Using Heron's formula, show that the equilateral triangle has the maximum area for any fixed perimeter. Hint: In xyz≤k.
maximum occurs when x=y=z

Answer: #1 108 #2 526. #3 512

Step-by-step explanation:

6*6=36 +12*6 = 108

22*3=66 + 20*23= 460 460+66=526

16*16=256 + 32*8=256 256+256=512

Answer:

1)  108 cm²

2)  906 ft²

3)  512 m²

Step-by-step explanation:

To calculate the area of each given composite figure, divide the figure into two rectangles and sum the area of the two rectangles.

[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]

Separate the figure into two rectangles by drawing a horizontal line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 6\cdot 6+12 \cdot 6\\&=36+72\\&=108\; \sf cm^2\end{aligned}[/tex]

Separate the figure into two rectangles by drawing a vertical line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 22\cdot23 + 20\cdot 20\\&=506+400\\&=906\; \sf ft^2\end{aligned}[/tex]

Separate the figure into two rectangles by drawing a horizontal line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 16\cdot16 + 32\cdot 8\\&=256+256\\&=512\; \sf m^2\end{aligned}[/tex]

Question 1

The probability of tossing a head is [tex]\frac{1}{2}[/tex], making the answer [tex]3\left(\frac{1}{2} \right)=\boxed{1.5}[/tex].

Question 2

[tex](3)(1/2)(1/2)(1/2)=\boxed{0.375}[/tex]

Answer:

total visitors = 247+79+36 = 362

probability of next person buying one or more costume = 115/362

hope it helps

Answer:

771 boys

Step-by-step explanation:

To find your answer, multiply 1285 by 0.6. Since 1 means 100%, 0.6 would be 60%. When you multiply these two things, you get 771, so that's your final answer. You can use a calculator probably to solve this problem.

The side length of each piece is given as follows:

[tex]20\sqrt{2}[/tex] inches.

The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.

For this problem, the parameters are given as follows:

Hence the side length of each piece is obtained as follows:

x² + x² = 40²

2x² = 40²

x² = 800

x = sqrt(2 x 400)

[tex]x = 20\sqrt{2}[/tex]

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The variance of the sample data is equal to 197.3463 kg^2.

In Statistics, the standard deviation of a data sample is the square root of the variance and as such, this given by the following mathematical expression:

Standard deviation, δ = √Variance

By making variance the subject of formula, we have the following:

Variance = δ²

By taking the square of standard deviation, the variance would be calculated as follows:

Variance = δ²

Variance = 14.0480²

Variance = 197.3463 kg^2.

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Answer:

The monthly payment for the loan is $3,308.10

Step-by-step explanation:

To find the monthly payment for the loan, we can use the standard formula for a fixed-rate mortgage loan:

M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1]

Where:

M = monthly payment

P = principal amount (the amount borrowed)

i = monthly interest rate (annual interest rate divided by 12)

n = total number of monthly payments (30 years * 12 months per year)

First, we need to calculate the monthly interest rate:

i = 8.5% / 12 = 0.00708333

Next, we can substitute the given values into the formula:

M = 450,000 [ 0.00708333(1 + 0.00708333)^360 ] / [ (1 + 0.00708333)^360 - 1]

Using a calculator, we can simplify this equation to find the monthly payment:

M = $3,308.10

Therefore, the monthly payment for the loan is $3,308.10

Because Alicia chose a random sample, the answer is (A).

Regardless of the population distribution's shape, the Central Limit Theorem states that as sample size rises, the sampling distribution of the sample means tends to resemble a normal distribution. 40 pairs of Brand A shoes and 33 pairs of Brand B shoes were randomly chosen by Alicia, meeting the Central Limit Theorem's minimum sample size criterion. Hence, even though the distribution of Brand A shoes contains outliers, the sampling distribution of the difference in sample means is roughly normal. The Central Limit Theorem can be applied without assuming a particular population size or sample size, provided that the sample is independent and random.

