A population begins with a single individual. In each generation, each individual in the population dies with probability 1/2 or doubles with probability 1/2. Let X_n denote the number of individuals in the population in the nth generation. Find the mean and variance of X_n.

The area of the figure is 18 mm².

The area is the amοunt οf space within the perimeter οf a 2D shape. It is measured in square units, such as cm², m², etc.

Yοu can think οf area as the area inside a given shape οr space. It refers tο hοw much space is taken up. The larger the shape, the larger the area and perimeter οf the shape will be. Nοt tο be cοnfused with vοlume, area οnly refers tο space taken up by a flat οr 2D οbject.

We have given the figure, with all right angles,

Draw an imaginary rectangle of 5 × 6, that covers up the whole figure.

Now,

The area of full rectangle - area of small rectangle = area of the figure

⇒ (6 × 5) - (4 × 3)

⇒ 30 - 12

⇒ 18 mm²

Thus, the area of the figure is 18 mm².

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Answer:

-1

Step-by-step explanation:

if x<0 then the value of x must be a negative number

If we substitute the value of -1 for x in the equality equation, we get.

[tex]1 < 1[/tex]

Which does not satisfy the inequality.

Hope this helps!

Brainliest is much appreciated!

Answer:

-1 because -1 + 2 < 1
1 < 1 which is false therefor not satisfying the inequality.

Answer:

You dis not add the option to check all that apply

Answer: 364 cm^3

Step-by-step explanation: Please mark brainliest and give thanks!

Answer:

[tex]364cm^3[/tex]

Hope this helps!

Brainliest and a like is much appreciated!

The expression x^4 - 16 is divided by x^3 + 2x^2 + 4x + 8 is x - 2

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

x^4 - 16 is divided by x^3 + 2x^2 + 4x + 8

Using the long division method of quotient, we have

x^3 + 2x^2 + 4x + 8 | x^4 - 16

The division steps are as follows

                                   x - 2

x^3 + 2x^2 + 4x + 8 | x^4 - 16

                                  x^4 + 2x^3 + 4x^2 + 8x

--------------------------------------------------------------

                                           -2x^3 - 4x^2 - 8x - 16

                                           -2x^3 - 4x^2 - 8x - 16

--------------------------------------------------------------

                                           0

Hence, the quotient is x - 2

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Answer: 150%

Step-by-step explanation:

Step-by-step explanation:

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Answer:

-------------------------

Use the distance formula to find the distance between two points.

Substitute coordinates and find the length of AB:

Question :-

Answer :-

[tex] \rule{200pt}{3pt}[/tex]

Solution :-

As per the provided information in the given question, we have been given that :-

To calculate the distance between the two points, we will apply the formula below :-

[tex] \bigstar \: \: \: \boxed{ \sf{ \: \: AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \: \: }}[/tex]

Substitute the given values into the above formula and solve for AB :-

[tex]\sf:\implies{ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}[/tex]

[tex]\sf:\implies{AB = \sqrt{( - 4 - 2)^2 + (0 - 4)^2}}[/tex]

[tex]\sf:\implies{AB = \sqrt{(-6)^2 + (-4)^2}}[/tex]

[tex]\sf:\implies{AB = \sqrt{36 + 16}}[/tex]

[tex]\sf:\implies\bold{AB = \sqrt{52} \approx 7.21 \: units}[/tex]

Therefore :-

[tex]\\[/tex]

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Have a great day! <33

A) To the nearest 0.1°, [tex]$\angle ABC\approx63.8\textdegree$[/tex] and B) AC is approximately 4.2 units long, correct to 1 decimal place.

A) We can use the law of cosines to solve for the angles of triangle ABC and then use the fact that opposite angles in a cyclic quadrilateral are supplementary to find angle ADC. Finally, we can use the law of cosines again to find AC.

