Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 159 disks are summarized as follows:


Shock Resistance

scratch resistance high low

high 70 9

low 16 5


Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. If a disk is selected at random, determine the following probabilities.


a. P(A)=86/100

b. P(B)=79/100

c. P(A')=7/50

d. P(A U B)=95/100

e. P(A' U B)= ???

Answer:

:)

Step-by-step explanation:

|z| = 0.669 < 1.96, we fail to reject the null hypothesis at any significance level up to α = 0.05.

In other words, we do not have enough evidence to conclude that the true proportion of executives who would agree with the statement is different from one-half.

The null hypothesis that the true proportion of all such executives.

Agree with this statement is one-half, we can use a two-sample z-test for proportions.

Let p be the true proportion of executives who would agree with the statement and let. \(\^p\) be the sample proportion.

Under the null hypothesis, the test statistic z is given by:

\(z = (\^p - 0.5) / \sqrt(0.5 \times 0.5 / n),\)

n is the sample size (104 in this case).

This test statistic follows a standard normal distribution under the null hypothesis.

To reject the null hypothesis at a significance level α, we need to find the critical values.\(z\alpha/2\) and \(-z\alpha/2\) such that \(P(Z > z\alpha/2) = \alpha/2\) and\(P(Z < -z\alpha/2) = \alpha/2\), where Z is a standard normal random variable.

The lowest level of significance at which the null hypothesis can be rejected against a two-sided alternative.

The smallest α such that \(|z| > z\alpha/2.\)

The two-sided alternative implies that we are interested in deviations from the null value of 0.5 in either direction.

Using the given information, we have:

\(\^p = 50/104 = 0.4808,\)

n = 104.

Substituting these values into the formula for z, we get:

\(z = (0.4808 - 0.5) / \sqrt(0.5 \times 0.5 / 104) = -0.669.\)

The critical value \(z\alpha/2\), we look up the corresponding value in the standard normal distribution table.

\(z\alpha/2\) such that \(P(Z > z\alpha/2) = \alpha/2\), or equivalently, \(P(Z < -z\alpha/2) = \alpha/2\). Since the standard normal distribution is symmetric, we can look up the value of zα/2 that satisfies. \(P(Z > z\alpha/2) = \alpha/2\), and then take its absolute value to find. \(-z\alpha/2.\)

Using a standard normal distribution table, we find that z0.025 = 1.96 (rounded to two decimal places).

The critical values for rejecting the null hypothesis at a significance level α are:

\(z\alpha/2\) = ±1.96.

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Using the normal approximation to the binomial, there is a 0.9713 = 97.13% probability that the sum of these variables is less than 60.

This problem is incomplete, but researching it on a search engine, it asks the probability that the sum of these Bernoulli variables is of less than 60.

The z-score of a measure X of a normally distributed variable with mean \(\mu\) and standard deviation \(\sigma\) is given by:

\(Z = \frac{X - \mu}{\sigma}\)

The binomial distribution is a series of n Bernoulli trials with p probability of a success on each trial, hence the parameters for the binomial distribution are given as follows:

n = 100, p = 0.5.

The mean and the standard deviation are given by:

Using continuity correction, the probability that the sum is less than 60 is P(X < 59.5), which is the p-value of Z when X = 59.5, hence:

\(Z = \frac{X - \mu}{\sigma}\)

Z = (59.5 - 50)/5

Z = 1.9

Z = 1.9 has a p-value of 0.9713.

Hence there is a 0.9713 = 97.13% probability that the sum of these variables is less than 60.

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The probability from the odds is 3/7

The value of the odds is given as

Odds = 3 to 4

Represent the odds as a fraction

So, we have the following representation

Odds = 3/4

To convert the odds to probability, we make use of the following equation

Probability = Odds/(1 + Odds)

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

Probability = (3/4)/(3/4 + 1)

Evaluate the sum

Probability = (3/4)/(7/4)

Evaluate the quotient

Probability = 3/7

Hence, the probability is 3/7

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Answer:

C

Step-by-step explanation:

Answer:

C- subtracted 2x from both sides

Step-by-step explanation:

when solving for x, you have to get ALL Xs on one side. so that would be the next step to solving x.

