Euler's method is based on the idea of walking along tangent lines of nearby solutions for short periods of time
a. true b. false

Repeat the process for different combinations of n and p values to explore the performance of the normal approximation under various scenarios.

To evaluate how well a normal distribution can approximate a binomial distribution using simulation, you can follow these steps:

1. Choose a combination of n (number of trials) and p (probability of success) from the given sets: n ∈ {10, 50, 100, 200} and p ∈ {0.01, 0.1, 0.5}.

2. Generate 100 samples of binomial distributions using the chosen n and p values: X ~ Binomial(n, p).

3. Calculate T for each sample using the formula[tex]T = Vn(X/n - p),[/tex] where[tex]Vn= \sqrt{(n / p(1 -p))[/tex]. This will result in a transformation that should approximate N(0, p(1 – p)) if n is large.

4. Perform the Kolmogorov-Smirnov test (ks.test) to compare the empirical distribution of T with the theoretical normal distribution N(0, p(1 – p)). The null hypothesis is that the two distributions are the same, and the alternative hypothesis is that they are different.

5. Check the p-value obtained from the Kolmogorov-Smirnov test. If the p-value is less than the significance level (α = 0.05), you can reject the null hypothesis and conclude that the deviation of T from N(0, p(1 – p)) can be detected. If the p-value is greater than α, you cannot reject the null hypothesis, and the normal distribution approximation is considered valid.

6. Repeat the process for different combinations of n and p values to explore the performance of the normal approximation under various scenarios.

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At a significance level of 0.03, the critical value is 1.711 (found using a t-distribution table with degrees of freedom equal to 20 and a significance level of 0.015). Since the calculated test statistic (2.568) is greater than the critical value, we reject the null hypothesis at a significance level of 0.03.

The degrees of freedom for the unequal variance test can be calculated using the formula:

[tex]df = [(s1^2/n1 + s2^2/n2)^2] / [((s1^2/n1)^2)/(n1-1) + ((s2^2/n2)^2)/(n2-1)][/tex]

Substituting the given values, we get:

[tex]df = [(15^2/27 + 9^2/15)^2] / [((15^2/27)^2)/(27-1) + ((9^2/15)^2)/(15-1)][/tex]

= 20.37

≈ 20 (rounded down to nearest whole number)

The decision rule for the 0.025 significance level and a right-tailed test is to reject the null hypothesis if the test statistic exceeds the critical value. The critical value can be found using a t-distribution table with degrees of freedom equal to 20 and a significance level of 0.025. From the table, we find the critical value to be 1.734.

The test statistic for the two-sample t-test with unequal variances can be calculated using the formula:

[tex]t = (x1 - x2) / sqrt(s1^2/n1 + s2^2/n2)[/tex]

Substituting the given values, we get:

[tex]t = (114 - 106) / sqrt(15^2/27 + 9^2/15)[/tex]

= 2.568

Using a significance level of 0.025 and degrees of freedom equal to 20, the critical value is 1.734. Since the calculated test statistic (2.568) is greater than the critical value, we reject the null hypothesis.

At a significance level of 0.03, the critical value is 1.711 (found using a t-distribution table with degrees of freedom equal to 20 and a significance level of 0.015). Since the calculated test statistic (2.568) is greater than the critical value, we also reject the null hypothesis at a significance level of 0.03. Therefore, we can conclude that there is evidence to suggest that the population mean of the first population is greater than that of the second population.

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Sierra can select 1 of each item to accessorize her outfit in 384 ways.

The number of ways Sierra can  elect one belt from 8 belts is 8. The number of ways she can  elect one handbag from 4 handbags is 4. And the number of ways she can  elect one choker from 12 chokers is 12. thus, the total number of ways Sierra can  elect one of each item is   Total number of ways =  8 x 4 x 12 =  384  

This means that Sierra has 384 different options to choose from when  opting  accessories to  round  her outfit. This gives her a wide range of choices to  produce a unique and  swish look that suits her  particular taste and preferences.  

In conclusion, the  addition rule of counting is an essential principle in combinatorics and provides a simple way to calculate the total number of  issues in a given situation. In this case, we used the  addition rule to determine the total number of ways Sierra could  elect one belt, one handbag, and one choker to accessorize her outfit.

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There is no point of diminishing returns for the sales function S(x) = 204 + 6.3x - 0.25x² in the interval 0 ≤ x ≤ 12.