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The correct answer is (C) Yes, because the size of each sample is at least 30.

The Central Limit Theorem (CLT) states that the sampling distribution of the mean of a sufficiently large number of independent and identically distributed random variables will be approximately normal, regardless of the population distribution, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

The Central Limit Theorem states that the sampling distribution of the difference in sample means will be approximately normal as long as the sample sizes are large enough, typically n≥30.

Therefore, the fact that Alicia's sample sizes are 40 and 33 respectively ensures that the sampling distribution is approximately normal, even with outliers in Brand A.

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x represents the increase in the length and width of the garden from the expression.

The area of Rectangle is length times of width.

Ishan garden is in the shape of a rectangle.

The length of the garden is 15 feet and width is 5 feet.

He plans to increase the length and width of his garden.

The total amount of fencing is given as expression

(x+15)+(x+5)+(x+15)+(x+5)

We have to check what the x represents in the expression.

The x represents the increase in the length and width of the garden.

Hence,  x represents the increase in the length and width of the garden.

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(a) Thus, they are 601.5 miles far from their destination.

(b) Thus, they are 692.5 miles far from their destination.

We know that speed, distance, and time all are in a relationship to each other. this relationship can be given as,

Speed*time = Distance

Given to us

Speed = 65 miles an hour

Don noticed that 5 hours into the trip they were 625 miles from the destination.

Distance traveled by them in 5 hours

We know that Don and Ana are traveling at a constant speed and they are traveling for 5 hours.

Distance = speed * time

Distance = 65 *5 = 320 miles.

Thus, the Distance traveled by them in 5 hours is 320 miles.

Total Distance from their destination

We know that Don and Ana are traveling on the freeway for the past 5 hours and they are still 625 miles away, therefore,

otal Distance from their destination

= Distance traveled by them in 5 hours + 650 miles

= 320 miles + 625miles

=945 miles

Thus, the distance traveled by then is 945 miles.

A.) Distance between them and destination after 5.3 hours since entering the freeway,

Distance traveled by them = speed x 5.3

                                           = 65*5.3   =  344.5

 Distance between them and the destination

= Total Distance from their destination - Distance traveled by them

= 945 miles - 344.5 miles

= 601.5 miles          

Thus, they are 601.5 miles far from their destination.      

B.) Distance between them and destination after 2.3 hours since entering the freeway,

Distance traveled by them = speed x 4.7

                             = 65 x 4.7 =  305.5 miles

Distance between them and the destination

= Total Distance from their destination - Distance traveled by them

= 945miles - 305.5 miles

= 639.5 miles

Thus, they are 692 miles far from their destination.

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The equation of the plane that passes through the point [tex]$(2,-1,3)$[/tex] and is perpendicular to the line [tex]$\mathbf{v}$[/tex] can be expressed in terms of a normal vector [tex]$\mathbf{n}$[/tex] and a point [tex]$\mathbf{p}$[/tex] on the plane as [tex]$\mathbf{n}\cdot(\mathbf{x}-\mathbf{p})=0$[/tex].

To find the normal vector [tex]$\mathbf{n}$[/tex], we must first find a vector parallel to the line [tex]$\mathbf{v}$[/tex]. We can do this by taking the cross product of two non-parallel vectors [tex]$\mathbf{u_1}$ and $\mathbf{u_2}$[/tex] that lie on the line. Let [tex]$\mathbf{u_1} = (1,2,1)$ and $\mathbf{u_2} = (3,4,3)$[/tex]. Then, [tex]$\mathbf{n}=\mathbf{u_1}\times\mathbf{u_2}=(-4,8,-4)$[/tex].The equation of the plane that passes through the point [tex]$(2,-1,3)$[/tex] and is perpendicular to the line [tex]$\mathbf{v}$[/tex] can be expressed in terms of a normal vector [tex]$\mathbf{n}$[/tex] and a point [tex]$\mathbf{p}$[/tex] on the plane as [tex]$\mathbf{n}\cdot(\mathbf{x}-\mathbf{p})=0$[/tex].