Hope this helps

The Gaussian curvature is K = (cos v) / (v₁² + v₂²), and the mean curvature is H = -1 / (2sqrt(v₁² + v₂²)).

Given the map f: M ⟶ R³ where f(v,θ) = (u cos v, u sin v, u²), and M = {(v₁, v₂) ∈ R² | 0 < v₁ < π}.a) The Weingarten map of a surface S can be obtained by differentiating the unit normal vector along any curve lying on the surface. Let r(u, v) be a curve on S. Then the unit normal vector at the point r(u, v) is given byN = (f_u × f_v) / ||f_u × f_v||Where f_u and f_v are the partial derivatives of f with respect to u and v respectively, and ||f_u × f_v|| denotes the norm of the cross product of f_u and f_v. Differentiating N along r(u, v) yields the Weingarten map of S.

b) To find the Gaussian and mean curvatures of S, we can use the first and second fundamental forms. The first fundamental form is given byI = (f_u · f_u)du² + 2(f_u · f_v)dudv + (f_v · f_v)dv²= u²(dv² + du²)

The second fundamental form is given byII = (f_uu · N)du² + 2(f_uv · N)dudv + (f_vv · N)dv²

where f_uu, f_uv and f_vv are the second partial derivatives of f with respect to u and v, and N is the unit normal vector. Using the formulas for the first and second fundamental forms, we can compute the Gaussian and mean curvatures of S as follows:

K = (det II) / (det I)H = (1/2) tr(II) / (det I)where det and tr denote the determinant and trace respectively. In this case, we have f_u = (-u sin v, u cos v, 2u) f_v

= (u cos v, u sin v, 0)f_uu

= (-u cos v, -u sin v, 0) f_uv = (cos v, sin v, 0)f_vv

= (-u sin v, u cos v, 0)N

= (u cos v, u sin v, -u) / u

= (cos v, sin v, -1)K = (cos v) / (u²) = (cos v) / (v₁² + v₂²)H

= -1 / (2u) = -1 / (2sqrt(v₁² + v₂²))

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Answer:

x>125

Step-by-step explanation:

Answer:

x>125

Step-by-step explanation:

The mean of the sampling distribution is still 810 grams, which is the same as the population mean. The standard deviation of the sampling distribution is 4 grams.

The mean of the sampling distribution for the average weight of a simple random sample can be considered an unbiased estimator of the population mean. In this case, since the population mean is 810 grams, the mean of the sampling distribution remains the same.

The standard deviation of the sampling distribution, on the other hand, is influenced by the sample size. The formula for the standard deviation of the sampling distribution of the sample mean is given by the population standard deviation divided by the square root of the sample size. In this case, the population standard deviation is 40 grams, and the sample size is 100. Therefore, the standard deviation of the sampling distribution is calculated as 40 grams divided by the square root of 100, which is 4 grams.

In summary, the mean of the sampling distribution for the average weight of simple random samples of 100 children born with ELBW is 810 grams, and the standard deviation is 4 grams.

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1. Sampling error refers to the discrepancy or difference between the characteristics of a sample and the characteristics of the population it represents. Representation, on the other hand, refers to the extent to which a sample accurately reflects the population. 2. The purpose of the null hypothesis is to serve as a basis for statistical hypothesis testing. It represents the assumption of no effect or no relationship in the population.

1. The criteria for a good hypothesis include being testable, specific, clear, and based on prior research or theoretical rationale. A good hypothesis should be formulated in a way that allows for empirical investigation and evaluation. It should provide a clear and concise statement about the expected relationship or difference between variables.

2. The research hypothesis is typically developed based on theoretical reasoning or prior empirical evidence. It represents the relationship or difference that the researcher is interested in investigating. The role of chance in relation to the research hypothesis is that statistical tests are used to determine the likelihood that the observed results are due to chance or reflect a true relationship. If the probability of obtaining the observed results by chance is low (below a predetermined threshold), it suggests support for the research hypothesis.

3. The null hypothesis always refers to the population because it represents the assumption that there is no effect or relationship in the population. Statistical hypothesis testing aims to make inferences about the population based on sample data. By comparing the sample data to the null hypothesis, researchers can evaluate whether the observed results are likely to occur due to chance or if they provide evidence to reject the null hypothesis and support the research hypothesis.

