Comfort Living Appliances has just released plans for a new flexible, yet light-weight vacuum cleaner. The proprietary component of this device is the pivot function of the handle. The plans have indicated that there remains the possibility of variation within this pivot function. The designers have specified that the height of this section should measure 44.550 mm in height but concede that anywhere between 44.500 and 44.645 is acceptable before the function interferes with the hose extender.

Let’s assume the designers have estimated a standard deviation for the pivot function of 0.019321 and assigned this a six-sigma process. What is the process capability? (Round the answer to 2 decimal places.)

Process capability:

Answer:

(f ￮ g)(- 1) = 11

Step-by-step explanation:

to evaluate ( f ￮ g)(- 1) find g(- 1) then substitute the value obtained into f(x)

g(- 1) means what is the value of g(x) when x = - 1

from the table when x = - 1 , g(x) = - 3

now evaluate f(- 3) , which means what is the value of f(x) when x = - 3

from the table when x = - 3 , f(x) = 11

Answer:

a. To find the velocity and acceleration of the rock at time t, we need to take the first and second derivatives of the height equation with respect to time:

s = 28t - 0.8t^2

v = ds/dt = 28 - 1.6t

a = dv/dt = -1.6

So the velocity of the rock at time t is v = 28 - 1.6t m/s, and its acceleration is a = -1.6 m/s^2.

b. To find how long it takes the rock to reach its highest point, we need to find the time at which the velocity is zero. We can set v = 28 - 1.6t = 0 and solve for t:

28 - 1.6t = 0

t = 17.5 seconds

So it takes 17.5 seconds for the rock to reach its highest point.

c. To find the maximum height reached by the rock, we can substitute t = 17.5 seconds into the height equation:

s = 28t - 0.8t^2

s = 28(17.5) - 0.8(17.5)^2

s = 245 meters

So the rock reaches a height of 245 meters.

d. To find how long it takes the rock to reach half its maximum height, we can set s = 245/2 = 122.5 meters and solve for t:

s = 28t - 0.8t^2

0.8t^2 - 28t + 122.5 = 0

t = 7.5 seconds or t = 20 seconds

So it takes the rock 7.5 seconds to reach half its maximum height.

e. To find how long the rock is aloft, we need to find the time at which it returns to the surface. We can set s = 0 and solve for t:

s = 28t - 0.8t^2

0 = 28t - 0.8t^2

t = 35 seconds

So the rock is aloft for 35 seconds.

Answer:

The mean is 22. So, as a percentage its 22%.

I don know what you're trying say for: 10) Ten Atlan h find be t Ex Fin​

a. The factored form of the expression x² - 3x - 28 is (x - 7)(x + 4).

b. The factored form of the expression x² + 3x - 28 is (x - 4)(x + 7).

c. The factored form of the expression x² + 12x - 28 is (x + 14)(x - 2).

d. The factored form of the expression x² - 28x - 60 is (x - 30)(x + 2).

a. To factorize the expression x² - 3x - 28, we can use the factoring method. We look for two numbers that multiply to give the constant term (-28) and add up to the coefficient of the linear term (-3). In this case, the numbers are -7 and 4.

Split the middle term (-3x) using -7x and 4x:

x² - 7x + 4x - 28

Group the terms and factor by grouping:

(x² - 7x) + (4x - 28)

x(x - 7) + 4(x - 7)

Factor out the common binomial (x - 7):

(x - 7)(x + 4)

b. For the expression x² + 3x - 28, we follow a similar process:

Split the middle term (3x) using 4x and -x:

x² + 4x - x - 28

Group the terms and factor by grouping:

(x² + 4x) + (-x - 28)

x(x + 4) - 1(x + 28)

Factor out the common binomial (x + 4):

(x + 4)(x - 7)

c. Moving on to x² + 12x - 28:

Split the middle term (12x) using 14x and -2x:

x² + 14x - 2x - 28

Group the terms and factor by grouping:

(x² + 14x) + (-2x - 28)

x(x + 14) - 2(x + 14)

Factor out the common binomial (x + 14):

(x + 14)(x - 2)

d. Finally, let's factorize x² - 28x - 60:

Split the middle term (-28x) using -30x and 2x:

x² - 30x + 2x - 60

Group the terms and factor by grouping:

(x² - 30x) + (2x - 60)

x(x - 30) + 2(x - 30)

Factor out the common binomial (x - 30):

(x - 30)(x + 2)

By following these steps, we can factorize the given expressions into their respective factored forms.