1.To determine the intervals where the function f(x) = √(2x + 3) is concave up or concave down, we need to find its second derivative and analyze its sign:
Step 1: Find the first derivative, f'(x):
f'(x) = (1/2)(2x + 3)^(-1/2) * (2)
Step 2: Find the second derivative, f''(x):
f''(x) = (1/4)(-1/2)(2x + 3)^(-3/2) * (2)
Step 3: Determine where f''(x) is positive (concave up) or negative (concave down):
Since the second derivative contains a negative sign, it is always negative, so the function is concave down on its entire domain.
The function f(x) = √(2x + 3) is concave down on its entire domain.
2. To find the point of diminishing returns for the sales function S(x) = 204 + 6.3x - 0.25x², we need to find its inflection point, where the second derivative changes sign:
Step 1: Find the first derivative, S'(x):
S'(x) = 6.3 - 0.5x
Step 2: Find the second derivative, S''(x):
S''(x) = -0.5
Step 3: Determine the inflection point:
Since the second derivative is constant and negative, it never changes sign, meaning there is no point of diminishing returns in the given interval.
There is no point of diminishing returns for the sales function S(x) = 204 + 6.3x - 0.25x² in the interval 0 ≤ x ≤ 12.

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Using partial derivative the solution of the given function is x^4 e^x²y .

To find the fyy, we need ti take the second partial derivative of f with respect to y to get the solution.

Thus, the first partial derivative of f with respect to y.

fy= df/dy

Now we can take second partial derivative;

fyy = d/dy(x² e^x²y)

fyy =  x² (d/dy e^x²y)

To find (d/dy e^x²y) we use chain rule;

(d/dy e^x²y) =   e^x²y d/dy (x²y)

(d/dy e^x²y) = x²e^x²y

Now substitute;

fyy =  x² (d/dy e^x²y)

fyy =  x² ( x²e^x²y )

fyy =  x^4 e^x²y

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According to Markov chain,

a) The limiting fraction of time that the chain spends at the state 1 is 3/7.

b) Pⁿ(1,1) converges as n tends to infinity, and the limiting value is the steady-state probability of being in state 1, which we have already calculated to be 3/7.

(i) To find the limiting fraction of time that the chain spends at the state 1, we need to find the steady-state probability of being in state 1. The steady-state probability of being in state i is the probability that the chain is in state i in the long run, i.e., as n tends to infinity, where n is the number of steps in the chain.

To find the steady-state probability, we need to solve the following system of equations:

π1 = π1(1+1) + π4(1)

π2 = π1(1+1)

π3 = π2(1+1)

π4 = π3(1+1)

where πi is the steady-state probability of being in state i. Solving these equations, we get π1 = 3/7, π2 = 2/7, π3 = 1/7, and π4 = 1/7.

(ii) To find whether Pⁿ(1,1) converges as n tends to infinity, we need to check if the chain is irreducible and aperiodic. A Markov chain is irreducible if it is possible to go from any state to any other state in a finite number of steps. A Markov chain is aperiodic if the chain does not have a regular pattern in the sequence of steps it takes to return to a state.

In this case, the Markov chain is irreducible and aperiodic since we can go from any state to any other state in a finite number of steps, and there is no regular pattern in the sequence of steps it takes to return to a state.

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Answer:

A natural number between 30-1 and 30-2 is a natural number between 999 and 970. One example is 987.

An integer between -11 and -10 is -10.

A whole number between 1 and 2 is 1.

A terminating decimal between -3.14 and -3.15 is -3.14.

The logistic model and its modification into a growth with threshold model, and you provided four possible differential equations to represent this modified model. The growth with threshold model has two features: 1) the population eventually dies out if the initial population lies below a certain threshold level P*, and 2) when the initial population level is above P*, it will approach the carrying capacity K in the long-term.

Considering these features and the given options, the correct differential equation to represent a growth with threshold model is:

Your answer: C. dP/dt = -P ( P - 7 ) ( P - 11 )

This equation represents the growth with threshold model because it has the desired properties: the population will eventually die out if P is below the threshold value (7 in this case) and will approach the carrying capacity (11 in this case) if P is above the threshold value.

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The estimated change in surface area when Joanne's weight decreases by 1 kg is approximately -0.001737 square meters.