Let angle ABC be x. Then, applying the law of cosines to triangle ABC, we have:

[tex]$AC^2=AB^2+BC^2-(2AB* BC*\cos(x))$[/tex]

Substituting the given values, we get:

[tex]$AC^2=5^2+4.5^2-2\cdot5\cdot4.5\cdot\cos(x)$[/tex]

Simplifying and solving for AC, we get:

[tex]$AC=\sqrt{3.125+11.25\cos(x)-10\cos^2(x)}$[/tex]

Next, applying the law of cosines to triangle BCD, we have:

[tex]$\cos(ADC)=\frac{3.4^2+4.5^2-(2*3.4*4.5*\cos(x))}{3.4*4.5}$[/tex]

Simplifying, we get:

[tex]$\cos(ADC)=\frac{29.15-15.3\cos(x)}{15.3}$[/tex]

Since ABCD is a cyclic quadrilateral, we have:

[tex]$\angle ADC=180\textdegree-\angle ABC=180\textdegree-x$[/tex]

Substituting this into the above equation and solving for [tex]$\cos(x)$[/tex], we get:

[tex]$\cos(x)=\frac{7.2}{15.3}$[/tex]

Using a calculator, we find that

[tex]$x\approx63.8\textdegree$[/tex].

Therefore, [tex]$\angle ADC\approx116.2\textdegree$[/tex].

B) Finally, substituting [tex]$x\approx63.8\textdegree$[/tex] into the expression for AC, we get:

[tex]$AC=\sqrt{3.125+11.25\cos(63.8\textdegree)-10\cos^2(63.8\textdegree)}$[/tex]

[tex]$AC\approx4.2$[/tex]

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Question:

ABCD is a cyclic quadrilateral with AB=5.6cm,BC=4.5cm,CD=3.4cm and AD=2.5cm.Calculate

a)B in ABC

b)AC?

The area of the regular hexagon shaped sandbox is found to be about 28.38 ft².

The sandbox is of the shape of regular hexagon, the area of the hexagon is given by the formula, 3√3a²/2, where, a is the side of the hexagon, the side of the hexagon is give to be 3 ft and the apothem is 3√3 ft.

Now, putting the value in the formula,

Area = 3√3(3)²/2

Area = 23.38 ft²

So, the area of the sandbox is found to be 23.38 ft².

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[tex]2x^{2} - 22x - 52 = 2(x^{2} - 11x - 26) = 2(x^{2} + 2x - 13x - 26) = 2[x(x + 2) - 13(x - 2)]\\[/tex]

[tex]= \bf 2(x + 2)(x - 13)[/tex]

Part A: −Δ[[tex]H2O2[/tex]]/Δt = Δ[[tex]H2[/tex]]/Δt + Δ[[tex]O2[/tex]]/Δt

Part B: −Δ[[tex]H2O2[/tex]]/Δt = Δ[[tex]O2[/tex]]/Δt

Part C: −Δ[[tex]N2O[/tex]]/Δt = 1/2 Δ[[tex]N2[/tex]]/Δt + Δ[[tex]O2[/tex]]/Δt

Part D: −Δ[[tex]N2O[/tex]]/Δt = Δ[[tex]O2[/tex]]/Δt

Part E: −Δ[[tex]N2[/tex]]/Δt = 1/2 Δ[[tex]NH3[/tex]]/Δt

Part F: −Δ[[tex]H2[/tex]]/Δt = Δ[[tex]NH3[/tex]]/Δt

Part G: −Δ[[tex]C2H5NH2[/tex]]/Δt = Δ[[tex]C2H4[/tex]]/Δt + Δ[[tex]NH3[/tex]]/Δt

Part H: −Δ[tex][C2H5NH2][/tex]/Δt = Δ[tex][NH3][/tex]/Δt

The rate of disappearance of [tex]H2O2[/tex]is equal to the sum of the rates of appearance of H2 and O2.

Part B: −Δ[[tex]H2O2[/tex]]/Δt = Δ[[tex]O2[/tex]]/Δt

The rate of disappearance of [tex]H2O2[/tex] is equal to the rate of appearance of O2.

Part C: −Δ[[tex]N2O[/tex]]/Δt = 1/2 Δ[[tex]N2[/tex]]/Δt + Δ[[tex]O2[/tex]]/Δt

The rate of disappearance of N2O is equal to half the rate of appearance of N2 plus the rate of appearance of O2.

Part D: −Δ[[tex]N2O[/tex]]/Δt = Δ[[tex]O2[/tex]]/Δt

The rate of disappearance of [tex]N2O[/tex]  is equal to the rate of appearance of O2.