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Answer:  It would be 80

Step-by-step explanation:

Answer:D. The graph is stretched horizontally and shifted 1 unit to the right.

Step-by-step explanation: The absolute value function |s| reflects the part of the graph below the x-axis to above the x-axis, resulting in a V-shape. The modification (s - 1)|s| shifts this V-shape 1 unit to the right and stretches it horizontally by a factor of 1. Therefore, option D is the correct answer.

A total of 7 Cyclists traveled more than 45 miles. 4 cyclists traveled 50 miles or more, and 3 cyclists traveled more than 54 1/4 miles.

A line plot is drawn with the cyclists' names plotted on the x-axis and their corresponding distances covered plotted on the y-axis.

The dots are then joined to form a line, revealing the relationship between the cyclist's names and their corresponding distance traveled. The cyclist names and their corresponding distances are given in the table below.

Name Distance (in miles) Gina 40 1/2 Rey 50 Faye 35 3/4 Mark 60 John 45 1/2 James 40 1/2 Albert 40 1/4 William 54 1/4 Nathan 40 Kate 40 1/4 Mary 60 Risa 40 Abi 35 3/4 Joan 50 1/2 1. The number of cyclists is 14. 2. The longest distance covered by a cyclist is 60 miles.

Mary and Mark both covered this distance.3. A total of 7 cyclists traveled less than 50 miles. 4. The most common distance traveled by cyclists is 40 1/2 miles. Gina, James, and Albert covered this distance. 5. 2 more cyclists traveled 45 miles than 55 1/4 miles. Only William traveled 55 1/4 miles, while John and James both traveled 45 1/2 miles.6.

A total of 7 cyclists traveled more than 45 miles. 4 cyclists traveled 50 miles or more, and 3 cyclists traveled more than 54 1/4 miles.

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The statistical analysis of the data using a two-sample t-test indicates;

(a) B. Yes, the Meters On data appears to have higher speeds

A. No, there does not appear to be any outlier

(b) H₀; [tex]\mu_{on}[/tex] = [tex]\mu_{off}[/tex]

H₁; [tex]\mu_{on}[/tex] > [tex]\mu_{off}[/tex]

The P-value for the test is about 0.037

A two-sample t-test is used to determine if there is a difference between the means of two independent groups of data.

(a) The average of the data values are;

Ramp meters on;

[tex]\mu_{on}[/tex] = (29+47+55+38+32+25+42+47+51+36+55+41+43+25+47)/15 ≈ 40.87

[tex]\mu_{off}[/tex] = (25+25+43+35+36+31+48+37+19+29+22+40+36+50+40)/15 ≈ 34.4

The higher average value for the speed with the Ramp meters on indicates;

B. Yes, the Meters On data appears to have higher speeds

The five number summary are;

Ramp Meters On

Min = 25, Max = 55, Q₁ = 32, Q₂ = 42, Q₃ = 47

IQR = 47 - 32 = 15

Outlier = 47 + 1.5 × 15 = 69.5

32 - 1.5 × 15 = 9.5

Therefore, there are no outliers for the Ramp Meters On

Ramp Meters Off

Min = 19, Max = 50, Q₁ = 25, Q₂ = 36, Q₃ = 40

IQR = 40 - 25 = 15

Outlier = 40 + 1.5 × 15 = 62.5

25 - 1.5 × 15 = 2.5

Therefore, there are no outliers for the Ramp Meters Off data

(b) The null hypothesis is; H₀; [tex]\mu_{on}[/tex] = [tex]\mu_{off}[/tex]

The alternative hypothesis is; H₁; [tex]\mu_{on}[/tex] > [tex]\mu_{off}[/tex]

(c) The two sample t-test, can be obtained using the formula;

[tex]t_{cal} = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2} ) } }[/tex]

Where; [tex]\bar{x}_1[/tex] = 40.87

[tex]\bar{x}_2[/tex] = 34.40

s₁ = 9.91

s₂ = 9.20

n₁ = n₂ = 15

Therefore; [tex]t_{cal}[/tex] = 1.8516. The degrees of 15 + 15 - 2 = 28, and the one-tailed hypothesis, indicates, using an online tool;

The p-value is less than 0.05, therefore, the result is significant.

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There were 8 slices in the whole pizza.

An equation means the formula that expresses the equality of two expressions by connecting them with the equals sign =.  Let us assume the number of slices in the whole pizza is "x".

Since Jim ate two slices which is 25% of the pizza, we will set up the equation:

2 = 0.25x

To solve for "x":

2 / 0.25 = x

x = 8.

