one side of the base, the height of the pyramid, and the slant height. The height of the pyramid is given as 8 feet, and half the length of the base is 7 feet.

Using the Pythagorean theorem:

[tex]Slant Height^2 = (Half Base Length)^2 + Height^2[/tex]

[tex]Slant Height^2 = 7^2 + 8^2Slant Height^2 = 49 + 64Slant Height^2 = 113Slant Height ≈ √113 ≈ 10.63 feet[/tex]

Now we can calculate the surface area of the tent:

Surface Area = 196 + (0.5 x 4 x 10.63)

Surface Area = 196 + (2 x 10.63)

Surface Area = 196 + 21.26

Surface Area ≈ 217.26 square feet

Since each bottle of waterproof spray covers 76 square feet, we can divide the total surface area of the tent by the coverage of each bottle to find the number of bottles needed:

Number of Bottles = Surface Area / Coverage per Bottle

Number of Bottles = 217.26 / 76

Number of Bottles ≈ 2.86

Therefore, Trent will need approximately 2.86 bottles of waterproof spray to cover his tent. Since we can't have a fraction of a bottle, he will need to round up to the nearest whole number. Therefore, Trent will need 3 bottles of waterproof spray to fully waterproof his tent.

for more such question on bottles visit

https://brainly.com/question/28855819

#SPJ8

Approximately 4.6 pounds of pecans are needed for one pecan pie.You should buy approximately 41.4 pounds of pecans to make 9 pecan pies.

To determine the number of pounds of pecans needed to make 9 pecan pies, we need to consider the amount of pecans required per pie and the number of pies we are making.

The recipe calls for 1 cup of pecans per pie, but we don't have a measuring cup available. However, we do have nutritional information from a bag of pecans, which states that there are 684 calories in 1 cup (99g) of pecans.

To find out how many pecans are in a pound, we can use the information that 1 cup of pecans weighs 99 grams. Since there are 454 grams in a pound, we can set up the following proportion:

1 cup (99g) = x pounds (454g)

Cross-multiplying, we get:

99g * x pounds = 1 cup * 454g

Simplifying, we have:

99x = 454

Dividing both sides by 99, we find:

x ≈ 4.5959 pounds

So, approximately 4.6 pounds of pecans are needed for one pecan pie.

Since we are making 9 pecan pies, we multiply the amount needed for one pie by the number of pies:

4.6 pounds/pie * 9 pies = 41.4 pounds

For more such questions on pounds visit:

https://brainly.com/question/12983359

#SPJ8

Answer:

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis, we can use the method of cylindrical shells.

The formula for the volume of a solid obtained by rotating a region bounded by the graph of a function f(x), the x-axis, x = a, and x = b about the y-axis is given by:

V = 2π ∫[a, b] x * f(x) dx

In this case, the function f(x) = (x - 4)^3, and the bounds of integration are a = 0 and b = 5.

Substituting these values into the formula, we have:

V = 2π ∫[0, 5] x * (x - 4)^3 dx

To evaluate this integral, we can expand the cubic term and then integrate:

V = 2π ∫[0, 5] x * (x^3 - 12x^2 + 48x - 64) dx

V = 2π ∫[0, 5] (x^4 - 12x^3 + 48x^2 - 64x) dx

Integrating each term separately:

V = 2π [1/5 x^5 - 3x^4 + 16x^3 - 32x^2] evaluated from 0 to 5

Now we can substitute the bounds of integration:

V = 2π [(1/5 * 5^5 - 3 * 5^4 + 16 * 5^3 - 32 * 5^2) - (1/5 * 0^5 - 3 * 0^4 + 16 * 0^3 - 32 * 0^2)]

Simplifying:

V = 2π [(1/5 * 3125) - 0]

V = 2π * (625/5)

V = 2π * 125

V = 250π

Therefore, the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis is 250π cubic units.

Answer: Therefore, the perimeter of the soccer field is 332 meters.