Multiple-Choice Questions:

6. c. techniques to be used

7. a. population

8. a. the null hypothesis

9. c. no sampling error

10. a. a one-tailed test

11. b. a two-tailed test

12. a. directional research hypothesis

13. b. nondirectional research hypothesis

14. a. directional research hypothesis

15. d. H1: X1 ≠ X2

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The correct expressions are sinC = 12/25.5 = 24/51, <A = 62 degrees and tan A = 1.875

The given figure is a right triangle with legs 12.0 and 22.5cm and the hypotenuse 25.5cm

For sinA, <A is the reference angle, hence the opposite side will be 22.5cm

sin A = opp/hyp

sinA = 22.5/25.5

For cos C, the adjacent side will be 22.5cm, hence the measure of cos C is 22.5/25.5

cosC = 22.5/25.5

Similarly, sinC = 12/25.5 = 24/51

<A = 180-(90 +38)

<A =180 - 128
<A = 62 degrees

tan A = opp/adj

tan A = 22.5/12 = 1.875

tan C = 12/22.5

tan C = 0.533

Hence the correct expressions are sinC = 12/25.5 = 24/51, <A = 62 degrees and tan A = 1.875

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Answer:

4/5 pages per minute.

Explanation:

For 3/5 pages in 3/4 minutes:

The unit rate is 4/5 pages per minute.

Answer:

6

Step-by-step explanation:

Answer:

Favourable outcomes = {1,4,6,8,9,10,12,14,15,16,18,20,21,22,24,25}

Number of favourable outcomes=16

Number of total possible outcomes=25

p(not a prime) = (number of favourable outcomes) / number of total possible outcomes

=16/25

Required Sample space of the following event ( Non- prime numbers between 1 to 25 )

\(\dashrightarrow\) 1, 4, 6, 8,9 10,12, 14, 15, 16,18, 20, 21, 22, 24, 25

Number of Favourable Outcomes = 16

Total number of outcomes = 25

Now as we know

Probability of an event =

\(\star\;\boxed{\sf{\pink{\dfrac{ Favourable\; number\;of\; outcomes}{Total\; number\;of\; outcomes}}}}\)

Probability = 16/25

Hence, probability that the number is not a prime number is 16/25.

The dimensions that minimize the cost of the material are a height of approximately 3.132 m and a radius of approximately 0.508 m.

Let's start by setting up some notation for the dimensions of the cylindrical container. Let the height of the container be h, and let the radius of the top and bottom be r. Then, the volume of the container is given by:

\(V =\pi r^2h\)

We want to minimize the cost of the material used to make the container. The cost is composed of two parts: the cost of the material used for the top and bottom, and the cost of the material used for the side. Let's compute these separately.

The cost of the material used for the top and bottom is given by the area of two circles with radius r, multiplied by the cost per square meter:

\(C1 = 2\pi r^2 * 20\)

The cost of the material used for the side is given by the area of the side of the cylinder, which is a rectangle with height h and length equal to the circumference of the base (which is 2πr), multiplied by the cost per square meter:

C2 = 2πrh * 15

The total cost is the sum of these two costs:

\(C = C1 + C2 = 2\pi r^2 * 20 + 2\pi rh * 15\)

We want to minimize this cost subject to the constraint that the volume is 10 \(m^3\):

\(V = \pi r^2h = 10\)

We can use the volume equation to eliminate h, obtaining:

\(h = 10/(\pi r^2)\)

Substituting this expression for h into the cost equation, we obtain:

\(C = 2\pi r^2 * 20 + 2\pi r * 15 * 10/(\pi r^2)\)

Simplifying, we have:

\(C = 40\pi r^2 + 300/r\)

To minimize this function, we take its derivative with respect to r and set it equal to zero:

\(dC/dr = 80\pi r - 300/r^2 = 0\)

Solving for r, we obtain:

\(r = (300/(80\pi ))^{(1/3)} = 0.508 m\)

To find the corresponding value of h, we can use the volume equation:

\(h = 10/(\pi r^2)\) ≈ 3.132 m

Therefore, the dimensions that minimize the cost of the material are a height of approximately 3.132 m and a radius of approximately 0.508 m.

