to total cost of 5 kg onion and 7 kg sugar is Rs 810e5 kg onion price is equals to 2 kg sugar then find the cost of 1 kg onion and cost of 3 kg sugar​

A p-value is a probability.

A p-value is the probability of obtaining a test statistic as extreme or more extreme.

The observed value, assuming the null hypothesis is true.

It ranges in value from 0 to 1 and represents the strength of evidence against the null hypothesis.

A p-value cannot be negative, as it is a probability and probabilities are always between 0 and 1.

A p-value also cannot be larger than 1, as it represents a probability.

A probability cannot exceed 1.

Finally, a p-value cannot be smaller than 0, as it represents a probability.

A probability cannot be negative.

the correct option is b. is a probability.

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The sides and the angle of the right triangle are a = 10√2, b = 10√2 and B = π / 4.

In this problem we need to determine the values of two sides and an angle of the right triangle. This can be done by means of the following properties:

A + B + C = π

sin A = a / c

cos A = b / c

tan A = a / b

Where:

If we know that A = π / 4, C = π / 2 and c = 20, then the missing angle and missing sides are, respectively:

B = π - π / 4 - π / 2

B = π / 4

cos (π / 4) = b / 20

b = 20 · cos (π / 4)

b = 10√2

sin (π / 4) = a / 20

a = 20 · sin (π / 4)

a = 10√2

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the likelihood of Erin getting at least one bullseye in 5 throws is 0.5563 or 55.63%.

To calculate the likelihood of Erin getting at least one bullseye, we need to first calculate the probability of her not getting a bullseye in a single throw. Since she scores a bullseye 15% of the time, the probability of her not getting a bullseye in a single throw is 85% (100% - 15%).

Using the probability of not getting a bullseye in a single throw, we can use the following formula to calculate the probability of not getting a bullseye in all 5 throws:

0.85 x 0.85 x 0.85 x 0.85 x 0.85 = 0.4437

Therefore, the probability of Erin not getting a bullseye in all 5 throws is 0.4437 or 44.37%.

To calculate the probability of Erin getting at least one bullseye in 5 throws, we can subtract the probability of her not getting a bullseye in all 5 throws from 1:

1 - 0.4437 = 0.5563

Therefore, the likelihood of Erin getting at least one bullseye in 5 throws is 0.5563 or 55.63%.

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The probability that Erin will get at least one bullseye in her 5 throws at the adventure arcade is approximately 55.63%.

To find the probability that she will get at least one bullseye in 5 throws, we can use the complementary probability.

This means we will first find the probability of her not getting a bullseye in all 5 throws, and then subtract that from 1.

Find the probability of not getting a bullseye (1 - bullseye probability)

1 - 0.15 = 0.85

Calculate the probability of not getting a bullseye in all 5 throws

0.85^5 ≈ 0.4437

Find the complementary probability (probability of at least one bullseye)

1 - 0.4437 ≈ 0.5563

So, the probability that Erin will get at least one bullseye in her 5 throws at the adventure arcade is approximately 55.63%.

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The terms arranged in order from smallest to biggest are: (-2)³, -√25, √11, 10, and 4² after comparing the values of the final numbers.

We shall first simplify the numbers to get their final values and then compare to which is smaller as follows:

4² = 4 × 4 = 16

-√25 = -5

10 = 10

√11 = 3.3166

(-2)³ = -2 × -2 × -2 = -8

In conclusion, we have by comparing the final values of the numbers the terms arranged from smallest to the biggest as: (-2)³, -√25, √11, 10, and 4².

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The points on the surface z² = xy + 16 closest to the origin are: (-4,4,0) and (4, -4, 0)

We know that the distance between an arbitrary point on the surface and the origin is d(x, y, z) = √(x² + y² + z²)

Using Lagrange multipliers,

L(x, y, z, λ) = x² + y² + z² + λ(z² - xy - 16)

We have partial derivatives.

