To indirectly measure the distance across a river, Houa stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Houa draws the diagram below to show the lengths and angles that she measured. Find PRPR, the distance across the river. Round your answer to the nearest foot.

The width and height of the rectangle for which the perimeter of the window will be as small as possible are (3sqrt(2), 2sqrt(2)).

What is the Lagrange multiplier?

The Lagrange multiplier, λ, measures the increase in the objective function (f(x, y) that is obtained through a marginal relaxation in the constraint (an increase in k). For this reason, the Lagrange multiplier is often termed a shadow price.

To use Lagrange multipliers to solve this problem, we can set up a function to minimize the perimeter of the window, subject to the constraint that the area of the window is fixed at 6 square meters.

Let x be the width of the rectangle, y be the height of the rectangle, and z be the Lagrange multiplier.

The function to minimize is the perimeter of the window, which is P = 2x + 2y + (2 * sqrt(x^2 + y^2))

The constraint is the area of the window, which is A = xy + (x * sqrt(x^2 + y^2))/2 = 6

The Lagrange equation is

L(x,y,z) = P + z(A-6) = 2x + 2y + (2 * sqrt(x^2 + y^2)) + z(xy + (x * sqrt(x^2 + y^2))/2 - 6)

To find the critical points, we will take the partial derivatives of L(x,y,z) with respect to x, y, and z, and set them equal to zero:

∂L/∂x = 2 + zy + zx * (x/sqrt(x^2 + y^2)) = 0

∂L/∂y = 2 + zx + zy * (y/sqrt(x^2 + y^2)) = 0

∂L/∂z = xy + (x * sqrt(x^2 + y^2))/2 - 6 = 0

Solving these equations simultaneously will give us the critical points of the function.

We will get two critical points (x,y) = (3sqrt(2), 2sqrt(2)), (0,2sqrt(2))

We should eliminate the point where x=0 as it does not fulfill the condition of the problem ( the width should be greater than 0)

Hence, the width and height of the rectangle for which the perimeter of the window will be as small as possible are (3sqrt(2), 2sqrt(2)).

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the cost of a broom is $62.50, and the cost of a mop is x + 15 = 62.5 + 15 = $77.50.

Therefore,

a) 1 mop cost $77.50

b) 1 broom cost $62.50

What is the system of equations?

A system of equations is a set of one or more equations involving a number of variables. The solutions to systems of equations are the variable mappings such that all component equations are satisfied in other words, the locations at which all of these equations intersect.

To find the cost of a mop and a broom, we can set up a system of equations using the information given:

Let x be the cost of a broom

Let y be the cost of a mop (which is $15 more than a broom)

From the problem, we know that:

y = x + 15 (since a mop costs $15 more than a broom)

5x + 7y = 855 (since 5 mops and 7 brooms cost a total of $855)

We can substitute the first equation into the second equation:

5x + 7(x + 15) = 855

5x + 7x + 105 = 855

12x = 750

x = 62.5

Hence, the cost of a broom is $62.50, and the cost of a mop is x + 15 = 62.5 + 15 = $77.50.

Therefore,

a) 1 mop cost $77.50

b) 1 broom cost $62.50

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The number of terms in the expansion of [tex](2x^3y^2)(2x^2y^3)[/tex] is 12.

When expanding the product of two polynomials, the number of terms in the expansion is equal to the product of the number of terms in each polynomial. In this case, the first polynomial has 1 term (2x^3y^2) and the second polynomial has 1 term (2x^2y^3), so the number of terms in the expansion will be 1*1=1.

A polynomial function is one in which the variable's non-negative integer powers or positive integer exponents are the only parts of the equation. Examples of polynomial functions include quadratic, cubic, and other equations. An example of a polynomial with an exponent of 1 is 2x+5.

However, the expression provided is not a polynomial, it is a number.

[tex]2x^3y^2 * 2x^2y^3 = 4x^5y^5[/tex]

It is just one term.

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If the equation that relates the Runner 2’s distance (in miles) with time (in minutes) is [tex]t=8.5d[/tex]  then the slope/rate of change of Runner 2 is 8.5 minutes for 1 mile  .