Step-by-step explanation:

To determine the perimeter of a soccer field with a length of 97 meters and a width of 69 meters, we can use the formula for the perimeter of a rectangle, which is given by:

Perimeter = 2 * (length + width)

Plugging in the values, we have:

Perimeter = 2 * (97 + 69)

Perimeter = 2 * 166

Perimeter = 332 meters

┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈

✦ The number is - 5

┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈

[tex]\begin{gathered} \; \sf{\color{pink}{Let \; the \; other \; number \; be \; (x)::}} \\ \end{gathered}[/tex]

Atq,,

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3 \times \bigg \lgroup \: x + ( - 7) \bigg \rgroup = - 36} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3 \times \bigg \lgroup \: x - 7 \bigg \rgroup = - 36} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x - 21 = - 36} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x = - 36 + 21} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x = - 15} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{x = \dfrac{\cancel{ - 15}}{\cancel{ \: 3}}} \qquad \bigg \lgroup \sf{Cancelling \: by \: 3} \bigg \rgroup \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink} :\dashrightarrow \underline{\color{pink}\boxed{\colorbox{black}{x = - 5}}} \: \pmb{\bigstar} \\ \\ \end{gathered}[/tex]

The answer is:

Work/explanation:

The product means we multiply two numbers.

Here, we multiply 3 and a number increased by -7; let that number be z.

So we have

[tex]\sf{3(z+(-7)}[/tex]

simplify:

[tex]\sf{3(z-7)}[/tex]

This equals -36

[tex]\sf{3(z-7)=-36}[/tex]

[tex]\hspace{300}\above2[/tex]

[tex]\frak{solving~for~z}[/tex]

Distribute

[tex]\sf{3z-21=-36}[/tex]

Add 21 on each side

[tex]\sf{3z=-36+21}[/tex]

[tex]\sf{3z=-15}[/tex]

Divide each side by 3

[tex]\boxed{\boxed{\sf{z=-5}}}[/tex]

The box plot that best represents the numerical data is: A. A box plot using a number line from 3 to 12.25 with tick marks every one-fourth unit. The box extends from 6.25 to 9.25 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 11. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.

In order to determine the five-number summary for the survey, we would arrange the data set in an ascending order:

4.5,6,6,6.5,7,7,7.5,8,8.5,9,9.5,10,11

Based on the information provided about the list of shoe sizes for a group of 13 people, we would use a graphical method (box plot) to determine the five-number summary for the given data set as follows:

Minimum (Min) = 4.5.

First quartile (Q₁) = 6.25.

Median (Med) = 7.5.

Third quartile (Q₃) = 9.25.

Maximum (Max) = 11.

Read more on boxplot here: brainly.com/question/29648407

#SPJ1

Answer:

$3606.44

Step-by-step explanation:

The question asks us to calculate the principal amount that needs to be invested in order to earn an interest of $6180 in 28 years at an annual interest rate of 6.12%.

To do this, we need to use the formula for simple interest:

[tex]\boxed{I = \frac{P \times R \times T}{100}}[/tex],

where:

I = interest earned

P = principal invested

R = annual interest rate

T = time

By substituting the known values into the formula above and then solving for P, we can calculate the amount that James needs to invest:

[tex]6180 = \frac{P \times 6.12 \times 28}{100}[/tex]

⇒ [tex]6180 \times 100 = P \times 171.36[/tex]     [Multiplying both sides by 100]

⇒ [tex]P = \frac{6180 \times 100}{171.36}[/tex]    [Dividing both sides of the equation by 171.36]

⇒ [tex]P = \bf 3606.44[/tex]

Therefore, James needs to invest $3606.44.

Answer:

Step-by-step explanation:

x=0.5, y=12.

x=3, y=2.

x=4, y=1.5.

x=6, y=1.

The rectangles that are similar are Rectangle 3 and Rectangle 4.

To determine which rectangles are similar, we need to compare their corresponding side lengths.

Rectangle 1:

Length: 3 cm

Height: 5 cm

Rectangle 2:

Length: 2.5 cm

Height: 5.5 cm

Rectangle 3:

Length: 2.5 cm

Height: 2 cm

Rectangle 4:

Length: 5 cm

Height: 4 cm

To determine similarity, we need to compare the ratios of the corresponding side lengths of the rectangles.

Comparing Rectangle 1 with Rectangle 2:

Length ratio: 3 cm / 2.5 cm = 1.2

Height ratio: 5 cm / 5.5 cm ≈ 0.91

The length ratio and height ratio are not equal, so Rectangle 1 and Rectangle 2 are not similar.

Comparing Rectangle 1 with Rectangle 3:

Length ratio: 3 cm / 2.5 cm = 1.2

Height ratio: 5 cm / 2 cm = 2.5

The length ratio and height ratio are not equal, so Rectangle 1 and Rectangle 3 are not similar.