[tex]L_x[/tex] = 2x - λy

[tex]L_y[/tex] = 2y - λx

[tex]L_z[/tex] = 2z + 2zλ

[tex]L_\lambda[/tex] = z² - xy - 16

Now we set each partial derivative to zero to find critical points.

[tex]L_x[/tex] = 0

2x - λy = 0

[tex]L_y[/tex] = 0

2y - λx = 0

After solving above equations simultaneously we get (x + y)(x - y) = 0

i.e., x = -y   OR   x = y

[tex]L_z[/tex] = 0

2z + 2zλ = 0

z = 0  OR  λ = 0

Consider [tex]L_\lambda[/tex] = 0

z² - xy - 16 = 0

-xy = 16                  ............(as z = 0)

when x = y then -y² = 16 which is not true.

So, consider x = -y

-(-y)y = 16

y² = 16

y = ±4

when y = 4 then we get x = -4

and when y = -4 then we get x = 4

Therefore, the closest points are:(-4,4,0) and (4, -4, 0)

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The theoretical and experimental probability of rolling a 5 are 1/6 and 4/15 respectively.

We will calculate the theoretical probability by substituting 30 for the number of favorable outcomes as the die is rolled 30 times with one option each for 30 rolls and 180 for total number of outcomes in theoretical probability formula.

P(Theoretical probability of rolling a 5) = 30/180

P(Theoretical probability of rolling a 5) = 1/6.

The experimental probability is calculated by substituting 8 for the number of time the event occurs and 30 for the total number of trials.

P(Experimental probability of rolling a 5)= 8/30

P(Experimental probability of rolling a 5) =4/15

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Answer: 4080

Step-by-step explanation:

First you have to find the area of the triangle. 24*10 = 240. 240/2 = 120. Then you multiply the area of the triangle and multiply it by 34. 120 * 34 = 4080. This means the answer is 4080

The probability that you rolled a 5, given that the number rolled was greater than 3, is 1/3 or approximately 0.333.

We need to find the probability that you rolled a 5, given that the number rolled was greater than 3. Let's break this down step by step:

1. Identify the total number of outcomes: Since it is a 6-sided dice, there are 6 possible outcomes (1, 2, 3, 4, 5, and 6).

2. Determine the number of outcomes greater than 3: The outcomes greater than 3 are 4, 5, and 6. There are 3 possible outcomes that satisfy this condition.

3. Identify the number of outcomes that result in rolling a 5: There is only 1 outcome that results in rolling a 5.

4. Calculate the probability: To find the probability, divide the number of outcomes that result in rolling a 5 (1) by the total number of outcomes greater than 3 (3).

Probability = (Number of outcomes with a 5) / (Number of outcomes greater than 3) = 1/3

So, the probability that you rolled a 5, given that the number rolled was greater than 3, is 1/3 or approximately 0.333.

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The probability that the number rolled was a 5, given that it was greater than 3, is [tex]$\frac{1}{3}$[/tex].

The number rolled was greater than 3, it must be either a 4, 5, or 6.

The probability that the number rolled was a 5, given that it was greater than 3.

Let [tex]$A$[/tex] be the event that the number rolled is a 5 and let [tex]$B$[/tex] be the event that the number rolled is greater than 3.

Then, we want to find. [tex]$P(A|B)$[/tex], the probability of [tex]$A$[/tex] given [tex]$B$[/tex].

By Bayes' theorem, we have:

Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.

The risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole.

One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference.

The probabilities involved in the theorem may have different probability interpretations.

Bayesian probability interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of related evidence.

Bayesian inference is fundamental to Bayesian statistics, being considered by one authority as; "to the theory of probability what Pythagoras's theorem is to geometry."

[tex]$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$[/tex]

[tex]$P(A) = \frac{1}{6}$[/tex], since there is only one way to roll a 5 on a 6-sided die.