The three runners are training for the marathon ,

The equation that relates the Runner 2’s distance (in miles) with time (in minutes) is ⇒ [tex]t=8.5d[/tex] ,

to cover the distance of 2 miles Runner2 will take [tex]t= 8.5 \times 2[/tex] minutes ;

So , [tex]t=17[/tex] minutes ;

the first point will be (2,17) .

and to cover the distance of 4 miles the runner 2 will take [tex]t= 8.5 \times 4[/tex]

So , [tex]t=34[/tex] minutes ;

the second point will be (4,34) .

we have to find the rate of change/slope of runner 2  ,

that can be written as : [tex]Slope = \frac{34-17}{4-2}[/tex]

On simplifying ,

we get ;

[tex]Slope = \frac{17}{2}[/tex] ;

So , [tex]Slope = 8.5[/tex].

Therefore , the rate of change of Runner 2 is 8.5 .

The given question is incomplete , the complete question is

Three runners are training for a marathon. One day, they all run about ten miles, each at their own constant speed. The equation that relates the Runner 2’s distance (in miles) with time (in minutes) is "t = 8.5d". Find the slope/rate of change of Runner 2 ?

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Yes, isosceles triangles have three congruent angles.

An isosceles triangle is a triangle with at least two congruent sides and two congruent angles. The two congruent sides and two congruent angles form an isosceles triangle. The two congruent angles are known as the base angles, and they are opposite to the two congruent sides. The third angle in an isosceles triangle must also be congruent to the two base angles in order for the triangle to be true. This is because the angles of a triangle must always sum up to 180 degrees. Therefore, isosceles triangles have three congruent angles. These angles are all equal in measure and have the same shape. The angles of each isosceles triangle can be different from other isosceles triangles, but the angles are still congruent to each other. Therefore, isosceles triangles have three congruent angles.

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You roll a 100 sided die with sides 1-100. The optimal strategy is to stop and cash out only if the dice shows 87 or higher , else keep spending $1 to roll again n again, leading to expected winning of $87.35.

The 100-sided die Lou Zocchi created is known by the trademark "Zocchihedron," and it made its debut in 1985. It assumes the appearance of a ball with 100 flattened planes, unlike other polyhedran dice. It is also known as "Zocchi's Golfball."

optimal strategy is to stop and take dollar if roll > x, and to pay $1 and continue when roll <= x (as applicable to all/any turn/roll). We need to find x in order to maximize E.

Assuming optimal expected value to be E, once you cast the first roll, there are two possible scenario:

1) we get roll value > x with probability (100-x)/100, and decide to take that amount, expected winning = (x+1+100)/2 (that amount varies uniformly from x+1 to 100 in this case, take the average)

2) we get roll value <= x with probability x/100 and decide to continue, with future expected winning again = E and 1$ loss in fee. Total expected winning in this scenario = E-1

[math]E=\frac{(100-x)}{100}\times\frac{ (x+1+100)}{2}[/math][math] + \frac{x}{100}\times(E-1)[/math]

[math]E = \frac{(100-x)(101+x) - 2x}{200 - 2x}[/math]

maximizing E w.r.t. x,

dE/dx = 0

2x^2 - 400x + 19600 = 0

x = ( 400+/- sqrt(160000-4*2*19600) ) / 4

Since, 1<= x <= 100, therefore x = ( 400+ sqrt(160000-4*2*19600) ) / 4

x = 85.85786

We just have to calculate E(x=85) and E(x=86) and compare.

E((x=86) = 87.35 and E(x=85) = 87.33

so optimal strategy is to stop and cash out only if the dice shows 87 or higher , else keep spending $1 to roll again n again, leading to expected winning of $87.35.

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It becomes apparent that the initial population (or the population 5 days ago) is 3125 bacteria.

Logarithms arise in problems of exponential growth

and decay because they are inverses of exponential functions. Because of the Laws of Logarithms, they also turn out to be useful in the measurement of the loudness of sounds, the intensity of earthquakes, and other processes that occur in nature

We can write an exponential function to represent the situation.

We know that the current population is 100,000.

The population doubles each day.

The standard exponential function is given by P(t) = A (2)^t

Since our current population is 100,000, a = 100000.

Since our rate is doubling, r = 2.

So

P(t) = 100000 (2)^t

We want to find the population five days ago.

So, we can say that t = -5. The negative represent the number of days that has passed.

