A rain gutter is made of sheets of metal 9 inches wide. The gutters have a 3 inch base and two 3 inch sides, folded up at an angle of . What angle will maximize the cross sectional area of the rain gutter?

The relative maximum of the function is (0, 4), and the relative minimum is (6, -152).

To find the relative extrema of the function f(x) = x^4 - 8x^3 + 4, we first take the derivative of the function:

f'(x) = 4x^3 - 24x^2

Then we set f'(x) = 0 to find the critical points:

4x^3 - 24x^2 = 0

4x^2(x - 6) = 0

This gives us two critical points: x = 0 and x = 6.

Next, we find the second derivative of f(x):

f''(x) = 12x^2 - 48x

We can use the Second-Derivative Test to determine the nature of the critical points.

For x = 0, we have:

f''(0) = 0 - 0 = 0

This tells us that the Second-Derivative Test is inconclusive at x = 0.

For x = 6, we have:

f''(6) = 12(6)^2 - 48(6) = 0

Since the second derivative is zero at x = 6, we cannot use the Second-Derivative Test to determine the nature of the critical point at x = 6.

To determine whether the critical points are relative maxima or minima, we can use the first derivative test or examine the behavior of the function around the critical points.

For x < 0, f'(x) < 0, so the function is decreasing.

For 0 < x < 6, f'(x) > 0, so the function is increasing.

For x > 6, f'(x) < 0, so the function is decreasing.

Therefore, we can conclude that the critical point at x = 0 is a relative maximum and the critical point at x = 6 is a relative minimum.

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Using graphs, we can see that the point (4,2) can be a coordinate where y will represent x.

The graph is simply a structured representation of the data. The numerical information gathered through observation is referred to as data.

If there is just one value of y (output) for every value of x, the relationship between x and y is said to be a function (input).

In other words, there can only be one value of y for each value of x.

Determine each plotted point's coordinates first:

(-4,4)

(-2,3)

(0,1)

(2, -1)

(3,0)

The following point cannot have any of the x-coordinates of the displayed points, which are -4, -2, 0, 2, and 3.

Options include:

A (0,1) →The relationship cannot be regarded as a function at this stage as the x-coordinate zero already has a corresponding value of y.

B (2,2) →Although there is already a value of y for the location x=2, the relationship cannot be regarded as a function at this point.

C (3,4) →Although there is already a value of y for the location x=3, the relationship cannot be regarded as a function at this point.

D (4,2) → The relationship will still be regarded as a function even though there are no points on the graph with the coordinates x=4 displayed.

Therefore, option D (4,2) is the point where y will represent x.

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The complete question:

Please see attached question

To sketch the phase plane and trajectories of both populations in the competing species model, plot the population of one species on the x-axis and the population of the other species on the y-axis. Then, plot the isoclines and use them to determine the direction and stability of the population trajectories.

The competing species model is a system of two differential equations that describe the population dynamics of two species competing for the same resources. To sketch the phase plane and trajectories, plot the population of one species on the x-axis and the population of the other species on the y-axis. Then, plot the isoclines, which are curves that represent the values of one species' population at which the other species' population does not change.

The isoclines are found by setting each differential equation to zero and solving for one population in terms of the other. For example, the isocline for species 1 is found by setting dN1/dt = 0 and solving for N2. The resulting equation gives the values of N2 at which the population of species 1 does not change. Plotting these curves on the phase plane divides it into regions where the population of each species increases or decreases.

The direction and stability of the population trajectories can be determined by analyzing the slope of the vector field, which represents the rate of change of the population at each point in the phase plane. Trajectories move in the direction of the vector field, and their stability depends on the curvature of the isoclines. If the isoclines intersect at a single point, it is a stable equilibrium where both populations coexist. If they intersect at multiple points, the stable equilibrium depends on the initial conditions of the populations. If they do not intersect, one species will eventually drive the other to extinction.

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--The question is incomplete, answering to the question below--

"Consider the competing species model, how to sketch the phase plane and the trajectories of both population"

Answer:

B

Step-by-step explanation:

5^2 is the small square, 4(3x4x1/2) are the 4 triangles

Answer: The answer would be B.

Step-by-step explanation:

Hello.

First, we know that the smaller square is 5, and to find the area of the big square, we need to square 5 to get the area. We also know that C wouldn't  be a viable option, so, our only remaining choices are A and B. We know that without the smaller square, there are 4 triangles, and the Area of a Triangle is: 1/2*b*h. So, this also takes A out as an option as well. After this, you will have your answer as B; 5^2 + 4(3 * 4 * 1/2)

(Or, you could have found the Area of the Triangles, and realize that neither A, nor C have those options, making B the answer by default.)