Comparing Rectangle 1 with Rectangle 4:

Length ratio: 3 cm / 5 cm = 0.6

Height ratio: 5 cm / 4 cm = 1.25

The length ratio and height ratio are not equal, so Rectangle 1 and Rectangle 4 are not similar.

Comparing Rectangle 2 with Rectangle 3:

Length ratio: 2.5 cm / 2.5 cm = 1

Height ratio: 5.5 cm / 2 cm = 2.75

The length ratio and height ratio are not equal, so Rectangle 2 and Rectangle 3 are not similar.

Comparing Rectangle 2 with Rectangle 4:

Length ratio: 2.5 cm / 5 cm = 0.5

Height ratio: 5.5 cm / 4 cm = 1.375

The length ratio and height ratio are not equal, so Rectangle 2 and Rectangle 4 are not similar.

Comparing Rectangle 3 with Rectangle 4:

Length ratio: 2.5 cm / 5 cm = 0.5

Height ratio: 2 cm / 4 cm = 0.5

The length ratio and height ratio are equal, so Rectangle 3 and Rectangle 4 are similar.

Therefore, the rectangles that are similar are Rectangle 3 and Rectangle 4.

for more such question on rectangles  visit

https://brainly.com/question/2607596

#SPJ8

Answer:

KL = 6

Step-by-step explanation:

We see that the length of IL includes IJ, JK, and Kl and is 26.

Since IL = 26 and IJ + JK + KL = IL, we can subtract the sum of the lengths of IJ and Jk from IL to find KL:

IL = IJ + JK + KL

26  = 9 + 11 + KL

26 = 20 + KL

6 = KL

Thus, the length of KL is 6.

We can confirm this fact by plugging in 6 for KL and checking that we get 26 on both sides of the equation when simplifying:

IL = IJ + JK + KL

26 = 9 + 11 + 6

26 = 20 + 6

26 = 26

Thus, our answer is correct.

The step length 'a' for the line search method at point x(k) = (1, -1) with the descent direction PL = (1, 0) is 0.5.

To compute the step length 'a' using the line search method, we can follow these steps:

1: Calculate the gradient at point x(k).

  - Given x(k) = (1, -1)

  - Compute the gradient ∇f(x₁,x₂) at x(k):

    ∇f(x₁,x₂) = (∂f/∂x₁, ∂f/∂x₂)

    ∂f/∂x₁ = 3x₁ + 1

    ∂f/∂x₂ = 2x₂ - 1

    Substituting x(k) = (1, -1):

    ∂f/∂x₁ = 3(1) + 1 = 4

    ∂f/∂x₂ = 2(-1) - 1 = -3

  - Gradient at x(k): ∇f(x(k)) = (4, -3)

2: Compute the dot product between the gradient and the descent direction.

  - Given PL = (1, 0)

  - Dot product: ∇f(x(k)) ⋅ PL = (4)(1) + (-3)(0) = 4

3: Compute the norm of the descent direction.

  - Norm of PL: ||PL|| = √(1² + 0²) = √1 = 1

4: Calculate the step length 'a'.

  - Step length formula: a = -∇f(x(k)) ⋅ PL / ||PL||²

    a = -4 / (1²) = -4 / 1 = -4

5: Take the absolute value of 'a' to ensure a positive step length.

  - Absolute value: |a| = |-4| = 4

6: Finalize the step length.

  - The step length 'a' is the positive value of |-4|, which is 4.

Therefore, the step length 'a' for the line search method at point x(k) = (1, -1) with the descent direction PL = (1, 0) is 4.

For more such questions on length, click on:

https://brainly.com/question/28322552

#SPJ8

Answer:

f(x) = 4x² - 2x + 11

f'(x) = 8x - 2

m = f'(3) = 8(3) - 2 = 24 - 2 = 22

41 = 22(3) + b

41 = 66 + b

b = -25

y = 22x - 25

The graph of the linear inequality is given by the image presented at the end of the answer.

The slope-intercept equation for a linear function is presented as follows:

y = mx + b

In which:

The function for this problem is given as follows:

y = -3x/4 + 2.

Hence the graph crosses the y-axis at y = 2, and when x increases by 4, y decays by 3.

The inequality is given as follows:

y < -3x/4 + 2.

Meaning that points below the dashed line are the solution to the inequality.

More can be learned about linear functions at https://brainly.com/question/15602982

#SPJ1