[tex]$P(B) = \frac{3}{6} = \frac{1}{2}$[/tex], since there are three outcomes (4, 5, or 6) that satisfy. [tex]$B$[/tex], out of a total of six possible outcomes.

[tex]$P(B|A)$[/tex], the probability of rolling a number greater than 3, given that the number rolled is a 5, note that. [tex]$B$[/tex] is true only if the number rolled is a 4, 5, or 6.

Since there is only one way to roll a 5, and only one of these three outcomes satisfies. [tex]$A$[/tex], we have:

[tex]$P(B|A) = \frac{1}{1} = 1$[/tex]

Substituting these values into Bayes' theorem, we get:

[tex]$P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{1 \cdot \frac{1}{6}}{\frac{1}{2}} = \frac{1}{3}$[/tex]

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the average speed of the plane in still air is s + 30.

Let's call the average speed of the plane in still air "s" (in miles per hour).

We can use the formula:

time = distance / speed

to find the time it takes the plane to travel from Salt Lake City to Oakland with the wind and against the wind.

With the wind:

time with wind = [tex]600 / (s + 30)[/tex]

Against the wind:

time against wind =[tex]600 / (s - 30)[/tex]

time against wind > time with wind

So we can set up an inequality:

[tex]600 / (s - 30) > 600 / (s + 30)[/tex]

Multiplying both sides by [tex](s - 30)(s + 30)[/tex], we get:

[tex]600(s + 30) > 600(s - 30)[/tex]

Expanding and simplifying, we get:

[tex]600s + 18000 > 600s - 18000[/tex]

Subtracting 600s from both sides, we get:

[tex]18000 > -18000[/tex]

This inequality is true for all values of s. In other words, there are no restrictions on the value of s that would make the return trip take longer than the trip with the wind.

Therefore, we can use the average of the two speeds (with and against the wind) to find the average speed of the plane in still air:

Average speed = [tex]2s(s + 30) / (s + 30 + s - 30)[/tex]

Simplifying, we get:

Average speed = [tex]2s(s + 30) / (2s)[/tex]

Canceling the common factor of 2s, we get:

Average speed = s + 30

We know that the distance from Salt Lake City to Oakland is 600 miles, and we can use the formula:

time = distance / speed

to find the time it takes the plane to travel this distance:

time = [tex]600 / (s + 30)[/tex]

We also know that the return trip (against the wind) takes longer, so we can set up another equation:

time return trip =[tex]600 / (s - 30)[/tex]

We can use these two equations to solve for s:

[tex]600 / (s + 30) = 600 / (s - 30)[/tex]

Cross-multiplying, we get:

[tex]600(s - 30) = 600(s + 30)[/tex]

Expanding and simplifying, we get:

[tex]600s - 18000 = 600s + 18000[/tex]

Subtracting 600s from both sides, we get:

[tex]-18000 = 18000[/tex]

This is not a valid equation, so there must be no solution.

However, we can still find the average speed of the plane in still air by using the equation we derived earlier:

Average speed = s + 30

So the average speed of the plane in still air is s + 30. We don't have a specific value for s, but we can say that the average speed is equal to the speed with the wind plus 30 (which is the speed of the wind).

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Hence, it is expected that 14 flights will arrive on time out of the 100 trials of this experiment.

The probability of an occurrence is a number used in mathematics to describe how likely it is that the event will take place. In terms of percentage notation, between 0% and 100% it is expressed as a number between 0 and 1, or . The higher the likelihood, the more likely it is that the event will take place.

 when we refer to an experiment or trial, we mean a random experiment. When difference  between a trial and an experiment, think of the experiment as a larger entity created by the fusion of several trials.

Unless otherwise stated,A trial is any specific outcome of a random experiment. In other words, a trial of the experiment is what we call when we conduct an experiment.

according to question, the number of on-time flights in 100 trials as a binomial random variable with parameters n = 100 (the number of trials) and p (the chance of success, i.e., a flight being on time), presuming that the probability of a flight being on time is the same in all trials.