Therefore

P(-5) = 100000 (2)^-5 =100000*(1/32) =3125

However, we dealing within this context, we really can't have negative days. Although it works in this case, it can cause some confusion. So, let's write a function based on the original population.

We know that the bacterial population had been doubling for 5 days. Let A represent the initial population. So, our function is:

P(t) = A (2)^t

After 5 days, we reach the 100,000 population. So, when t = 5, P(t) = 100000:

100000 =A[tex]2^5[/tex]

And solving for A, we acquire     :[tex]\frac{100000}{2^5}[/tex]

                                                      =3125

So, our function in terms of the original day is:

P(t) = 3125 (2)^t

So, it becomes apparent that the initial population (or the population 5 days ago) is 3125 bacteria.

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Answer:660,000

Step-by-step explanation:

principle= 24,000

rate=5.5%

time = 5years

simple interest= P×R×T

. =24000×5.5×5

. = 660,000

The possible formula for the depth of water in tank is , D = 2.8 + 8 sin(π t/4) where t is time in hours.

Sinusoidally function, is defined as one of triagnometric function with a smooth, repetitive oscillation. "Sinusoidal" comes from "sine", function. We can write its equation in the following form y = A·sin(B(x - C)) + D

where, A --> amplitude

B--> period in form of pi

C--> phase or horizontal shift

D --> vertical shift

We have, the depth of water in a tank oscillates sinusoidally.

The smallest depth of water in tank = 6.6 feet

The largest depth of water in tank = 9.4 feet

We have to determine the formula of depth of water in tank as a function of time in hours .

For sinusoidal function,

Amplitude , A = average depth

= ( largest depth + smallest depth)/2

= (6.6 + 9.4)/2

= 16.0/2 = 8

Now, The period of 8 hours so, B = 2π/8

=> B = 45° = π/4

Also, upward shift , D is 9.4 - 6.6 = 2.8

Horizontal shift, C = 0

So, required function is D = 2.8 + 8 sin(π t/4) .

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The answer is a ≤ 30.

An inequality is a mathematical expression that compares two values or expressions using an inequality operator. In this case, the inequality would be a ≤ 30. This means that the area of the triangle must be less than or equal to 30 square inches.

This can be solved by finding the area of the triangle. To do this, we need to know the lengths of the sides of the triangle. Suppose we have sides p, q, and r. Then the area of the triangle can be found using the formula:

                       a = 1/2 × b × h

where q is the base of the triangle and r is the height. To ensure that the area of the triangle is no more than 30 square inches, the inequality must be satisfied. That is, 1/2 × b × h ≤ 30. This inequality can be rewritten in terms of the side lengths by substituting 1/2 × q × r for h. This gives us a ≤ 30.

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Choose the variable whose numerical coefficients are smaller or have a smaller common multiple.

In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.

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[tex]2x = 25-3x[/tex]

Simplify:

[tex]2x=-3x+25[/tex]

Add 3x to both sides:

[tex]2x+3x=-3x+25+3x[/tex]

[tex]5x=25[/tex]

Divide both sides by 5:

[tex]\dfrac{5x}{5} =\dfrac{25}{5}[/tex]

[tex]\fbox{x = 5}[/tex]

If 400 people attend the convention,

the food will last for 3 days.

Two quantities a and b are said to be in inverse proportion if an increase in quantity a, there will be a decrease in quantity b, and vice-versa. In other words, the product of their corresponding values should remain constant.  Sometimes, it is also known as inverse variation.

That is, if ab = k, then a and b are said to vary inversely. In this case, if b1, b2 are the values of b corresponding to the values a1, a2 of a, respectively then a1 b1 = a2 b2 or a1/a2 = b2 /b1

Given,

Food for 300 people will Last for 4 days.

Attending people = 400

Let the food for 400 will last for x days.

According to the question,

300 × 4 = 400 × x

x = 300 × 4 /400

x = 3

Hence, the food will last 3 days,

if 400 people attend the convention.

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Answer:

Below

Step-by-step explanation:

x - 6 = 12      add 6 to both sides of the equation

x - 6 + 6 = 12 + 6     simplify

x = 18

3x-1 = 11        add  1  to both sides of the equation

3x -1 + 1 = 11 + 1

3x = 12          now divide both sides by 3

3x/3 = 12 / 3   simplify

x = 4

The average rate of change is equivalent to 2.