Hope this helps, (and maybe brainliest?)

Answer:

10 in.

Step-by-step explanation:

V = LWH

100 = L × 2.5 × 4

L = 10

Answer:

Step-by-step explanation:

To calculate the interest earned by Linda for the first year, we can use the formula:

A = P(1 + r/n)^(nt)

Where A is the amount after t years, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

For the first year, we have:

A = $50,000(1 + 0.06/1)^(1*1) = $53,000

So, the interest earned by Linda for the first year is:

Interest = $53,000 - $50,000 = $3,000

For the second year, we can use the same formula with t = 2:

A = $50,000(1 + 0.06/1)^(1*2) = $56,180

Interest = $56,180 - $53,000 = $3,180

For the third year, we can use the same formula with t = 3:

A = $50,000(1 + 0.06/1)^(1*3) = $59,468.80

Interest = $59,468.80 - $56,180 = $3,288.80

Now, to calculate the interest earned by Bob for each of the first three years, we can use the formula:

Interest = Prt

Where P is the principal amount, r is the annual interest rate, and t is the time in years.

For the first year, we have:

Interest = $50,0000.061 = $3,000

For the second year, we have:

Interest = $50,0000.061 = $3,000

For the third year, we have:

Interest = $50,0000.061 = $3,000

As we can see, Linda earns more interest than Bob for each year, as her interest is compounded annually, while Bob's interest is simple interest. Therefore, the answer is:

Linda earns more.

Answer:

Linda earns $9550.8 interest and bob earns $9000 interest

Step-by-step explanation:

Linda takes compound interest: C.I. = Principal (1 + Rate)Time − Principal

interest= 50,000(1+6/100)³

=59550.8 - 50000

Linda earns $9550.8 interest in 3 years.

bob takes simple interest: S.I = prt/100

interest = 50,000*6*3/100

Bob earns $9000 in 3 years.

thus, Linda earns more interest than bob.

Answer:

258 m/s

250 m/s

Step-by-step explanation:

s = 2t³

s(5) = 2(5)³ = 250

s(8) = 2(8)³ = 1024

average = (s(8) - s(5))/(8 - 5) = (1024 - 250)/3 = 258 m/s

s(5) = 2(5)³ = 250 m/s

The probability of a pitcher throwing exactly 5 pitches in an at-bat is 0.1 or 10%.

What is probability and how is it calculated?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain. The probability of an event A is calculated as the ratio of the number of outcomes that correspond to event A to the total number of possible outcomes.

Calculating probability of a pitcher throwing exactly 5 pitches :

To calculate the probability of a pitcher throwing exactly 5 pitches in an at-bat, we need to add up the frequencies of all the at-bats that have exactly 5 pitches. From the given table, we see that there are 8 at-bats that have exactly 5 pitches.

The total number of at-bats is the sum of the frequencies of all pitch counts.

Total number of at-bats = 12+16+32+12+8 = 80

Therefore, the probability of a pitcher throwing exactly 5 pitches in an at-bat is:

P(5) = Frequency of 5-pitch at-bats / Total number of at-bats

P(5)= 8/80 = 0.1 or 10%

Hence, the probability that a pitcher will throw exactly 5 pitches in an at-bat is 10%.

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Answer:

  (x -14)² +(y -7)² = 1²

Step-by-step explanation:

You want the equation of the circle that represents the border of a logo centered 14 m right and 7 m up from the lower left corner of a soccer field. The logo is 2 m in diameter.

The equation of a circle with center (h, k) and radius r is ...

  (x -h)² +(y -k)² = r²

Since the origin of the coordinate system is the lower left corner of the field, the center is located at (h, k) = (14, 7). The diameter of 2 m means the radius is 1 m. Using these values in the equation, it becomes ...

  (x -14)² +(y -7)² = 1²

Tom's fortnightly income is $3000.

In mathematics, an average is a measure that represents the central or typical value of a set of numbers. There are several types of averages commonly used, including the mean, median, and mode.

To calculate Tom's fortnightly income, we need to divide his yearly salary by the number of fortnights in a year:

Fortnightly income = Yearly salary / Number of fortnights in a year

Fortnightly income = $78000 / 26 = $3000

Therefore, Tom's fortnightly income is $3000.

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question - Calculate the  Tom's fortnightly income and yearly salary by the number of fortnights in a year .