We can now calculate the equation of the plane as [tex]$\mathbf{n}\cdot(\mathbf{x}-\mathbf{p})=0$[/tex], where [tex]$\mathbf{x}=(x,y,z)$ and $\mathbf{p}=(2,-1,3)$[/tex]. Thus, the equation of the plane is [tex]$-4x+8y-4z+17=0$.[/tex]

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The equations of the lines passing through the specified points in slope-intercept form are;

The slope-intercept form of the equation of a line is; y = m·x + c

Where;

m = The slope of the line

c = The y-intercept

The points are; (2, 5), and (6, 2)  

The slope and the y-intercept of the line are found as follows;

Let m represent the slope and let c represent the y-intercept, we get the following system of equations;

5 = 2·m + c...(1)

2 = 6·m + c...(2)

Therefore;

The difference between the two equations are;

5 - 2 = 2·m - 6·m + (c - c)

3 = -4·m

m = -3/4

5 = 2·m + c

Therefore;

5 = 2 × (-3/4) + c

c = 5 + 2 × (3/4) = 6.5

The points are; (-1, 3) amd (0, -2)

The equation are;

3 = -1·m + c...(1)

-2 = 0×m + c...(2)

Therefore;

c = -2

3 = -1·m + c

3 = -1·m - 2

m = (3 + 2)/(-1) = -5

The equation is therefore;

The points are; (-2, -4) and (2, 1)

The equation are;

-4 = -2·m + c...(1)

1 = 2·m + c...(2)

Subtracting equation (2) from equation (1), we get;

-4 - 1 = -2·m - 2·m = -4·m

-5 = -4·m

m = -5/(-4) = 1.25

m = 1.25

-4 = -2·m + c

Therefore;

-4 = -2 × 1.25 + c

-4 = -2.5 + c

c = -4 + 2.5 = -1.5

c = -1.5

The equation is therefore; y = 1.25·x - 1.5

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Answer: I think that the answer is the fourth option. Yes. All conditions are met. I might be wrong but that's my best guess.

Step-by-step explanation: The athletes were assigned randomly, so it can't be option one. I'm not totally sure what a ten percent condition is, but I didn't see anything in the question about it. The Normal/Large Sample condition is met. And it looks like all of the conditions are met.

Answer: 11/3

Step-by-step explanation: 2 1/5= 11/5

11/5 ÷ 3/5= 11/3

(keep change and flip) change the division sign to multiplication and the 3/5 would become 5/3

write it down visually

Answer: 3 2/3

In order to solve this we must turn our fractions into whole numbers.

2 1/5 as a whole number is 2.20

3/5 as a whole number is .6

Next we must divide our two whole numbers into one another.

2.20/.6 is 3.66666667

666666667 as a fraction is 2/3.

This means that 3/5 can fit into 2 1/5 3 whole times and 2/3 of a whole. This means you will need 3 whole boxes and 2/3 of a box.

I hope this helps & Good Luck <3 !!!

The equation of the linear function represented by the table is y=-x+9. Therefore, option A is the correct answer.

The general equation of a straight line is y=mx+c, where m is the gradient, and y = c is the value where the line cuts the y-axis. This number c is called the intercept on the y-axis.

The given table is

  x        y

 -5       14

 -2       11

  1        8

  4       5

From the given table, the coordinate points are (-5, 14) and (-2, 11)

Here, slope(m) =(11-14)/(-2+5)

= -1

Substitute m=-1 and (x, y)=(-5, 14) in y=mx+c, we get

14=-1(-5)+c

c=9

So, the equation is y=-x+9

Therefore, option A is the correct answer.

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To verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.