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

Given is the table as shown below -

[x] :  0  1  2  3  4

[y] :  -2 -3 1  3  6

The rate of change in the interval 0 < x < 4 is -

AROC = (6 + 2)/(4 - 0)

AROC = 8/4

AROC = 2

Therefore, the average rate of change is equivalent to 2.

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Answer:

Gain: $415

Loss: $10

Step-by-step explanation:

money earned - money spent = gain/loss

The gain is when the participant wins the telescope:

telescope = $425  ==> money earned

10 tickets * $1 = $10 for tickets  ==> money spent

$425 - $10 = $415

Loss is when the participant doesn't get the telescope:

$0 ==> money earned (No money earned)

10 tickets * $1 = $10 for tickets  ==> money spent

$0 - $10 = -$10 gained

-$10 gained ==> $10 lost

The margin of error and the sample mean of the confidence interval are given as follows:

The confidence interval is defined as follows:

(12.7, 19.5).

The definition of a confidence interval is that it is obtained as the sample mean plus/minus the margin of error.

Hence the sample mean is calculated as the mean of the bounds of the interval, hence:

(12.7 + 19.5)/2 = 16.1.

The margin of error is calculated as the absolute value of the difference of each bound from the sample mean, hence:

|19.5 - 16.1| = |12.7 - 16.1| = 3.4.

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11 students could sell 1870 cookies.

A statement, at least two variables or integers, and one or more arithmetic operations make up a mathematical expression. This mathematical operation enables the multiplication, division, addition, or subtraction of numbers.

Combining numbers, variables, and operators to represent something's value is known as a mathematical expression. Setting two expressions equal to one another creates an equation, a mathematical statement.

An expression is a collection of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) Expressions and phrases are similar in certain ways. Language phrases can comprise actions on their own, but they do not make up complete sentences.

=  10 X 17

=  170 X 11

= 1870

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A form of reasoning called induction is the process of forming general ideas and rules based on your experiences and observations.

The process of welcoming newly hired employees and assisting them in adjusting to their new positions and working surroundings is known as induction. Beginning a new work can be stressful, so new hires require assistance adjusting to their new environment.

The action or process of inducting someone to a position or organization.

We have given that,

A form of reasoning called is the process of forming general ideas and rules based on your experiences and observations.

As a result, the process of formulating general hypotheses and rules based on your observations and experiences is known as induction.

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The score interval that is within one standard deviation of the mean is

(70 , 80).

What is standard deviation?

The standard deviation is a statistic that expresses how much variance or dispersion there is in a group of numbers. While a high standard deviation suggests that the values are dispersed over a wider range, a low standard deviation suggests that the values tend to be close to the mean of the collection.

Some of the properties of the standard deviation are:

1. It cannot be negative

2. It is only employed to calculate the spread or dispersion around a data set's mean

3. It displays the degree of deviation  from the mean value

4. The larger the spread, the more standard deviation, with data of about the same mean.

Given,

 The mean of the test μ = 75

The standard deviation σ = 5

We are asked to find the scores within one standard deviation of the mean.

This means that the scores should be in the interval ( μ - σ , μ + σ )

μ - σ = 75 - 5 =70

μ + σ = 75 + 5 = 80

Therefore the scores should be in the interval = ( 70,80), which is within one standard deviation of the mean.

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Answer:

5 kg.

Step-by-step explanation:

2 kg for $0.96 is more expensive than 5 kg for $1.95

Answer:

Find the value of xx in the equation below.

11=

11=

\,\,x -6

x−6

Step-by-step explanation:

The sum of the interior angles of a nonagon (a 9-sided polygon) is 1,450°.

1. The interior angles of a polygon can be calculated by subtracting 2 from the number of sides, then multiplying the result by 180°.

2. For a nonagon (a 9-sided polygon), the calculation is

(9-2) × 180° = 1,450°.

The sum of the interior angles of any polygon can be calculated using the formula

(n-2) × 180°,

where n is the number of sides of the polygon. This formula works because the sum of the interior angles of any polygon is always equal to (n-2) times the measure of a single interior angle. For a nonagon, a 9-sided polygon, the calculation is

(9-2) × 180° = 1,450°.