The rate at which the distance between the car and the truck is changing when the car is 9 m from the intersection and the truck is 40 m from the intersection is about 19.187 m/s

The rate of change of a function is an indication of how fast the function's output changes per unit change in the input of the function.

Let y represent the distance of the car from the intersection, and let x represent the distance of the truck from the intersection, we get;

The distance between the car and the truck, d, can be obtained from Pythagorean Theorem as follows;

d² = y² + x²

2·d·d/dt = 2·y·dy/dt + 2·x·dx/dt

dy/dt = The speed of the car = 48 km/h = 40/3 m/s

dx/dt = The speed of the truck = 60 km/h = 50/3 m/s

y = The distance of the car from the intersection = 9 m

x = The distance of the truck from the intersection = 40 m

d² = 9² + 40² = 1681

d = √(1681) = 41

d = 41 m

Therefore;

2×41×d/dt = 2×9 × 40/3 + 2 × 40 × 50/3 = 4720/3

d/dt = 4720/3/(2 × 41) = 2360/123 ≈ 19.187

d/dt ≈ 19.187 m/s

The rate of change distance between the car and the truck d/dt is about 19.187 m/s

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The new water bill after 5% increase is $126.

The ratio that may be stated as a fraction of 100 is called as a percentage in mathematics. To compute a percentage of a number, divide it by its whole and then multiply it by 100. The percentage therefore refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.

Given bill of water =$120

Increment in percentage=5%

New water bill= 120+120×5/100

=$126

Hence, The new water bill after 5% increase is $126.

% increase = Increase in bill ÷ Old bill × 100.

5%=Increase in bill/120×100

Increase=5×120/100

=6

New bill=Old bill+ increase=120+6=126

hence, the new water bill after 5% increase is $126.

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The probability of obtaining a reading less than 0.35° C is approximately 35%.

Accοrding tο the prοbability fοrmula, the likelihοοd οf an event οccurring is equal tο the ratiο οf the number οf favοurable οutcοmes tο the tοtal number οf οutcοmes. Prοbability οf an event οccurring P(E) = The number οf favοurable οutcοmes divided by the tοtal number οf οutcοmes.

The readings at freezing οn a set οf thermοmeters are nοrmally distributed, with a mean (x) οf 0°C and a standard deviatiοn (μ) οf 1.00°C. We want tο knοw hοw likely it is that we will get a reading that is less than 0.35°C.

To solve this problem, we must use the z-score formula to standardise the value:

[tex]$Z = \frac{x - \mu}{\sigma}[/tex]

Z = standard score

x = observed value

[tex]\mu[/tex] = mean of the sample

[tex]\sigma[/tex] = standard deviation of the sample

Here

x = 0.35° C

[tex]\mu[/tex] = 0° C

[tex]\sigma[/tex] = 1.00°C

Using the values on the formula:

[tex]$Z = \frac{0.35 - 0}{1}[/tex]

Z = 0.35

The probability of obtaining a reading less than 0.35° C is approximately 35%.

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The standard normal areas are given as follows:

The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Considering the second bullet point, the areas are given as follows:

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V is 5.92 centimetres above the pyramid's base at its highest point.

A pyramid is a 3D pοlyhedrοn with the base οf a pοlygοn alοng with three οr mοre triangle-shaped faces that meet at a pοint abοve the base. The triangular sides are called faces and the pοint abοve the base is called the apex. A pyramid is made by cοnnecting the base tο the apex. Sοmetimes, the triangular sides are alsο called lateral faces tο distinguish them frοm the base. In a pyramid, each edge οf the base is cοnnected tο the apex that fοrms the triangular face.

Give the altitude the letter h. Next, we have:

tan(60) = h/2

Simplifying, we get:

h = 2 tan(60) = 2 √(3)

The Pythagorean theorem yields the following:

[tex]$\begin{align}{{V O^{2}+O F^{2}=V F^{2}}}\\ {{V O^{2}+1^{2}=6^{2}}}\\ {{V O^{2}=35}}\end{align}$[/tex]

Taking the square root of both sides, we get:

VO ≈ 5.92 cm

Rounding to three significant figures, we get:

VO ≈ 5.92 cm

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Answer:

Step-by-step explanation:

2

IV individuals from different cultures

DV the belief that dreams have meaning.

Independent Variable: Culture

Belief in the meaning of dreams is a quasi-independent variable (since it cannot be manipulated or assigned randomly)

The response to whether or not dreams have meaning is the dependent variable.