To prove that,

[tex]limn → ∞ \frac{2n+1}{5n+4} = \frac{2}{5} [/tex]

we use the limit definition then we need to prove the statement

[tex]∀ε>0,∃N∈Ns.t.n>N⟹ \frac{2n+1}{5n + 4} [/tex]

we already have,

[tex]∣ \frac{2n+1}{5n+4}− \frac{2}{5} ∣=∣− \frac{3}{5(5n+4)} ∣= \frac{3}{5(5n+4)} [/tex]

For every positive ε, we choose that N=⌈1/ε⌉, where we are using the ceiling function to locate the smallest integer number larger than the fraction 1/ε. This gives us a natural number, proving that the limit's quantified statement definition is accurate:.

[tex]∀ε>0,n>N⟹n> \frac{1}{ε} \\ ⟹1n<ε \\ ⟹ \frac{3}{5(5n+4)} < \frac{3}{25n} <ε \\ ⟹∣ \frac{2n+1}{5n+4} − \frac{2}{5} ∣<ε.[/tex]

Hence, proved.

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Answer: Without the coupon, Mariana would have to pay $6.25 per pound of chicken, so for 5 pounds, the cost would be:

5 * $6.25 = $31.25

With the coupon, she would receive a discount of $3.25, so the final cost would be:

$31.25 - $3.25 = $28.00

To write an expression for the cost to buy p pounds of chicken, we can use the formula:

C = 6.25p - 3.25

where C is the cost in dollars and p is the number of pounds of chicken purchased. Note that the expression assumes that at least one pound is purchased, as indicated in the question.

Step-by-step explanation:

1. Rounding each price per dozen to nearest penny will be $1.6 and $5.4

2. The percent of increase between these 2 years will be

A percentage simply has to do with the a value or ratio which can be stated as a fraction of 100. It should be noted that when we want to we calculate a percentage of a number, we simply divide it and then multiply the value that is gotten by 100.

The percent of increase between these 2 years will be:

= (5.4 - 1.6) / 1.6 × 100

= 3.8/1.6 × 100

= 237.5%

The reason for the change in price of eggs in that time was due to the increase in the feeding of poultry animals.

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The volume of the set of stairs is 65 cubic feet.

In mathematics, the volume of an object is the amount of three-dimensional space that it occupies. It is a measure of the total amount of space that a solid object or a container can hold.

The volume of an object is usually measured in cubic units such as cubic meters ([tex]m^3[/tex]), cubic centimeters ([tex]cm^3[/tex]), or cubic feet (ft³), depending on the units of measurement used for its dimensions. The formula for calculating the volume of an object varies depending on its shape.

For simple shapes like cubes, rectangular prisms, and cylinders, the formulas for calculating their volumes are:

Volume of a cube = length x width x height

Volume of a rectangular prism = length x width x height

Volume of a cylinder = π x radius² x height

Where π (pi) is a mathematical constant approximately equal to 3.14, and the dimensions are all measured in the same units.

For irregular shapes, the volume can be determined using mathematical equations, computer modeling, or physical measurements. In real-world applications, volume is used in a variety of fields such as engineering, architecture, physics, and chemistry to describe the size, capacity, or amount of a substance or material.

To calculate the volume of the set of stairs, we need to find the volume of each rectangular block that makes up the stairs and then add them up. Let's first find the dimensions of each rectangular block:

The first block has dimensions 2 feet by 5 feet by 1 foot.

The second block has dimensions 8 feet by 5 feet by 1 foot.

To find the volume of each block, we multiply its length by its width by its height. So, the volume of the first block is:

Volume of first block = length x width x height

= 5 feet x 5 feet x 1 foot

= 25 cubic feet

The volume of the second block is:

Volume of second block = length x width x height

= 8 feet x 5 feet x 1 foot

= 40 cubic feet

Therefore, the total volume of the set of stairs is:

Total volume = volume of first block + volume of second block

= 25 cubic feet + 40 cubic feet

= 65 cubic feet

Therefore, the volume of the set of stairs is 65 cubic feet.