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Answer:

The greatest number of cutlery holders Layla can stock is 15 holders.

Step-by-step explanation:

You can put 2 of each type of cutlery in each holder.

12 forks use 6 holders.

18 knives use 9 holders.

6+9=15

The greatest number of cutlery holders Layla can stock is 15 holders.

Answer: 9

Step-by-step explanation: If the ratio is 3:4, then figure what we multiply 4 by to get 12. That would be 3, so then multiply 3 by 3 to get 9, which is the number of boys

Let's create a vector F(x, y, z) with at least two variables in its components:

F(x, y, z) = (xy + 2z)i + (yz + 3x)j + (xz + y)k

Now, let's find the gradient, divergence, and curl of this vector:

1. Gradient (∇F):

The gradient of a vector is given by the partial derivatives of its components with respect to each variable. For our vector F(x, y, z), the gradient is:

∇F = (∂F/∂x)i + (∂F/∂y)j + (∂F/∂z)k

Calculating the partial derivatives:

∂F/∂x = yj + zk

∂F/∂y = xi + zk

∂F/∂z = 2i + xj

Therefore, the gradient ∇F is:

∇F = (yj + zk)i + (xi + zk)j + (2i + xj)k

2. Divergence (div F):

The divergence of a vector is the dot product of the gradient with the del operator (∇). For our vector F(x, y, z), the divergence is:

div F = ∇ · F

Calculating the dot product:

div F = (∂F/∂x) + (∂F/∂y) + (∂F/∂z)

Substituting the partial derivatives:

div F = y + x + 2

Therefore, the divergence of F is:

div F = y + x + 2

3. Curl (curl F):

The curl of a vector is given by the cross product of the gradient with the del operator (∇). For our vector F(x, y, z), the curl is:

curl F = ∇ × F

Calculating the cross product:

curl F = (∂F/∂y - ∂F/∂z)i - (∂F/∂x - ∂F/∂z)j + (∂F/∂x - ∂F/∂y)k

Substituting the partial derivatives:

curl F = (z - 3x) i - (z - 2y) j + (y - x) k

Therefore, the curl of F is:

curl F = (z - 3x)i - (z - 2y)j + (y - x)k

That's it! We have calculated the gradient (∇F), divergence (div F), and curl (curl F) of the given vector F(x, y, z) by finding the partial derivatives, performing dot and cross products, and simplifying the results.

Given:

Lamont has purchased 20 trading cards.

He wants to have at least 50 trading cards.

To find:

The inequality for the number of trading cards Lamont needs and solve it.

Solution:

Let x be the number of trading cards Lamont needs.

He has 20 trading cards. So,

Total cards = x + 20

It is given that, he wants to have at least 50 trading cards. It means, total card must be greater than or equal to 50.

\(x+20\geq 50\)

Subtract 20 from both sides.

\(x+20-20\geq 50-20\)

\(x\geq 30\)

Therefore, the required inequality is \(x+20\geq 50\) and solution is \(x\geq 30\).

The number of trading cards that Lamont needs will be at least 30 trading cards.

From the information given, we are informed that Lamont has purchased 20 trading cards and wants to have at least 50 trading cards.

Therefore, the number that will be needed more will be:

= 50 - 20 = 30

Therefore, he'll need at least 30 more cards.

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By finding the average rate of change, we can see that:

a) The cost decreases.

b) The cost increases.

To check that, we need to find the average rate of change on the interval.

Remember that for function f(x) on an interval (a, b), the average rate of change is:

R = (f(b) - f(a))/(b - a)

Here the cost function is:

c(n) = 0.25*n² - 10n + 800

a) In the interval [10, 15] the average rate of change is given by:

R = (c(15) - c(10)/(15 - 10)

Where:

c(15) = 0.25*15^2 - 10*15 + 800 = 706.25

c(10) = 0.25*10^2 - 10*10 + 800 = 725

Then the average rate of change is:

R = (706.25 - 725)/(15 - 10) = -3.75

This means that between 10 and 15 light fixtures, the cost is decreasing.

b) Now we have the interval [20, 25], so let's do the same ting:

c(20) = 0.25*20^2 - 10*20 + 800 = 700

c(25) = 0.25*25^2  - 10*25 + 800 = 706.25

Here the average rate of change is:

R = (706.25 - 700)/(25 - 20) = 1.25

It is positive, which means that the cost is increasing.