An experimental investigation typically contains three types of variables: independent variables, dependent variables, and controlled variables.

A compared to the rest of the country. Because the variable levels are pre-existing, it is not possible to assign participants to groups at random.

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Answer:

1/6

Step-by-step explanation:

5/6 - 4/6 = 1/6

Answer:

Step-by-step explanation:

To subtract these fractions, we need to have a common denominator. In this case, the denominators 5 and 6 do not match, so we have to find the least common multiple (LCM) of 5 and 6, which is 30.

Then, we can convert both fractions so that they have a denominator of 30:

5/6 = 25/30

4/6 = 20/30

Now we can subtract the numerators:

25/30 - 20/30 = 5/30

Simplifying the result to its lowest terms, we have:

5/30 = 1/6

Therefore, 5/6 - 4/6 = 1/6.

Answer: 6y + 6

Step-by-step explanation:

To simplify the expression 8(3/4y -2) + 6(-1/2+4) + 1, we can follow the order of operations (PEMDAS):

First, we simplify the expression within parentheses, working from the inside out:

6(-1/2+4) = 6(7/2) = 21

Next, we distribute the coefficient of 8 to the terms within the first set of parentheses:

8(3/4y -2) = 6y - 16

Finally, we combine the simplified terms:

8(3/4y -2) + 6(-1/2+4) + 1 = 6y - 16 + 21 + 1 = 6y + 6

Therefore, the expression 8(3/4y -2) + 6(-1/2+4) + 1 is equivalent to 6y + 6.

Graph D of the given function has the smallest value for b.

As per name signifies, exponents are used in exponential functions. But take note that an exponential function does not have a constant as its base and a variable as its exponent. One of the following forms can be used for an exponential function.

f (x) = aˣ

y=f(x) >0

f(x)=abˣ , where a>0

So, f(x)=abˣ

When, b<1 f(x) decreases

When, b>1 f(x) increases and the larger the b the steeper the graph

So, graph of D is increasing and is steepest

So, graph D has the smallest value for b.

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a. X= number of unbroken eggs in a standard carton. b. Y= number of absent students on first day of a course. c. U= number of swings before hitting a golf ball. d. X= length of a rattlesnake.

e. Z= royalties earned from selling a first edition of 10,000 textbooks. f. Y= pH of a soil sample. g. X= tension (psi) of a tennis racket. h. X= total coin tosses required for three individuals to get a match.

a. X can take on values 0, 1, 2, 3, 4, 5, 6, as there can be zero to six unbroken eggs in a standard egg carton. X is a discrete random variable.

b. Y can take on values 0, 1, 2, 3, ..., n, where n is the total number of students on the class list. Y is a discrete random variable.

c. U can take on values 1, 2, 3, .... U is a discrete random variable.

d. X can take on any positive real value, as the length of a rattlesnake can vary continuously. X is a continuous random variable.

e. Z can take on any non-negative real value, as the amount of royalties earned can be any non-negative amount. Z is a continuous random variable.

f. Y can take on any value between 0 and 14, as the pH of a soil sample can range from 0 to 14. Y is a continuous random variable.

g. X can take on any positive real value, as the tension at which a tennis racket has been strung can vary continuously. X is a continuous random variable.

h. X can take on values 3, 4, 5, 6, ... as there must be at least three coin tosses and the tosses must continue until a match is obtained. X is a discrete random variable.

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The polynomial function with the given zeros and numeric value at x = 1 is given as follows:

f(x) = x^4 - 7x³ + 16x² - 8x - 32

The zeros of the polynomial function are given as follows:

Then the linear factors of the function are given as follows:

According to the Factor Theorem, the function with leading coefficient a can be defined as a product of it's linear factors are follows:

f(x) = a(x + 1)(x - 4)(x - 2 - 2i)(x - 2 + 2i).

f(x) = a(x² - 3x - 4)(x² - 4x + 8)

f(x) = a(x^4 - 7x³ + 16x² - 8x - 32).

When x = 1, y = -30, hence the leading coefficient a is obtained as follows:

-30 = a(1 - 7 + 16 - 8 - 32)

-30a = -30

a = 1.

Hence the function is:

f(x) = x^4 - 7x³ + 16x² - 8x - 32

The problem asks for the polynomial function with the given zeros and numeric value at x = 1.

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The z-scores that represent the middle 50% of the standard normal distribution are between -0.6745 and 0.6745, the lowest 5% of the standard normal distribution is -1.645, and the 90% of the standard normal distribution is between -1.645 and 1.645.