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Answer:

Step-by-step explanation:

Answer:

The volume of the set of stairs is 65 cubic feet.

What is volume ?

In mathematics, the volume of an object is the amount of three-dimensional space that it occupies. It is a measure of the total amount of space that a solid object or a container can hold.

The volume of an object is usually measured in cubic units such as cubic meters (), cubic centimeters (), or cubic feet (ft³), depending on the units of measurement used for its dimensions. The formula for calculating the volume of an object varies depending on its shape.

For simple shapes like cubes, rectangular prisms, and cylinders, the formulas for calculating their volumes are:

Volume of a cube = length x width x height

Volume of a rectangular prism = length x width x height

Volume of a cylinder = π x radius² x height

Where π (pi) is a mathematical constant approximately equal to 3.14, and the dimensions are all measured in the same units.

For irregular shapes, the volume can be determined using mathematical equations, computer modeling, or physical measurements. In real-world applications, volume is used in a variety of fields such as engineering, architecture, physics, and chemistry to describe the size, capacity, or amount of a substance or material.

According to given information :

To calculate the volume of the set of stairs, we need to find the volume of each rectangular block that makes up the stairs and then add them up. Let's first find the dimensions of each rectangular block:

The first block has dimensions 2 feet by 5 feet by 1 foot.

To find the volume of each block, we multiply its length by its width by its height. So, the volume of the first block is:

Volume of first block = length x width x height

= 5 feet x 5 feet x 1 foot

= 25 cubic feet

The volume of the second block is:

Volume of second block = length x width x height

= 8 feet x 5 feet x 1 foot

= 40 cubic feet

Therefore, the total volume of the set of stairs is:

Total volume = volume of first block + volume of second block

= 25 cubic feet + 40 cubic feet

= 65 cubic feet

Therefore, the volume of the set of stairs is 65 cubic feet.

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Equation:

[tex]\frac{3}{2}[/tex](4x – 1) – 3x = [tex]\frac{5}{4}[/tex]– (x + 2)

Solving for x

[tex]\frac{3}{2}[/tex] × 4x - [tex]\frac{3}{2}[/tex](-1) -3x=  [tex]\frac{5}{4}[/tex]– (x + 2)

⇒6x + [tex]\frac{3}{2}[/tex] -3x=  [tex]\frac{5}{4}[/tex]– x - 2

⇒6x -3x + x =  [tex]\frac{5}{4}[/tex] - 2 -  [tex]\frac{3}{2}[/tex]

⇒4x = [tex]\frac{5-8-6}{4}[/tex]

⇒4x = [tex]\frac{-9}{4}[/tex]

⇒x= [tex]\frac{-9}{16}[/tex]

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The radius of the smallest circle is 5 units

Three concentric circles are given in which the two largest circles have radii of 12 and 13. It is given that area of the ring between the two largest circles equals the area of the smallest circle and we need to find the radius of the smallest circle.

Let the radius of the smallest circle be r.

Area of smallest circle = πr²

Area of outermost circle = π(13)² = 169π

Area of middle circle = π(12)² = 144π

Area of the ring between the two largest circles = 169π - 144π = 25π

Area of smallest circle = Area of the ring between the two largest circles

                                          πr² = 25π

                                            r² = 25

                                            r = 5

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The correct option for mentioned function is is A. (-1,1).

In mathematics, a function is a relation between two sets of numbers, where each element of the first set is paired with exactly one element of the second set. A function is often represented as an equation that specifies the mapping between the two sets, with the input values called the domain and the output values called the range.

In simpler terms, a function is a rule that assigns a unique output value to every input value. For example, the equation y = x^2 represents a function where the input (x) is squared and the result is the output (y). If x = 3, then y = 9. A function can be represented as a table, graph, or equation.

Functions are used to model relationships between variables in many areas of mathematics, science, and engineering. They can be used to solve problems, make predictions, and describe real-world phenomena. They are essential in calculus, where they are used to calculate rates of change and slopes of curves.