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The missing measure, h (height), for a right circular cone with a radius of 5 and a volume of 100 is approximately 3.82.

A cone is simply a 3-dimensional geometric shape with a flat base and a curved surface pointed towards the top.

The volume of a cone is expressed as;

V = (1/3) × π × r² × h

Where r is the radius of the base, h is the height of the cone and π is constant pi.

Given that:

Volume V = 100

Radius r = 5

Height h =?

Plug the given values into the above equation and solve for height h:

V = (1/3) × π × r² × h

100 = (1/3) × π × 5² × h

3 × 100 = 3 × (1/3) × π × 25 × h

300 = π × 25 × h

300 = 25π × h

h = 300 / 25π

h = 12/π

h = 3.82 units

Therefore, the height is approximately  3.82 units.

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Answer: b

Step-by-step explanation:

Answer:

∠ T = 46°

Step-by-step explanation:

the secant- secant angle T is half the difference of the measures of the intercepted arcs , that is

∠ T = [tex]\frac{1}{2}[/tex] (DR - ML) = [tex]\frac{1}{2}[/tex] (136 - 44)° = [tex]\frac{1}{2}[/tex] × 92° = 46°

The function g(x) = x + 3 is a component of the composite function.

In this question we find the case of a composite function, of which one of its components must be deducted. This can be done by means of algebra properties. First of all, the composition of functions is defined below:

f ° g (x) = f [g(x)]

First, write the entire function:

√(x + 5)

Second, derive the expression g(x):

√(x + 5)

√[(x + 3) + 2]

f(x) = √(x + 2), g(x) = x + 3

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There are a total of 45,144,000 (3 x 26 x 26 x 10 x 10) different character possibilities given the provided criteria.

To calculate the number of character possibilities, we'll break down each criterion and multiply the number of options for each position.

For the first position, we have 3 possibilities: the digits 1, 2, or 3.

For the second and third positions, we have 26 possibilities each, representing the 26 letters of the alphabet.

For the fourth and fifth positions, we have 10 possibilities each, representing the digits 0 through 9.

To find the total number of possibilities, we multiply the number of options for each position:

3 x 26 x 26 x 10 x 10 = 45,144,000.

Therefore, there are a total of 45,144,000 different character possibilities when considering the given criteria of having the options of 1, 2, or 3 for the first position, the 26 letters of the alphabet for the second and third positions, and the digits 0 through 9 for the fourth and fifth positions.

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There is no solution for the given Angle measures and side length of the right-angled triangle.

In a right-angled triangle ABC, the given angle measures are:∠A = 40°, ∠C = 90° and AB = 7.5 cm.Law of sines:Sine rule, also known as the law of sines, is a theorem in trigonometry that defines the relationship between the angles and sides of a triangle. This law is generally used in trigonometry when we have a triangle with angle measures and a side length which are known or have been given.

Law of cosines:The law of cosines, also known as the cosine formula or the cosine rule, relates to the lengths of the sides of a triangle to the cosine of one of its angles. In geometry, it is a statement that explains the relationship between the sides and angles of a triangle.Let ∠B = x

We have the side adjacent to ∠A which is AB and the side opposite to ∠B which is BC.For the ∆ABC using sine rule:AB/sin ∠C = BC/sin ∠A=> AB/sin 90 = BC/sin 40=> AB = BC.sin 40⇒ AB = 7.5 sin 40 ≈ 4.83 cmNow we can use cosine rule:AC² = AB² + BC² - 2 AB BC cos ∠C=> AC² = 7.5² + BC² - 2 × 7.5 × BC × 0=> AC² = 7.5² + BC²⇒ AC² = 56.25 + BC²Also, we have the angle opposite to the hypotenuse of the right-angled triangle.

Using the cosine rule:cos ∠C = AB/ACcos 90 = AB/AC=> 0 = AB/AC⇒ AB = 0 (which is not possible)

Therefore, there is no solution for the given angle measures and side length of the right-angled triangle.

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Joe made the first mistake in the proof by making the assumption that line l is parallel to AB without providing any evidence or justification for this claim. This suspicion subverts the ensuing thinking and conclusion of the proof.

Joe's first error in his proof was assuming, without any evidence or justification, that line l is parallel to line AB. In order to construct a parallel line, specific information or conditions are required, such as the use of particular construction techniques or predetermined angles.