The mean and variance of a standard normal distribution are both 0. A z distribution is another name for this.

Yes, here are the z-scores for the given standard normal areas:

A. Middle 50%: The area between the 25th and 75th percentiles corresponds to the middle 50% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-scores that correspond to those percentiles as follows:

-0.6745 is the z-score corresponding to the 25th percentile.

The 75th percentile z-score is 0.6745.

As a result, the z-scores representing the middle 50% of the standard normal distribution range between -0.6745 and 0.6745.

B. Lowest 5%: The area to the left of the 5th percentile corresponds to the lowest 5% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-score that corresponds to that percentile as follows:

The z-score associated with the fifth percentile is -1.645.

As a result, the z-score representing the bottom 5% of the standard normal distribution is -1.645.

C. Middle 90%: The area between the 5th and 95th percentiles corresponds to the middle 90% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-scores that correspond to those percentiles as follows:

The z-score associated with the fifth percentile is -1.645.

The z-score associated with the 95th percentile is 1.645.

As a result, the z-scores representing the middle 90% of the standard normal distribution range between -1.645 and 1.645.

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The correct numerical expression for the given statement "multiply the sum of 7 and 6 by the sum of 4 and 5" is (7 + 6) x (4 + 5).

Mathematical statements are called expressions if they have at least two words that are related by an operator and contain either numbers, variables, or both. There are two sorts of expressions in mathematics: numerical expressions, which only contain numbers, and algebraic expressions, which also include variables. A mathematical expression is a phrase that has a minimum of two numbers or variables and at least one mathematical operation.

The given statement, "multiply the sum of 7 and 6 by the sum of 4 and 5", can be represented as:

sum of 7 and 6 = (7 + 6)

sum of 4 and 5 = (4 + 5)

multiply: (7 + 6) x (4 + 5)

Hence, the correct numerical expression for the given statement is (7 + 6) x (4 + 5).

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Answer:

(7 + 6) x (4 + 5).

Step-by-step explanation:

hope this helps!

The eigenvalues of the reflection about a plane in R3 are 1 and -1, with corresponding eigenvectors lying on the plane and perpendicular to the plane, respectively. Therefore, the transformation is diagonalizable with an eigenbasis consisting of these eigenvectors.

Consider a reflection about a plane V in R3. Let's denote this linear transformation by T.

We know that any vector v in R3 can be decomposed uniquely into a sum of two vectors, one in V and one in the orthogonal complement of V. Let's denote these subspaces by V and V⊥, respectively. Then we have:

R3 = V ⊕ V⊥

Since T reflects vectors across the plane V, any vector in V will be fixed by the transformation, while any vector in V⊥ will be flipped across the plane.

Let's consider a vector v in V. Since T fixes v, we have:

T(v) = v

This means that v is an eigenvector of T with eigenvalue 1.

Now let's consider a vector u in V⊥. Since T flips u across the plane V, we have:

T(u) = -u

This means that u is an eigenvector of T with eigenvalue -1.

Since any vector in R3 can be written as a sum of a vector in V and a vector in V⊥, we have shown that every vector in R3 is an eigenvector of T, and the corresponding eigenvalues are 1 and -1.

To find an eigenbasis, we need to find a basis for R3 consisting of eigenvectors of T. We have already shown that every vector in R3 is an eigenvector, so the standard basis {e1, e2, e3} is an eigenbasis. Therefore, T is diagonalizable.

The eigenvalues are λ1 = 1 and λ2 = -1, and the corresponding eigenvectors are {v} and {u}, where v is any nonzero vector in V and u is any nonzero vector in V⊥.

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Probability that precisely 5 people will respond that they would feel comfortable is 0.0808

Probability that more than 5 people will respond that they would feel comfortable is0.1061

Probability that at most 5 people will respond that they would feel comfortable is 0.9747

Probability is a way to gauge how likely something is to happen. Several things are difficult to forecast with absolute confidence.

35 percent of households claim that having $50,000 in savings would make them feel comfortable. Ask 8 homes that were chosen at random if they would feel comfortable if they had $50,000 in savings.

Binomial conundrum with p(secure) = 0.35 and n = 8.

the likelihood that the number of people who claim they would feel comfortable is

(a) The number exactly five is equal to ⁸C₅ (0.35)5×(0.65)×3=binompdf(8,0.35,5) = 0.0808.

(b) more than five = 1 - binomcdf(8,0.35,4) = 0.1061

(c) at most five = binomcdf(8,0.35,5) = 0.9747.

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The relationship between AB and BC is given as follows:

AB > BC.