Part E: −Δ[[tex]N2[/tex]]/Δt = 1/2 Δ[[tex]NH3[/tex]]/Δt

The rate of disappearance of N2 is equal to half the rate of appearance of[tex]NH3.[/tex]

Part F: −Δ[[tex]H2[/tex]]/Δt = Δ[[tex]NH3[/tex]]/Δt

The rate of disappearance of H2 is equal to the rate of appearance of NH3.

Part G: −Δ[[tex]C2H5NH2[/tex]]/Δt = Δ[[tex]C2H4[/tex]]/Δt + Δ[[tex]NH3[/tex]]/Δt

The rate of disappearance of[tex]C2H5NH2[/tex] is equal to the sum of the rates of appearance of[tex]C2H4[/tex] and [tex]NH3.[/tex]

Part H: −Δ[tex][C2H5NH2][/tex]/Δt = Δ[tex][NH3][/tex]/Δt

The rate of disappearance of [tex]C2H5NH2[/tex] is equal to the rate of appearance of [tex]NH3.[/tex]

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High order thinking:

PART A:

If each friend buys 11 more action figures, then each will have x + 11 action figures. Since the total action figures is 120, we can write the equation:

[tex]5(x+11)=120[/tex]

PART B:

Solve for x. Divide both sides by 5.

[tex]x+11=24[/tex]

Substract 11 from both sides:

[tex]x=13\\[/tex]

Each friend originally had 13 action figures each.

Assessment practice:

11: Let Kevin Earns X amount

So, Jason earns = 2x - 32.50

= 212.50

2x = 2120 x 50 +32.40 = 245.00/2

= 122.5

Answer: Kevin Earns $122.50

12: Subtract 6. Then multiply by 2.

[tex]\frac{1}{2} x+6=18\\\\\frac{1}{2}x = 18-6\\\frac{1}{2}x=12\\\frac{1}{2}x*2=12*2\\x=24[/tex]

Thanks,
Eddie E.

Answer:

10)

a) 5x + 5(11) = 120

b) x = 13

11) $392.50

12) Letter D

Step-by-step explanation:

10:

5x + 5(11) = 120

5x + 55 = 120  Subtract 55 from both sides

5x + 55 - 55 = 120 - 55

5x = 65  Divide both sides by 5

[tex]\frac{5x}{5}[/tex] = [tex]\frac{65}{5}[/tex]

x = 13

11:

Let j = Jason's earnings

Let k = Kevin's earnings

j = 212.50

k = 2j - 32.50

substitute 212.50 for j in the second equation and solve for k

k = 2(212.50)- 32.50

k = 425 - 32.50

k = 392.50

Kevin earned $392.50.

12:

18 = [tex]\frac{1}{2}[/tex] x + 6   Subtract 6 from both sides

18 - 6 = [tex]\frac{1}{2}[/tex] x + 6 - 6

12 = [tex]\frac{1}{2}[/tex] x  Multiply both sides by 2

12(2) = [tex]\frac{1}{2}[/tex] x ([tex]\frac{2}{1}[/tex])   ([tex]\frac{2}{1}[/tex] is another name for 2)

24 = x

To solve we subtract 6 first and then multiply by 2.  Letter D.

Helping in the name of Jesus.

Answer:

1/6

Step-by-step explanation:

I am assuming you mean the probability of rolling a 3 on a die.

If you are rolling only 1 die, with the numbers 1 through 6, then the chance of landing on any number is 1/6.

For example, the chance of landing on a 2 or 5 are equal, both 1/6.

Therefore, the chance of landing on a die is 1/6.

Hope this helped!

The required volume of the given sphere is 904.32 cm³.

A sphere is a geometrical object that resembles a two-dimensional circle in three dimensions.

In three-dimensional space, a sphere is a collection of points that are all located at the same distance from a single point.

The radius of the sphere is denoted by the letter r, and the specified point represents its center.

All of the points on a circle are equally spaced apart from the center along a plane, but all of the points on a sphere are equally spaced apart from the center along any of the axes.

So, we must ascertain the sphere's volume. With a radius of 6 cm, we have:

V = 4/3 π r^3

V = 4/3 x 3.14 x 6^3

V = 4/3 x 3.14 x 216

V = 904.32  

Therefore, the required volume of the given sphere is 904.32 cm³.