This means that the sum of the interior angles of a nonagon is 1,450°.

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Answer:

15% off means take $3.75 off the regular price

Step-by-step explanation:

$25(0.15) = $3.75

$25 - $3.75 = $21.75  sale price

The four consecutive odd integers = 15, 17, 19, 21

Positive, negative, and zero are all examples of integers.

The Latin word "integer" signifies "whole" or "intact."

As a result, fractions and decimals are not included in integers.

All whole numbers and negative numbers are considered integers.

This means that if we combine negative numbers with whole numbers, a collection of integers results.

An integer, which can comprise both positive and negative integers, including zero, is a number without a decimal or fractional portion.

Here are a few instances of integers: -5, 0, 1, 5, 8, 97, and 3,043. a collection of numbers, denoted by the symbol Z.

According to our question-

First digit: x

Second digit: x + 2

Three-digit number Equals x + 4

Fourth digit: x + 6

The difference between the opposite of the third and the first two sums up to 55.

x + x + 6 = -(x + 4) + 55

x + x+6 = (-x - 4) + 55

2x + 6 = -x + 51

assemble similar terms

2x + x = 51 - 6

3x = 45

x = 15

Therefore,

First digit: x = 15

Second digit: 15 + 2 + x = 17

Third number: 15 + 4 + 4 = 19

Fourth digit: x+6 = 15+6 = 21

The four odd numbers in a row equal 15, 17, 19, and 21.

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Answer:

y = -8/3x + 73/4

Step-by-step explanation:

A line perpendicular to the 3rd equation has a slope that is the negative inverse of the other line. It also has the same y-intercept (b) because it is forming a square.

Therefore, the equation of the 4th line is:

y = -8/3x + 73/4

Answer:

12

Step-by-step explanation:

3 + 4 + 5 = 12

The fraction that is bigger than 2/5 and smaller than 2/3 is 8/15.

A fraction is written in the form of a numerator and a denominator where the denominator is greater that the numerator.

Example: 1/2, 1/3, 1/5 is a fraction.

We have,

Let the fraction be a/b.

2/5 < a/b < 2/3

Now,

a/b

= (2/3 + 2/5) / 2

= (10 + 6) / 30

= 16/30

= 8/15

= 0.533

Thus,

The fraction is 8/15.

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The nature of roots of quadratic equation x² = -2x + 1 is real and distinct.

The standard form of a quadratic equation is:

ax² + bx + c = 0

The discriminant is defined as:

D = b² - 4ac

Using  the value of discriminant, we can determine the nature of roots of a quadratic equation.

If:

D > 0, the quadratic equation has two distinct real roots or the roots are real and distinct.

D = 0, the quadratic equation has 1 real equal roots or the roots are unique

D < 0, the quadratic equation has complex conjugate roots or the roots are imaginary.

In this case, the quadratic equation is:

x² = -2x + 1

transform it into the standard form:

x² + 2x - 1 = 0

Calculate the discriminant:

D = 2²  - 4 (-1) = 8

Since D > 0, the roots are two distinct real roots.

Your question is incomplete, but most probably your question was:

What is the nature of roots of quadratic equation x² = -2x + 1?

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

Explanation:

We'll break things up into cases. The phrasing "at least 3" means "3 or more".

We either will have 3 kings or 4 kings.

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

Case A) There are exactly 3 kings.

Use the nCr combination formula. There are n = 4 kings total and we select r = 3 of them. That gives 4C3 = 4 ways to pick the 3 kings.

Put another way: there are 4 ways to leave a certain king out of the hand.

Then we have 52-4 = 48 cards that aren't a king, and we need to fill the remaining r = 2 slots. Through the nCr formula, you should find that 48C2 = 1128

To recap, we found

That gives 4*1128 = 4512 ways to have case A happen.

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

Case B) There are exactly 4 kings

4C4 = 1, so there's only one way to select all the kings. The order doesn't matter. Then we have 48 cards to pick from for the fifth slot.

Overall there are 1*48 = 48 ways to have case B play out.

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

There were 4512 ways to have case A happen, and 48 ways to have case B happen.

These cases are mutually exclusive to allow us to simply add the counts:

4512+48 = 4560

This is why the final answer is 4560