We can estimate the surface area of Joanne as S =

[tex]0.02235(150)^0.42246(80)^0.51456[/tex]

≈ 2.232 square meters. To estimate how much her surface area changes after her weight decreases by 1 kg, we need to find the derivative of S with respect to w and then multiply it by -1 (since we are considering a decrease in weight).

Using the chain rule and the power rule, we get: dS/dw =

[tex]0.02235(0.51456)(150)^0.42246(80)^(-0.48544)[/tex]

= 0.001737 This means that for every 1 kg decrease in Joanne's weight, her surface area is estimated to decrease by approximately 0.001737 square meters according to the given model.

This is only an estimate and may not reflect the actual change in Joanne's surface area, as the model is based on certain assumptions and may not be applicable to all patients. Other factors such as body composition and medical history may also affect the dosage of chemotherapy needed for a particular patient.

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Answer:

Let's assume that Holly has x trophies. When you don't know the amount someone has you can always use x in your equation because later on in the calculation you'll figure it out anyways

Emma has 7 fewer trophies than Holly. So Emma has x - 7 trophies.

Together, they have 27 trophies. So we can write an equation:

x + (x-7) = 27

Simplifying and solving for x, we get:

2x - 7 = 27

2x = 34

x = 17

So Holly has 17 trophies, and Emma has 17 - 7 = 10 trophies in total!

A company has total profit function P(x) = -2x² + 3003 - 22331, where is the number of items (or production level.)

The break-even production level(s). 7532.07 items

Now, let's dive into the question at hand. The company's total profit function is given as P(x) = -2x² + 3003x - 22331, where x is the number of items produced. To find the break-even production level, we need to find the value of x that makes the profit equal to zero.

In other words, we need to solve the equation -2x² + 3003x - 22331 = 0 for x. There are a few ways to do this, but one common method is to use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

In this case, a = -2, b = 3003, and c = -22331, so we have:

x = (-3003 ± √(3003² - 4(-2)(-22331))) / 2(-2) x ≈ 1492.93 or x ≈ 7532.07

These are the two break-even production levels, rounded to two decimal places. What this means is that the company needs to produce at least 1492.93 items or at most 7532.07 items to cover all its costs and make a profit of zero.

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Let f(x,y) = 144. The set of points where f is continuous is A. Whole of R², since f(x,y) is a constant function and constant functions are continuous everywhere.

The set of points where f is continuous is A. Whole of R². This is because f(x,y) is a constant function, meaning it is continuous at every point in the plane. There are no points of discontinuity, including (0,0), as the value of f is the same everywhere. Therefore, option B is incorrect. Additionally, options C and D are also incorrect as they only include points on one of the axes, while f is continuous everywhere in the plane.

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The particular solution that satisfies the initial condition is:

[tex]xy - 3y^2 = -2.[/tex]

To find the particular solution of the given differential equation, we can use the method of integrating factors.

First, we need to rearrange the equation in the standard form:

(x - 6y)dx - (2x - 9y)dy = 0

Multiply both sides by a suitable integrating factor, which is given by:

IF = e(-∫(6/x - 9/2)dy) = e[tex]^(9/2 ln(x)[/tex]- 6y) = x[tex]^(9/2)e^(-6y)[/tex]

Using this integrating factor, we can rewrite the equation as:

[tex]x^(9/2)e^(-6y)(x - 6y)dx - x^(9/2)e^(-6y)(2x - 9y)dy = 0[/tex]

The left-hand side of this equation is the product rule of (xy - 3y^2), so we can rewrite it as:

[tex]d(xy - 3y^2) = 0[/tex]

Integrating both sides, we get:

[tex]xy - 3y^2 = C[/tex]

To find the particular solution that passes through the point (1, 1), we can substitute x = 1 and y = 1 into the above equation and solve for C:

[tex]1(1) - 3(1)^2 = C[/tex]

C = -2

Therefore, the particular solution that satisfies the initial condition is:

[tex]xy - 3y^2 = -2.[/tex]

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The provided equation is true for any y = f(x), which makes any function a solution. However, given the specific points x = 1 and y = 1, we find that the particular solution to the equation is y = x.

The given equation is a first order homogeneous differential equation that we can solve using a substitution method. Let's substitute v = y/x, or y =vx, such that dy = vdx+ xdv.

Therefore, the particular solution of the equation is y = 1x, or y = x.