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The midpoint of A and B is (Xa , Yb) = (-4,1) and  the midpoint of C and D is (Xc , Yd) = (4 , 6)

What is midpoint?

A(-3,5) and B(-5, -3)

(Xa , Yb) = [(x₁ + x₂ / 2) , (y₁ + y₂ / 2)]

= [(-3 + (-5) / 2) , (5 +(-3) / 2)]

= (-8/2 , 2/2)

(Xa , Yb) = (-4,1)

Hence, the midpoint of A and B is (Xa , Yb) = (-4,1).

C(7,9) and D(1,3)

(Xc , Yd) = [(x₁ + x₂ / 2) , (y₁ + y₂ / 2)]

= [(7+1/2) , (9+3 / 2)]

= ( 8/2 , 12/2)

(Xc , Yd) = (4 , 6)

Hence, the midpoint of C and D is (Xc , Yd) = (4 , 6).

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It will take 585 billion years to solve the puzzle tower of Hanoi puzzle.

What is hanoi of tower?

The Tower of Hanoi is a mathematical game or puzzle that uses three rods and a number of disks that can slide onto any rod. It is also known as The Problem of the Benares Temple, the Tower of Brahma, Lucas' Tower, or simply the pyramid puzzle. The disks are stacked on one rod at the start of the problem, the smallest at the top and roughly forming a conical shape.

Using the Tower of Hanoi as little as possible

In one version of the problem, 64 golden disks are being used by Brahmin priests to solve it.

Minimum moves required if you had 64 golden disks \(=2^{64}-1\)

The problem would take around 585 billion years to solve if every move took a single second.

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Answer:

80º and 100º

Step-by-step explanation:

The probability of selecting two blue marbles without replacement is 1/6, and the probability of selecting three blue marbles without replacement is 1/35. The fewest number of marbles that must have been in the bag before any were drawn is 36.

Let's assume there are x marbles in the bag. The probability of selecting two blue marbles without replacement can be calculated using the following equation: (x - 1)/(x) * (x - 2)/(x - 1) = 1/6. Simplifying this equation gives (x - 2)/(x) = 1/6. Solving for x, we find x = 12.

Similarly, the probability of selecting three blue marbles without replacement can be calculated using the equation: (x - 1)/(x) * (x - 2)/(x - 1) * (x - 3)/(x - 2) = 1/35. Simplifying this equation gives (x - 3)/(x) = 1/35. Solving for x, we find x = 36.

Therefore, the fewest number of marbles that must have been in the bag before any were drawn is 36.

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this is how

Step-by-step explanation:

gusto ko lang magka points

The Sturm separation theorem guarantees that between any consecutive zeros of Sin(2x) + Cos(2x) and 8sin(2x) - cos(x) + i*sin(x), there is exactly one zero. The given differential equation y'' + (x + i)y = 6 has an infinite number of positive zeros for every real solution.

The Sturm separation theorem states that if a real-valued polynomial has consecutive zeros between two intervals, then there is exactly one zero between those intervals.

Consider the polynomial P(x) = Sin(2x) + Cos(2x) - Zero. Let Q(x) = 8sin(2x) - cos(x) + i*sin(x). We need to show that between any consecutive zeros of P(x), there is exactly one zero of Q(x).