Joe's proof's subsequent reasoning and conclusions become questionable if the parallel relationship between the lines is not established. For the purpose of establishing that the sum of the interior angles of the triangle equals 180 degrees, the congruence or equality of angles must be demonstrated using the parallel line assumption.

Joe's proof cannot be considered accurate or valid because it lacks a solid logical foundation for the parallel line assumption.

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To configure the emergency trauma center within the given 1,500 square feet, consider the dimensions of equipment items such as patient beds, medical carts, examination tables, imaging equipment, surgical equipment, workstations, and storage areas.

To configure the emergency trauma center within the given 1,500 square feet, we need to consider the dimensions of the equipment items and plan an efficient layout. Here is a list of equipment items and their dimensions:

Patient beds: These typically have dimensions of around 3 feet by 6 feet. Depending on the number of beds required, we can allocate sufficient space for them, keeping in mind the need for walkways around each bed.

Medical carts: These can vary in size, but a common dimension is around 2 feet by 3 feet. Allocate a designated area or storage space for the medical carts, ensuring easy access and mobility.

Examination tables: These usually have dimensions of approximately 2.5 feet by 5 feet. Plan a separate area for examination rooms where the tables can be placed along with other necessary equipment.

Imaging equipment: Such as X-ray machines or CT scanners, which may require a designated room or area with specific shielding and safety considerations. Ensure enough space for patient accessibility and the equipment's operation.

Surgical equipment: Depending on the specific surgical procedures performed, allocate adequate space for surgical tables, anesthesia carts, surgical instruments, and any required sterilization equipment.

Workstations: Create workstations for medical staff, including computers, desks, and storage areas for paperwork and supplies. Consider the need for privacy and comfortable working conditions.

Storage areas: Allocate space for storing medical supplies, medications, and other equipment that may not be immediately needed but should be easily accessible.

When planning the layout, consider factors such as patient flow, accessibility, and safety regulations. It's essential to create a functional and efficient floor plan that allows for smooth operations and optimal patient care within the given square footage.

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The value of the angle B from the trigonometric ratios is 53.13 degrees

The Pythagorean theorem is a powerful tool for solving various problems involving right triangles. It allows us to find the length of a missing side in a right triangle when the lengths of the other two sides are known. It is also used to identify whether a triangle is a right triangle or not.

We have that;

TanB = 4/3

B= Tan-1(4/3)

B = 53.13 degrees

Hence we are going to have by the use of tan that the angle is 53.1 degrees

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Answer: D. Rhombus

Step-by-step explanation: Based on the given image, the quadrilateral can be classified as a rhombus. A rhombus is a quadrilateral with all sides of equal length. In the given image, all four sides of the quadrilateral appear to be congruent, which satisfies the definition of a rhombus.

Answer:

the answer is 16 please give me brainlest

The ratios expressed in their simplest form are as follows:

Simplifying ratios involves reducing the given numbers to their lowest possible terms. Let's break down the calculations for each ratio:

1. Simplifying 15:35:

To find the greatest common divisor (GCD) of 15 and 35, we look for the largest number that evenly divides both 15 and 35.

The factors of 15 are: 1, 3, 5, 15.

The factors of 35 are: 1, 5, 7, 35.

The largest common factor between 15 and 35 is 5.

Divide both numbers by the GCD:

15 ÷ 5 = 3

35 ÷ 5 = 7

The simplified ratio is 3:7.

2. Simplifying 49:35:

Finding the GCD of 49 and 35:

The factors of 49 are: 1, 7, 49.

The factors of 35 are: 1, 5, 7, 35.

The largest common factor between 49 and 35 is 7.

Divide both numbers by the GCD:

49 ÷ 7 = 7

35 ÷ 7 = 5

The simplified ratio is 7:5.

3. Simplifying 1.3:45:

Since 1.3 is a decimal, it cannot be simplified further. The ratio remains as 1.3:45.

4. Simplifying 5mm:10cm:

Since 1cm is equivalent to 10mm, we can convert 10cm to 100mm.

Divide both the numerator and denominator by 5:

5mm ÷ 5 = 1mm

100mm ÷ 5 = 20mm

The simplified ratio is 1:20.

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Answer:

64

Step-by-step explanation:

given 2 similar figures with ratio of heights a : b , then

ratio of volumes = a³ : b³

here ratio of heights is 1 : 4

ratio of volumes = 1³ : 4³ = 1 : 64

that is the volume of B is 64 times larger than the volume of A

Answer:

x = 52.7°

Step-by-step explanation:

To find the value of x in the given right triangle, we can use the tangent trigonometric ratio.