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Probability of getting at least 34 successes in a sample of 83 with a population proportion of 0.47 using the normal approximation is approximately 0.1056 or 10.56%.

To use the normal approximation, we need to check if the sample size and the population proportion satisfy the conditions of the Central Limit Theorem. In this case, since n*p = 39.01 and n*(1-p) = 43.99 are both greater than 10, we can assume that the sampling distribution of X is approximately normal.

To find P(X ≥ 34), we can use the normal distribution with mean[tex]µ = n*p = 39.01[/tex] and standard deviation σ = sqrt(n*p*(1-p)) = 4.01. Then, we need to standardize the value of X using the formula z = (X - µ) / σ, which gives:

z = (34 - 39.01) / 4.01 = -1.25

Using a standard normal table or calculator, we can find the probability of z being less than -1.25, which is equivalent to the probability of X being greater than or equal to 34, as:

P(Z < -1.25) = 0.1056

Therefore, the probability of getting at least 34 successes in a sample of 83 with a population proportion of 0.47 using the normal approximation is approximately 0.1056 or 10.56%.

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Answer:

Step-by-step explanation:

Answer is

(x+4) (2x+9)

(3x+13)

3/13

x =4.3

The rate at which the surface area is increasing when there are 288pi cubic inches of helium in the balloon is 4 * pi inches^2/second.

We are required to determine the rate at which the surface area of a spherical balloon is increasing when there are 288pi cubic inches of helium in the balloon.

1. First, we need to find the radius (r) of the balloon when the volume is 288pi cubic inches. The formula for the volume (V) of a sphere is:

V = (4/3) * pi * r^3

2. Plug in the given volume and solve for r:

288pi = (4/3) * pi * r^3

(3/4) * (288) = r^3

r = 6 inches

3. Next, we need to find the formula for the surface area (A) of a sphere:

A = 4 * pi * r^2

4. Now, let's differentiate the volume formula and the surface area formula with respect to time (t) to find dV/dt and dA/dt:

dV/dt = 4 * pi * r^2 * dr/dt

dA/dt = 8 * pi * r * dr/dt

5. We are given that the balloon is being filled at a rate of 2 cubic inches per second (dV/dt = 2). We can plug this into the dV/dt formula and solve for dr/dt:

2 = 4 * pi * (6)^2 * dr/dt

dr/dt = 1/36 inches per second

6. Finally, plug in the radius (r = 6) and dr/dt into the dA/dt formula:

dA/dt = 8 * pi * 6 * (1/36)

dA/dt = 4 * pi inches^2/second

So, the rate at which the surface area is increasing is 4 * pi inches^2/second.

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The differentiation of the given function is [tex]\frac{dx}{dy} =\frac{ (sec^2(x+y) - x - 2y)}{  (y - sec^2(x+y))}[/tex]

Given the equation[tex]xy + y^2 = tan(x+y)[/tex], we want to find the derivative[tex]dx/dy.[/tex]

First, let's differentiate both sides of the equation with respect to y:

[tex]\frac{d(xy)}{dy} +\frac {d(y^2)}{dy} =\frac {d(tan(x+y))}{dy}[/tex]

Using the product rule for the first term and the chain rule for the last term, we get:

(x * dy/dy + y * dx/dy) + 2y = (sec^2(x+y)) * (dx/dy + 1)

Since dy/dy = 1, we can simplify the equation to:

[tex]x + y *\frac{ dx}{dy} + 2y = (sec^2(x+y)) * (\frac{dx}{dy} + 1)[/tex]

Now, we want to solve for dx/dy:

[tex]y * \frac{dx}{dy} - (sec^2(x+y)) * \frac{dx}{dy} = (sec^2(x+y)) - x - 2y[/tex]

Factor out dx/dy:

[tex]dx/dy * (y - sec^2(x+y)) = sec^2(x+y) - x - 2y[/tex]
Finally, divide both sides by[tex](y - sec^2(x+y))[/tex]to isolate dx/dy:

[tex]\frac{dx}{dy} =\frac{ (sec^2(x+y) - x - 2y)}{  (y - sec^2(x+y))}[/tex]

And that's your answer!

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The value of f(5) for the given function is -6.

What is a sequence?

A sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms).

A heuristic, on the other hand, is an approach to problem-solving that seeks to come up with a workable answer quickly rather than promising an ideal or even accurate solution.