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Complete question:

Find the volume of the sphere.

Either enter an exact answer in terms of \piπpi or use 3.143.143, point, 14 for \piπpi and round your final answer to the nearest hundredth.

There are 15 true/false questions on the test and 5 multiple choice questions on the test.

Equations can be written in many forms, but they all have the same basic structure: an expression on the left-hand side of the equals sign, and an expression on the right-hand side of the equals sign, with the equals sign itself indicating that the two expressions are equal.

Let the number of true/false questions be x and the number of multiple choice questions be y. We know that:

x + y = 20 (since there are 20 questions in total)

3x + 11y = 100 (since the test is worth 100 points)

We can solve this system of equations by substitution or elimination. Here, we'll use substitution:

x + y = 20 -> x = 20 - y

Substitute x = 20 - y into the second equation:

3x + 11y = 100

3(20 - y) + 11y = 100

60 - 3y + 11y = 100

8y = 40

y = 5

So there are 5 multiple choice questions on the test. To find the number of true/false questions, we can substitute y = 5 back into x + y = 20:

x + y = 20

x + 5 = 20

x = 15

Therefore, there are 15 true/false questions on the test and 5 multiple choice questions on the test.

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Answer:

Step-by-step explanation:

The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.

We are given that the line passes through the point (-4, 2) and has a slope of -1. This means that we can substitute the values of the point and slope into the equation and solve for b.

y = mx + b

2 = (-1)(-4) + b

2 = 4 + b

b = -2

Now we know the slope and y-intercept of the line, so we can write the equation in slope-intercept form:

y = -x - 2

Therefore, the equation of the line that passes through the point (-4, 2) and has a slope of -1 is y = -x - 2.

The general expressions for sin(2θ) and cos(2θ) are Sin(2θ) = 2sinθcosθ, and cos(2θ) = cos2θ − sin2θ.

The formula for deriving sin(2θ) and cos(2θ) is as follows:

To derive the sin(2θ) formula, use the following formula: sin (2θ) = 2sinθcosθ

And to derive the cos(2θ) formula, use the following formula: cos(2θ) = cos2θ − sin2θ

From the compound angle formulas, we know that:

Sin (α + β) = sinαcosβ + cosαsinβ, and Cos (α + β) = cosαcosβ − sinαsinβ

We may derive sin(2θ) from the above formulas by putting α = β = θ, which gives us:

Sin (2θ) = sinθcosθ + sinθcosθSin (2θ) = 2sinθcosθcos(2θ) can be derived from the above formula by following these steps:

Cos (2θ) = cosθcosθ − sinθsinθCos (2θ) = cos2θ − sin2θ

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Answer:

s is not inversely proportional to t

Step-by-step explanation:

s is not inversely proportional to t.  I had responded that they were, based on the fact that as s went up, t went down.  But the question was not simply is there an inverse relationship, but are they inversely proportional.

The term proportional means that the relationship between s and t is a constant.  That is:

                                        t = s*(1/x)

Let's rewrite that to y*x = k and then check the numbers.  See the attached spreadsheet.  If the relationship were inversel proportioanl, thaen the product of t*s would be a contant for the series.  The third set is different from the first two.  The data has an is inverse relationship, but it is NOT proportional.

The simplified expression, expressed in expanded form, is (3m - y)/(12m + 2y).

To simplify the expression (6m² - 5my - y²)/(12m + 2y), we can factor the numerator and denominator, if possible, and then simplify the expression by canceling out common factors.

The numerator can be factored as follows:

6m² - 5my - y² = (3m - y)(2m + y)

The denominator can also be factored by factoring out a common factor of 2:

12m + 2y = 2(6m + y)

Now we can substitute these factorizations back into the original expression:

(6m² - 5my - y²)/(12m + 2y) = [(3m - y)(2m + y)]/[2(6m + y)]

We can now cancel out the common factor of (2m + y) in the numerator and denominator:

[(3m - y)(2m + y)]/[2(6m + y)] = (3m - y)/(2(6m + y))

Expanding this expression, we get:

(3m - y)/(2(6m + y)) = (3m - y)/(12m + 2y)

Therefore, the simplified expression, in expanded form, can be written as (3m - y)/(12m + 2y).