[tex]\boxed{\begin{minipage}{7 cm}\underline{Tangent trigonometric ratio} \\\\$\sf \tan(\theta)=\dfrac{O}{A}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\end{minipage}}[/tex]

The unknown angle is x, so θ = x.

The side opposite the angle measures 21 units, so O = 21.

The side adjacent the angle measures 16 units, so A = 16.

Substitute the values into the ratio and solve for x:

[tex]\tan x=\dfrac{21}{16}[/tex]

     [tex]x=\tan^{-1}\left(\dfrac{21}{16}\right)[/tex]

     [tex]x=52.6960517...[/tex]

     [tex]x=52.7^{\circ}\; \sf (nearest\;tenth)[/tex]

Therefore, the value of x is 52.7°.

The equation for the missing angle is

Tan x = 21 / 16

The problem will be solved using Trigonometry - SOH CAH TOA

SOH for Sin = Opposite / Hypotenuse

CAH for Cos = Adjacent / Hypotenuse

TOA for Tan = Opposite / Adjacent

The given part is

opposite  = 21

Adjacent = 16

using TOA

Tan x = 21 / 16

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The surface area of the figures are

Pyramid = 96 ft²

Square base prism = 552 ft²

Cube = 216 ft²

Total surface area = 756 ft²

To find the surface area of different shapes, you can use specific formulas depending on the shape.

Here are the formulas used for each shape:

Pyramid:

Surface Area = Base Area + (1/2 × Perimeter × Slant Height)

Base Area =  6² = 36 ft²

Perimeter = 4(6) = 24 ft

Slant Height = √3² + 4² = √9 + 16 = √25 = 5 ft

Surface Area = 36 + (1 / 2 × 24 × 5) = 96 ft²

Square base prism:

Surface Area = 2(Base Area) + (Perimeter of Base × Height)

Base Area = 36

Perimeter of Base = 4(6)= 24

Height = 20

Surface Area = 2(36) + (4 * 6) * 20 = 552 ft²

Cube

The surface area of the cube  = 6 x length² = 6 × 6² = 6 × 36 = 216 ft²

Total surface area

Total surface area = 96  ft² + 552  ft² + 216  ft²= 864 ft²

corrected total surface area = 864 ft² - 36 ft²  - 36 ft² - 36 ft² = 756 ft²

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Answer:

[tex]\sin(2\arccos(-\frac{3}{5}))=-\frac{24}{25}[/tex]

Step-by-step explanation:

Let [tex]\sin(2\arccos(-\frac{3}{5}))=\sin(2\theta)=2\sin\theta\cos\theta[/tex] and [tex]\theta=\arccos(-\frac{3}{5})[/tex] so that [tex]\cos\theta=-\frac{3}{5}[/tex]. Now we'll need to find [tex]\sin\theta[/tex] having known [tex]\cos\theta[/tex]:

[tex]\displaystyle \cos\theta=\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{-3}{5}\\\\\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{\sqrt{5^2-(-3)^2}}{5}=\frac{\sqrt{25-9}}{5}=\frac{\sqrt{16}}{5}=\frac{4}{5}[/tex]

Therefore, [tex]2\sin\theta\cos\theta=2(\frac{4}{5})(-\frac{3}{5})=2(\frac{-12}{25})=-\frac{24}{25}[/tex], which makes [tex]\sin(2\arccos(-\frac{3}{5}))=-\frac{24}{25}[/tex]

The trigonometric identity cos a(2sec a + tan a)(sec a - 2tan a) = 2cos a - 3tan a

Trigonometric identities are equations that contain trigonometric ratios.

Given the trigonometric identity cos a(2sec a + tan a)(sec a - 2tan a) = 2cos a - 3tan a, we need to show that Left hand side (L.H.S) equals Right hand side (R.H.S). We proceed as follows

L.H.S =  cos a(2sec a + tan a)(sec a - 2tan a)

Expanding the brackets, we have that

cos a(2sec a + tan a)(sec a - 2tan a) =  cos a(2sec² a -  4tan asec a + tan a sec a - 2tan² a)

= cos a(2sec² a -  3tan a sec a - 2tan² a)

Now using the trigonometric identity 1 + tan² a = sec² a, we have that

cos a(2sec² a -  3tan a sec a - 2tan² a) = cos a(2(1 + tan² a) -  3tan a sec a - 2tan² a)

= cos a(2 + 2tan² a -  3tan a sec a - 2tan² a)