When an algorithmic solution is not viable, useful, or effective, heuristics are frequently applied.

Given that, f(1)=6 and  f(n)=f(n−1)−3 thus we have:

f(2) = f(1) - 3 = 6 - 3 = 3

f(3) = f(2) - 3 = 3 - 3 = 0

f(4) = f(3) - 3 = 0 - 3 = -3

f(5) = f(4) - 3 = -3 - 3 = -6

Hence, the value of f(5) for the given function is -6.

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The angles in degrees are

180 degrees

90 degrees

90 degrees

90 degrees

270 degrees

180 degrees

What is Rotation in Transformation?

In mathematical terminology, rotation is a special kind of transformation that implicates rotating an item about a designated point or an axis.

During the rotational process, each individual point of the item moves in a manner resembling a circular pattern around the constant point or axis. The space between these entities remains static; yet, the angle between them varies.  

Rotation can be either clockwise or counterclockwise, and is typically measured in degrees or radians.

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Answer:

1.73

Step-by-step explanation:

To find the cost for a 43.25-mile section of highway with a toll of $0.04 per mile, we can simply multiply the number of miles by the toll rate per mile.

Cost = Miles * Toll rate per mile

Given:

Miles = 43.25

Toll rate per mile = $0.04

Plugging in the values into the formula:

Cost = 43.25 * $0.04

Cost = $1.73

The distribution of sample means has a mean of 86.2 and a standard deviation of approximately 6.83.

a. The mean of the distribution of sample means is equal to the population mean (µ). In this case, µ = 86.2.

b. The standard deviation of the distribution of sample means, also known as the standard error (SE), can be calculated using the formula:

SE = σ / √n

Where σ is the population standard deviation and n is the sample size. In this case, σ = 54.2 and n = 63.

SE = 54.2 / √63 ≈ 6.83

So, the standard deviation of the distribution of sample means is approximately 6.83 (rounded to two decimal places).

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Answer:  66% of 110 is 72.6

Step-by-step explanation:

Because i said so

The expected value of XY is 2.67.

We can start by using the definition of expected value:

E(XY) = Σxy × P(X=x, Y=y)

where Σ denotes the sum over all possible values of x and y, and P(X=x, Y=y) is the joint probability of X=x and Y=y.

Since X and Y are positive integers, the possible values for X and Y are {1, 2, 3}. We can calculate the joint probabilities as follows:

P(X=1, Y=1) = P(X=1) × P(Y=1) = (1/3) × (1/3) = 1/9

P(X=1, Y=2) = P(X=1) × P(Y=2) = (1/3) × (1/3) = 1/9

P(X=1, Y=3) = P(X=1) × P(Y=3) = (1/3) ×(1/3) = 1/9

P(X=2, Y=1) = P(X=2) × P(Y=1) = (1/3) × (1/3) = 1/9

P(X=2, Y=2) = P(X=2) × P(Y=2) = (1/3) × (1/3) = 1/9

P(X=2, Y=3) = P(X=2) × P(Y=3) = (1/3) × (1/3) = 1/9

P(X=3, Y=1) = P(X=3) × P(Y=1) = (1/3) × (1/3) = 1/9

P(X=3, Y=2) = P(X=3) × P(Y=2) = (1/3) × (1/3) = 1/9

P(X=3, Y=3) = P(X=3) × P(Y=3) = (1/3) × (1/3) = 1/9

Next, we need to find the values of X and Y that satisfy X^2 + Y < 13. These are:

(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2)

For each of these pairs, we can calculate the value of XY:

(1, 1): XY = 1

(1, 2): XY = 2

(1, 3): XY = 3

(2, 1): XY = 2

(2, 2): XY = 4

(2, 3): XY = 6

(3, 1): XY = 3

(3, 2): XY = 6

Finally, we can substitute these values into the definition of expected value:

E(XY) = Σxy × P(X=x, Y=y)

= (11/9) + (22/9) + (31/9) + (21/9) + (41/9) + (61/9) + (31/9) + (61/9)

= 2.67

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The area under the standard normal curve between z=â1.16 and z=â0.03 is 0.3663.

To find the area under the standard normal curve between z = -1.16 and z = -0.03, we will use the following steps:

1. Look up the z-values in the standard normal distribution table (or use a calculator or online tool that provides the corresponding probability values).
2. Subtract the probability value for z = -1.16 from the probability value for z = -0.03.
3. Round your answer to four decimal places.