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[tex]\textit{vertex of a vertical parabola, using coefficients} \\\\ y=x^2-1\implies y=\stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{+0}x\stackrel{\stackrel{c}{\downarrow }}{-1} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right) \\\\\\ \left(-\cfrac{ 0}{2(1)}~~~~ ,~~~~ -1-\cfrac{ (0)^2}{4(1)}\right) \implies \left( - \cfrac{ 0 }{ 2 }~~,~~-1 - \cfrac{ 0 }{ 4 } \right) \\\\\\ \left( 0 ~~~~ ,~~~~ -1 +0 \right)\implies (0~~,~-1)[/tex]

Clara's school is 7 miles west of her house and 3 miles south of her friend Scott's house. Therefore, 5 miles far does Clara bicycle each day.

Given that:

Clara's school is 7 miles west of her house and

3 miles south of her friend Scott's house.

Total distance of Clara's house from her school is 7 miles

Therefore, distance walked by Clara's from home

     = (7 - 3) miles

     = 5 miles.

Therefore, After school, she bicycles from her school to Scott's house. Before dinner, she bicycles home on a bike path that goes straight from Scott's house to her own house. 5 miles far does Clara bicycle each day.

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By using the graphs above, a graph that represent h(x), given that function h(x) = f(x) + g(x) include the following: A. graph A.

In Mathematics, the general form of a quadratic function is modeled by the following mathematical expression;

y = ax² + bx + c

Where:

Next, we would write quadratic functions that represent both f(x) and g(x) in standard form and with a leading coefficient of 1 as follows;

f(x) = (x + 3)(x + 1)

f(x) = x² + 3x + x + 3

f(x) = x² + 4x + 3

For the function g(x), we have the following:

g(x) = -(x - 3)(x - 1)

g(x) = -(x² - 3x - x + 3)

g(x) = -x² + 4x - 3

Therefore, a function that represent h(x) can be calculated as follows;

h(x) = f(x) + g(x)

h(x) = x² + 4x + 3 -x² + 4x - 3

h(x) = (x² - x²) + (4x + 4x) + (- 3 + 3)

h(x) = 8x

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Answer:

40°

Step-by-step explanation:

The angles of a triangle will always sum to 180°.

Therefore, we can say that

[tex](x) + (x+6) + (2x+14) = 180\\4x + 20 = 180\\4x = 160\\x = 40[/tex]

The value of x° = 40°.

Answer:

0.1

Step-by-step explanation:

100 x 10 = 1000. Essentially, you just add a zero to 100. so you take a zero out of .01. SO your answer is 0.1.

Answer: 0.1 fluid ounce

explanation:

since 100 drops of a liquid have 0.01 fluid ounces

1 drop of a liquid has (0.01/100)× 1 fluid ounces = 0.0001 ounces

so 1000 drops of a liquid have (1000×0.0001)= 0.1 fluid ounce

Probability of X Wins with Biased Wheel: 0.0378

X, the Number of Winning Rounds: 36

Net Profit from X Wins: $1,260

Probability of X Wins with Biased Wheel: 0.0378 (rounded to 4 significant figures)

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The cosine of θ is the ratio of the length of the adjacent side to the length of the hypotenuse

The cosine of an angle is a trigonometric function that relates the length of the adjacent side of a right triangle to the length of the hypotenuse. Specifically, it is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

In mathematical terms, if we have a right triangle where one of the angles is labeled as theta (θ), then the cosine of theta is given by the formula:

cos(θ) = adjacent side / hypotenuse

1) Cos R =30/34 =15/17

Cos S = 16/34 = 8/17

2) Cos R = 24/26 = 12/13

Cos S = 10/26 = 5/13

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Answer:

≈ 50 cm²

Step-by-step explanation:

Use the equation [tex]A= \frac{1}{4} \pi d^{2}[/tex] Where d is the diameter.

Answer:

The answer that you're looking for is approximately 50 (rounded), In terms of π, it is 16π.

Step-by-step explanation:

In order to find the area you need to use the formula: Area = [tex]\pi r^{2}[/tex].

Since The Diameter is double the amount of the radius you need to make sure to divide the diameter by 2 and replace "r" in the equation with the equation given.

8/2 gives you 4.  Now you have the equation Area of Circle =  [tex]\pi 4^{2}[/tex].