Collecting like terms in the equation, we have that

= cos a(2 + 2tan² a - 2tan² a -  3tan a sec a)

= cos a(2 + 0 -  3tan a sec a)

= cos a(2 -  3tan a sec a)

Expanding the bracket, we have that

= 2cos a -  cos a × 3tan a sec a

=  2cos a -  cos a × 3tan a/cos a  (since sec a = 1/cos a)

= 2cos a - 1 × 3tan a

= 2cos a - 3tan a

= R.H.S

So, since L.H.S = R.H.S,

cos a(2sec a + tan a)(sec a - 2tan a) = 2cos a - 3tan a

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[tex](x - 1)(x - 2)(x - 6) \\ ({x}^{2} - 2x - x + 2)(x - 6) \\ ( {x}^{2} - 3x + 2)(x - 6) \\ {x}^{3} - 6 {x}^{2} - 3 {x}^{2} + 18x + 2x - 12 \\ {x}^{3} - 9 {x}^{2} + 20x - 12 [/tex]

PLEASE GIVE ME BRAINLIEST

Where the above mean and standard deviation is given, the approximately 47.72% of the years have a snowfall between 31 and 41 inches in this town. Thus the nearest answer is (Option B)

Standardizing the values of 31 and 41 inches, we must use the following formula

Z = (X - μ) / σ

Where -  

For 31 inches -  

Z1 = (31 - 41) / 5 = -2

For 41 inches  -

Z2 = (41 - 41) / 5 = 0

The probability of the snowfall being   less than or equal to 31 inches is the same as the probability ofthe Z-score being less than or equal to -2, which is approximately 0.0228.

P  (31 ≤ X ≤41) = P(X ≤ 41) - P (X  ≤ 31)

= 0.5 -0.0228

= 0.4772

Finally, we convert this probability to a percentage by multiplying by 100  -

Percentage = 0.4772 * 100

= 47.72%

Hence , approximately 47.72% of the years have a snowfall between 31 and 41 inches in this town.

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Vertically stretch the cubic parent function, we multiply the function by the vertical stretch factor, a. The new equation of the function is y = a(x^3).  

If you vertically stretch the cubic parent function, the equation of the new function is y = a(x^3) where a is the vertical stretch factor. A cubic parent function is a polynomial function with a degree of three and an equation of y = x^3.

A vertical stretch is a transformation that increases the distance between each point on a graph and the x-axis. This transformation does not change the shape of the graph, but it affects the height of the graph.

To vertically stretch the cubic parent function, we multiply the function by a vertical stretch factor, a. Therefore, the equation of the new function is y = a(x^3).

For example, let's say we want to vertically stretch the cubic parent function y = x^3 by a factor of 2. To do this, we multiply the function by 2, which gives us y = 2(x^3). This means that the new function is twice as tall as the original function.

On the other hand, if we want to vertically shrink the cubic parent function y = x^3 by a factor of 1/2, we divide the function by 1/2. This gives us y = (1/2)(x^3), which means that the new function is half as tall as the original function.

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The value of the integral is [tex]\int\limits^{10}_4 {f(x)} \, dx = 6[/tex]

The average rate of change over the interval [4, 10] is 3/2

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

[tex]f(x) = \left\{ \begin{array}{lr} -4, & \text{if } 4 \le x < 7\\ 5, & \text{if } 7 \le x < 10 \end{array}[/tex]

The integral expression is given as

[tex]\int\limits^{10}_4 {f(x)} \, dx[/tex]

So, we have

[tex]\int\limits^{10}_4 {f(x)} \, dx = \int\limits^{10}_4 {[5 - 4]} \, dx[/tex]

Evaluate the difference

[tex]\int\limits^{10}_4 {f(x)} \, dx = \int\limits^{10}_4 {[1]} \, dx[/tex]

Integrate

[tex]\int\limits^{10}_4 {f(x)} \, dx = [x]\limits^{10}_4[/tex]

Expand

[tex]\int\limits^{10}_4 {f(x)} \, dx = 10 - 4[/tex]

Evaluate the difference

[tex]\int\limits^{10}_4 {f(x)} \, dx = 6[/tex]

Here, we have

[tex]f(x) = \left\{ \begin{array}{lr} -4, & \text{if } 4 \le x < 7\\ 5, & \text{if } 7 \le x < 10 \end{array}[/tex]

The interval is [4, 10]

So, we have

Average rate = (5 + 4)/(10 - 4)

Evaluate

Average rate = 3/2

Hence, the average rate of change is 3/2

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