Two angles are defined as supplementary angles when the sum of their measures is of 180º.

The supplementary angles for this problem are given as follows:

By the law of sines, we have that:

AB/sin(110º) = BC/sin(70º).

As sin(110º) > sin(70º), the inequality for this problem is given as follows:

AB > BC.

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Answer:

AB>BC

Step-by-step explanation:

AI-generated answer

Based on the given information, the relationship between AB and BC depends on the measure of angle ZADB. If mZADB is 110°, we can determine the relationship as follows:

Since triangle ABD and triangle CBD share side AB, the larger the angle ZADB, the longer the side AB will be compared to side BC. Therefore, if mZADB is 110°, we can conclude that AB is greater than BC.

In summary, when mZADB is 110°, the relationship between AB and BC is:

AB > BC.

The real solutions of the following equation are 5 and 1

[ Graph is attached below ]

To solve the given equation graphically we need to find the coordinate points that pass through the graph.

This can be done by taking 'x' values and solving for them 'y' values.  

Now draw the graph using the above coordinates and find the solution as shown below.

Here we have

x³ - 6x²+ 5x = 0  

Let y = x³ - 6x²+ 5x  

To draw the graph find the coordinates of points as follows

At x = 0 =>  y = (0) + (0) + (0) = 0

At x = 1 => y = (1)³ - 6(1)² + 5(1) = 0

At x = -1 => y = (-1)³ - 6(-1)² + 5(-1) = - 12

At x = 2 => y = (2)³ - 6(2) + 5(2) = 6

From the above calculation,

The coordinates of the points to draw the graph are (0, 0), (1, 0), (-1, -12), and (2, 6)

Here the solutions of the graph are the x-coordinate of points where the graph cuts the x-axis

From the figure, the graph will cut the x-axis at 1 and 5  

Therefore,

The real solutions of the following equation are 5 and 1

[ Graph is attached below ]

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(a) The function x as a function of t is t = 250ln(499x/998)

(b) Half the population will have heard the rumor approximately 109.86 units of time after it was started.

(a) To solve the differential equation dt/dx = kx(N−x), we can separate the variables and integrate

dt/dx = kx(N−x)

dt/(N-x) = kx dx

Integrating both sides, we get

t = -1/k × ln(N-x) - 1/k × ln(x) + C

where C is the constant of integration.

To find C, we can use the initial condition that two people start the rumor at t=0, so x=2:

0 = -1/k * ln(N-2) - 1/k * ln(2) + C

C = 1/k * ln(N-2) + 1/k * ln(2)

Substituting C back into the equation, we get:

t = -1/k * ln(N-x) - 1/k * ln(x) + 1/k * ln(N-2) + 1/k * ln(2)

Simplifying, we get

t = 1/k * [ln((N-2)x/(2(N-x)))]

Substituting k=1/250 and N=1000, we get:

t = 250ln(499x/998)

(b) We want to find the time t when half the population has heard the rumor, so x = N/2 = 500. Substituting this into the equation we obtained in part (a), we get

t = 250ln(499(500)/998) = 250ln(249/499)

t ≈ 109.86

Therefore, half the population will have heard the rumor approximately 109.86 units of time after it was started.

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The Butler family used their sprinkler for 30 hours and the Phillips family used their sprinkler for 25 hours.

Let's solve the problem with algebra.

Let x represent the number of hours the Butlers used their sprinkler, and y represent the number of hours the Phillips family used their sprinkler. We are aware of the following:

The Butler family's sprinkler had a water output rate of 25 L per hour, so the total amount of water they used is 25x.

The Phillips family's sprinkler had a water output rate of 40 L per hour, so the total amount of water they used was 40y.

The sprinklers were used by the families for a total of 55 hours, so x + y = 55.

The total amount of water produced was 1750 L, so 25x + 40y = 1750.

Using these equations, we can now solve for x and y.

First, we can solve for one of the variables in terms of the other using the equation x + y = 55. For instance, we can solve for x as follows:

x = 55 - y

When we plug this into the second equation, we get:

25(55 - y) + 40y = 1750

We get the following results when we expand and simplify:

1375 - 25y + 40y = 1750

15y = 375

y = 25

As a result, the Phillips family ran their sprinkler for 25 hours. We get the following when we plug this into the equation x + y = 55:

x + 25 = 55

x = 30

As a result, the Butlers used their sprinkler for 30 hours.

As a result, the Butler family sprinkled for 30 hours and the Phillips family sprinkled for 25 hours.

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