Following the rules of PEMDAS we do exponents since there is no parenthesis.

[tex]4^{2}[/tex] is the same as 16. In terms of pi, you just put pi next to your result giving 16π.

However, if you want to find out normally you can multiply with either 3.14 or π.

Both cases will give you different decimals, but when rounded to the nearest whole number they all give you 50.

Area of Circle  = [tex]\pi r^{2}[/tex].
Area of Circle  = [tex]\pi 4^{2}[/tex]
Area of Circle  = 16π
Area of Circle = 50 (rounded).

I hope this was helpful!

Pizza is therefore consumed in the following sequence, from best to worst: J. Antonio, Marcus, and Joe

A sequence in mathematics is a group of numbers that are organized in a particular order and that adhere to a particular pattern or formula. Sequences can be endless or finite and have a variety of uses in many disciplines, including probability theory, analysis, and number theory. Sequences can be defined recursively, where each term is defined in terms of the terms that came before it, or they can be defined using a formula that produces each term of the sequence. For instance, using the equation a = 2n, where n is a positive integer and a1 = 2 is the first term, one can determine the sequence of even numbers. If n is greater than 1, it can also be defined recursively as a1 = 2, and a = an-1 Plus 2.

given

Half of Joe's pie was consumed.

The result of multiplying the numerator and denominator by 3/8 is as follows:

5/8 × 3/3 = 15/24 = 5/8

We must change 3/4 into a comparable fraction with a denominator of 24 in order to compare this fraction to Joe and Marcus' fractions (the least common multiple of 3 and 8).

The result of multiplying the numerator and divisor by 6/6 is as follows:

3/4 × 6/6 = 18/24

Joe: 2/3

Marcus, 5/8 equals 15/24.

Thomas: 18/24

We can transform the fractions to decimals and arrange them in ascending order:

Joe: 2/3 = 0.666...

Marcus, 15/24 equals 0.625

Anthony: 18/24 equals 0.75

Pizza is therefore consumed in the following sequence, from best to worst: J. Antonio, Marcus, and Joe .

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The jogger's momentum is 512 kg·m/s.

The momentum of an object is defined as the product of its mass and velocity.

Mathematically, momentum (p) can be expressed as:

p = m x v

where m is the mass of the object and v is its velocity.

Momentum is an important concept in physics because it describes the quantity of motion an object possesses. The momentum of an object can be changed by applying a force to it for a certain period of time. This change in momentum is called impulse and is equal to the force multiplied by the time it acts on the object.

In this case, the jogger's mass is given as 64 kg and their velocity is given as 8 m/s. To find their momentum, we can simply multiply these values:

p = m x v

p = 64 kg x 8 m/s

p = 512 kg*m/s

Therefore, the momentum of the jogger is 512 kg*m/s.

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Answer:

(-8/3, 19/27) and (3, -17/2).

Step-by-step explanation:

To find the coordinates of the points on the graph of ƒ(x) = ½x³ − ¹⁄₂x² − 8x + 7 where the gradient is 4, we need to find the points where the derivative of ƒ(x) is equal to 4.

First, we need to find the derivative of ƒ(x):

ƒ'(x) = 3/2x² - x - 8

Next, we need to set ƒ'(x) = 4 and solve for x:

3/2x² - x - 8 = 4

3/2x² - x - 12 = 0

Multiplying both sides by 2 to eliminate the fraction:

3x² - 2x - 24 = 0

Factoring the quadratic equation:

(3x + 8)(x - 3) = 0

So x = -8/3 or x = 3.

Now we can find the corresponding y-coordinates:

When x = -8/3:

ƒ(-8/3) = 1/2(-8/3)³ - 1/2(-8/3)² - 8(-8/3) + 7 = 19/27

So one point on the graph with gradient 4 is (-8/3, 19/27).

When x = 3:

ƒ(3) = 1/2(3)³ - 1/2(3)² - 8(3) + 7 = -17/2

So another point on the graph with gradient 4 is (3, -17/2).

Therefore, the coordinates of the points on the graph of ƒ(x) = ½x³ − ¹⁄₂x² − 8x + 7 where the gradient is 4 are (-8/3, 19/27) and (3, -17/2).