Step 1: Look up the z-values in the standard normal distribution table.
- For z = -1.16, the corresponding probability value is 0.1230.
- For z = -0.03, the corresponding probability value is 0.4893.

Step 2: Subtract the probability values.
Area = P(-0.03) - P(-1.16) = 0.4893 - 0.1230

Step 3: Round the answer to four decimal places.
Area = 0.3663

So, the area under the standard normal curve between z = -1.16 and z = -0.03 is approximately 0.3663.

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There are 38,400 ways to seat the four couples in a row such that no person sits next to their significant other. This can be answered by the concept of Permutation and combination.

To solve this problem, we can use the concept of permutations and derangements. First, we need to find the total ways of seating the couples without restrictions and then subtract the ways where at least one couple is sitting together.

Total ways to seat the couples without restrictions: There are 8 people (4 couples), so there are 8! (8 factorial) ways to seat them.

Now, we'll find the number of ways where at least one couple sits together. For each of the 4 couples, consider them as a single unit. We have 5 units (4 couple-units and 4 single people), so there are 5! ways to arrange these units. However, within each couple-unit, there are 2! ways to arrange the individuals, so we need to multiply by 2!⁴.

Ways with at least one couple together: 5! × (2!)⁴

To find the number of ways where no person sits next to their significant other, we subtract the ways with at least one couple together from the total ways:

Desired seating arrangements: 8! - (5! × (2!)⁴)

Calculating the values: 40320 - (120 × 16) = 40320 - 1920 = 38,400

So, there are 38,400 ways to seat the four couples in a row such that no person sits next to their significant other.

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Answer:

The far right graph is correct.

Step-by-step explanation:

Answer: below!

Step-by-step explanation:

9: tan51 = 8/x

x = 8/(tan51) ≅ 6.478

10: sin24 = x/31

.: x = 31sin24 ≅ 12.609

11: You've done it correctly!

12: tanx = 25/78

.: x = [tex]tan^-^1(\frac{25}{78} )[/tex] ≅ 17.77°

15: Correct!

16: sin25 = 15/(XZ)

.: XZ = 15/sin25 ≅ 35.5

Since triangle WXZ is right, the pythagorean theorem tells us that WZ = [tex]\sqrt{22^2 + 35.5^2}[/tex] ≅ 41.76

Now, using the Law of Sines, we can say that sin90/WZ = sinW/XZ

this means 1/41.76 = sinW/35.5

W = [tex]sin^-^1(\frac{35.5}{41.76} )[/tex] ≅ 58.2°

17. Draw this image. You'll see a right triangle, with angles 90-75-15. Since the side adjacent to the 75 deg angle is 6, we can solve for the length of the ladder. in essence:

cos75 = 6/x, where x is the length of the ladder

.: x = 6/cos75 ≅ 23.18 feet

Answer:

4 people per taxi

Step-by-step explanation:

60 divided by 15

Answer:4

Step-by-step explanation:

Your basically asking what 60 divided by 15 is or how many times can 15 be added to get 60.

15 can be added 4 times before reaching 60 So the anwser for your equation is 4.

The calculated prediction of the population in 50 years is 164856.53

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

Initial population, a = 95400

Rate of increase, r = 1.1%

Using the above as a guide, we have the following:

Population function, f(x) = a * (1 + r)^x

substitute the known values in the above equation, so, we have the following representation

f(x) = 95400 *(1 + 1.1%)^x

In 50 years, we have

f(50) = 95400 *(1 + 1.1%)^50

Evaluate

f(50) = 164856.53

Hence, the population in 50 years is 164856.53

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Antiderivative of f(x) = 1/x is ln|x| + C. Antiderivative of g(x) = 11/x is 11 ln|x| + C. Antiderivative of h(x) = 5 - 4/x is 5x - 4 ln|x| + C.

The antiderivative of f(x) = 1/x can be found using the natural logarithm function:

∫ 1/x dx = ln|x| + C

where C is the constant of integration.

The antiderivative of g(x) = 11/x can be found similarly:

∫ 11/x dx = 11 ln|x| + C

where C is the constant of integration.

The antiderivative of h(x) = 5 - 4/x can be found by integrating each term separately:

∫ 5 dx - ∫ 4/x dx = 5x - 4 ln|x| + C

where C is the constant of integration.

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