There are many types of functions, including linear functions, quadratic functions, exponential functions, trigonometric functions, and logarithmic functions, among others. Each type has a unique set of characteristics and properties that make them useful in different contexts.

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The required  the distance from point A to point B is approximately 855.31 feet.

The angle of elevation is the angle created between the line of sight and the horizontal. The angle created is an angle of elevation if the line of sight is upward from the horizontal line.

Let's call the distance from point A to the lighthouse "x" and the height of the lighthouse "h" (h = 111 feet). We can use the tangent function to relate the angle of elevation to the distance and height:

tan(5°) = h/x

Solving for x, we get:

x = h/tan(5°) ≈ 1268 feet

Now let's call the distance from point B to the lighthouse "y". We can use the same tangent function to relate the angle of elevation from point B:

tan(15°) = h/y

Solving for y, we get:

y = h/tan(15°) ≈ 412.69 feet

The distance between points A and B is just the difference between x and y:

1268 - 412.69 ≈ 855.31 feet

So the distance from point A to point B is approximately 855.31 feet.

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Complete question:

A boat is heading towards a lighthouse, whose beacon-light is 111 feet above the water. From point A, the boat's crew measures the angle of elevation to the beacon, 5º, before they draw closer. They measure the angle of elevation a second time from point B at some later time to be 15°. Find the distance from point A to point B. Round your answer to the nearest tenth of a foot if necessary.​

In hyperbola, If a is negative, left does the parabola open.

What is an example of a hyperbola?

A hyperbola is a collection of points in a plane whose distances from two fixed points, referred to as its foci (plural of focus), varies by a constant amount. For instance, the picture displays a hyperbola with foci at (-3,0) and (3,0). The point at illustrates its constant distance as being 4. (-3,-2.5).

                                Open and closed hyperbolas are the two different varieties. A hyperbola widens on the x-axis at one end and the y-axis at the other.

The equation below describes a parabola.

                  x= av

If a is negative, left does the parabola open.

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The explicit formula for the nth term of an arithmetic sequence with a11=40 and a27=112 is an = 4.5n - 9.5.

To find the explicit formula for an arithmetic sequence, you need to know two terms in the sequence. Using the formula, you can find the common difference, and then use one of the terms to find the first term. With those values, you can write the explicit formula for any term in the sequence.

The explicit formula for an arithmetic sequence is given by:

an = a1 + (n - 1)d

where a1 is the first term, d is the common difference, and n is the term number.

To find the common difference d, we can use the fact that a27 = a1 + 26d = 112 and a11 = a1 + 10d = 40.

Subtracting the second equation from the first, we get:

16d = 72

So, d = 4.5.

Now we can use a11 = 40 to find a1:

a11 = a1 + 10d = 40

a1 + 45 = 40

a1 = -5

Therefore, the explicit formula for this arithmetic sequence is:

an = -5 + 4.5(n - 1)

Simplifying, we get:

an = 4.5n - 9.5

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Answer:

3)

Step-by-step explanation:

2 lines establish a plane (it is the same plane P as mentioned in the description).

a line being perpendicular (having a right angle = 90°) to 2 other lines means it is perpendicular to the plane these 2 lines are establishing.

as it is impossible for a line to be perpendicular to 2 other lines in the same plane, if these 2 other lines are intersecting. it would only be possible, if these 2 lines were parallel.

The measure of the height of the triangle is 3 / 8 units.

The branch of mathematics that sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry.

The trigonometric functions, also known as circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths.

Given that the triangle has sides, 3 / 4 cm, 1 cm, and 5 / 4 cm.

The angle of the triangle is,

tanθ = ( 1 ) / ( 3 / 4 )

tanθ = 1.33

θ = 53.6°

The height will be,

h = ( 3 / 4 ) x sin53.4

h = 0.6

h = 4 / 5 cm

Therefore, the measure of the height of the triangle is 3 / 8 units.