First, let's find the zeros of P(x):

Sin(2x) + Cos(2x) = Zero

=> Sin(2x) = -Cos(2x)

=> Tan(2x) = -1

=> 2x = -π/4 + nπ, where n is an integer

=> x = (-π/8) + (nπ/2), where n is an integer

Now, let's find the zeros of Q(x):

8sin(2x) - cos(x) + isin(x) = Zero

=> 8sin(2x) - cos(x) = -isin(x)

=> (8sin(2x) - cos(x))^2 = (-i*sin(x))^2

=> (8sin(2x))^2 - 2(8sin(2x))(cos(x)) + (cos(x))^2 = sin^2(x)

=> 64sin^2(2x) - 16sin(2x)cos(x) + cos^2(x) = sin^2(x)

=> 63sin^2(2x) - 16sin(2x)cos(x) + cos^2(x) - sin^2(x) = 0

Now, let's observe the zeros of P(x) and Q(x). We can see that for every zero of P(x), there is exactly one zero of Q(x) between any two consecutive zeros of P(x). This satisfies the conditions of the Sturm separation theorem.

2. The given differential equation is y'' + (x + i)y = 6. We need to show that every real solution of this equation has an infinite number of positive zeros.

Let's assume that y(x) is a real solution of the given equation. Since the equation has complex coefficients, we can write the solution as y(x) = u(x) + i*v(x), where u(x) and v(x) are real-valued functions.

Substituting y(x) = u(x) + iv(x) into the differential equation, we get:

(u''(x) + iv''(x)) + (x + i)(u(x) + iv(x)) = 6

(u''(x) - v''(x) + xu(x) - xv(x)) + i*(v''(x) + u''(x) + xv(x) + xu(x)) = 6

Since the real and imaginary parts of the equation must be equal, we have:

u''(x) - v''(x) + xu(x) - xv(x) = 6

v''(x) + u''(x) + xv(x) + xu(x) = 0

Now, let's consider the real part of the equation:

u''(x) - v''(x) + xu(x) - xv(x) = 6

Assuming u(x) is a solution, we can apply Sturm separation theorem to show that there exist an infinite number of positive zeros of u(x). This is because the equation has a positive coefficient for the x term, which implies that the polynomial u''(x) + xu(x) has an infinite number of positive zeros.

Since the Sturm separation theorem applies to the real part of the equation, and the real and imaginary parts are interconnected, it follows that every real solution y(x) of the given equation has an infinite number of positive zeros.

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True. It's because the capacitance between two plates depends on the electric field between them, which is naturally geometry-dependent; if you have charges in different places, you get a different electric field.

A capacitor's capacitance is affected by the area of the plates, the distance between the plates, and the dielectric's ability to support electrostatic forces.
The capacitance of a parallel plate capacitor (depending on its geometry) is given by the formula C=Ad C = A d, where C is the capacitance value, A is the area of each plate, d is the distance between the plates, and is the permittivity of the material between the parallel capacitor's plates.
The curvature of the plates indicates whether the plates are spherical or cylindrical. As a result, the only factor that has no effect on the capacitance of the capacitor is the type of material used to make the plates.

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Answer:

Step-by-step explanation:

Note that

\(\frac{\sqrt{5}}{3}=\frac{5\sqrt{5}}{15}\\\frac{3}{5}=\frac{9}{15}\)

Since

\((5\sqrt{5})=125 > 81 = 9^2, \text{ then }\\\frac{5\sqrt{5}}{15} >\frac{9}{15}\)

So

\(\frac{3}{5} < \frac{\sqrt{5}}{3}\)

Now clearly

\(\frac{\sqrt{5}}{3} < 1 < 27, 5 < 97 < 97\pi\)

Therefore

\(\frac{3}{5} < \frac{\sqrt{5}}{3} < 27,5 < 97\pi\)

In total, Menaha traveled 97km 1000m (97.1km).

Distance is a numerical measurement of how far apart two objects, points, or places are in space. Distance can be measured in linear units such as meters, kilometers, feet, miles, etc. It can also be measured in angular units such as degrees or radians.

Distance can also refer to the space between two points in time, such as the time between two events. Distance can be used to measure physical distance, time, or even emotional distance.
To calculate this, the two distances must be added together.
The train distance is =86km 520m (86.52km)

and the car distance is =11km 480m (11.48km).
When added together, =86km 520m+11km 480m = 97.52km.
However, since the distances are measured in km and m,
it is necessary to convert the measurements into a single unit of measurement.
To do this, the measurements must be converted into metres.
The train distance is 86,520 metres

And the car distance is 11,480 metres.
When added together,

the total distance is 97,000 metres (97km).

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