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Answer:

              The true statements to be selected are the following:

"The slope of Function A is greater than the slope of Function B."

"The y-intercept of Function A is greater than the y-intercept of Function B."

Step-by-step explanation:

              Based on the points that were provided for Function A: (-5, -11), (-4, -8), and (2, 10), we can create a linear function to represent the function. The slope of the function is 3, and the y-intercept would be 4, so the linear function of Function A would be y = 3x + 4. And the function of Function B is, as given, y = 2x - 4.

              Now, we can compare the two functions and determine which statements are correct. Function A has a slope of 3 and a y-intercept of (0, 4), whilst Function B has a slope of 2 and a y-intercept of (0, -4). As both the slope and y-intercept of Function A is greater than the slope and y-intercept of Function B, the correct statements to select would be "The slope of Function A is greater than the slope of Function B." and "The y-intercept of Function A is greater than the y-intercept of Function B."

Have a great day! Feel free to let me know if you have any more questions :)

The futures price F0 must be equal to the present value of the expected future spot price, which is [tex]S0e^{(r-q)} T[/tex]. Hence, equation (5.3) is true.

We can use a similar strategy to Footnote 2 to show that equation (5.3) is correct.

Consider an investor who wants to invest in an asset with spot price S0, which pays a continuous dividend yield of q, and simultaneously take a short position in a futures contract with maturity T, which has a futures price of F0. The investor buys one unit of the asset for S0 and sells a futures contract for F0.

At maturity T, the futures contract will be settled at the spot price of the asset, ST. If ST > F0, the investor makes a profit of ST - F0 on the asset, but incurs a loss of F0 - ST on the futures contract. If ST < F0, the investor incurs a loss of F0 - ST on the asset, but makes a profit of ST - F0 on the futures contract.

Now, suppose that equation (5.3) does not hold, i.e., [tex]F0 \neq S0e^{(r-q)} T[/tex]. If [tex]F0 > S0e^{(r-q)} T[/tex], then the investor can buy the asset for S0, sell a futures contract for F0, and invest the difference [tex](F0 - S0e^{(r-q)} T)[/tex] at the risk-free rate r. At maturity T, the investor will receive ST from the asset, F0 from the futures contract, and [tex](F0 - S0e^{(r-q)} T)e^r(T-t)[/tex] from the investment. The total cash inflow will be [tex]ST + F0 + (F0 - S0e^{(r-q)} T)e^r(T-t).[/tex] But this is greater than the initial outflow of S0, which means that the investor can make a riskless profit, violating the no-arbitrage principle.

Similarly, if[tex]F0 < S0e^{(r-q)} T[/tex], then the investor can short-sell the asset for S0, buy a futures contract for F0, and borrow the difference [tex](S0e^{(r-q)} T - F0)[/tex] at the risk-free rate r. At maturity T, the investor will receive ST from the short sale, F0 from the futures contract, and [tex](S0e^{(r-q)} T - F0)e^r(T-t)[/tex]from the borrowing. The total cash inflow will be [tex]ST + F0 + (S0e^{(r-q)} T - F0)e^r(T-t)[/tex]. But this is greater than the initial outflow of S0, which means that the investor can make a riskless profit, violating the no-arbitrage principle.

Therefore, to avoid arbitrage opportunities, The futures price F0 must be equal to the present value of the expected future spot price, which is [tex]S0e^{(r-q)} T[/tex]. Hence, equation (5.3) is true.

for such more question on contract

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Step-by-step explanation:

first Pythagoras to get PT :

85² = 77² + PT²

7225 = 5929 + PT²

PT² = 1296

PT = 36

second Pythagoras to get ST :

39² = PT² + ST² = 36² + ST²

1521 = 1296 + ST²

ST² = 225

ST = 15

third Pythagoras to get TR :

97² = ST² + TR² = 15² + TR²

9409 = 225 + TR²

TR² = 9184

TR = 95.83318841... ≈ 96