The expected number of on-time flights in 100 trials is E(X) = np if the same of a flight being on time is p. Given that E(X) = 15, we determine p ,

E(X) = n p = 15 n = 100

p = [tex]\frac{E(X)}{n} = \frac{15}{100}[/tex] = 0.15

Therefore, it is probability that 0.15 %of flights will arrive on time.

To determine the expected  number of trials from a total of 100

Using the probability mass function of the binomial distribution, we can get the expected probability of trials out of 100 that result in precisely 15 flights departing on time:

[tex]P(X = 15)=(100 choose 15) * 0.15^{15} * 0.85^{85}[/tex]

We can calculate this 0.144 get  using a calculator.

therefore it is expected that 14 flights will arrive on time out of the 100 trials of this experiment. It should be noted that while this is an expected value, random fluctuation may cause the actual number of on-time flights in each trial to deviate somewhat from this figure.

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The value of x in the intersecting chords that extend outside circle is 5

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

intersecting chords that extend outside circle

Using the theorem of intersecting chords, we have

4 * (x + 6 + 4) = 6 * (x - 1 + 6)

Evaluate the like terms

So, we have

4 * (x + 10) = 6 * (x + 5)

Using a graphing tool, we have

x = 5

Hence. the value of x is 5

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Tim was driving for 5 hours before he caught up to Dora.

The answer is (a) 5 hours.

To solve this problem, we can use the formula:
distance = rate × time
Let's denote the time Tim drove as t hours.

Since Dora started driving one hour earlier, her driving time would be (t + 1) hours.
Dora's distance: 75 kph × (t + 1)
Tim's distance: 90 kph × t
Since Tim catches up to Dora, their distances will be equal:
75(t + 1) = 90t
Now we can solve for t:
75t + 75 = 90t
75 = 15t
t = 5.

The answer is (a) 5 hours.

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Answer:

[tex]h(x)=2x^2-12x+9[/tex],  [tex]p(x)=-5x^2-20x-5[/tex]

Step-by-step explanation:

To get to the standard form of a quadratic equation, we need to expand and simplify. Recall that standard form is written like so:

[tex]ax^2+bx+c[/tex]

Where a, b, and c are constants.

Let's expand and simplify h(x).

[tex]2(x-3)^2-9=\\2(x^2+9-6x)-9=\\2x^2+18-12x-9=\\2x^2+9-12x=\\2x^2-12x+9[/tex]

Thus, [tex]h(x)=2x^2-12x+9[/tex]

Let's do the same for p(x).

[tex]-5(x+2)^2+15=\\-5(x^2+4+4x)+15=\\-5x^2-20-20x+15=\\-5x^2-5-20x=\\-5x^2-20x-5[/tex]

Thus, [tex]p(x)=-5x^2-20x-5[/tex]

Therefore, the roots of the polynomial equation x² + x - 72 = 0 are -9 and 8.

A quadratic equation is a type of polynomial equation of the second degree, which means it has one or more terms in which the variable is raised to the power of two, but no higher powers.Quadratic equations can have zero, one, or two real solutions, depending on the values of a, b, and c. These solutions are also called the roots or zeros of the equation.

Here,

To find the roots of the polynomial equation x² + x - 72 = 0, we can set y = 0 and solve for x. This is equivalent to finding the x-intercepts of the graph of the function f(x) = x² + x - 72.

We can use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.

In this case, a = 1, b = 1, and c = -72, so we have:

x = (-1 ± √(1² - 4(1)(-72))) / (2(1))

x = (-1 ± √(1 + 288)) / 2

x = (-1 ± √(289)) / 2

x = (-1 ± 17) / 2

Therefore, the roots of the polynomial equation x² + x - 72 = 0 are:

x = -9 or x = 8

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The value of angle GFH is 18°

A circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident.

A theorem in circle geometry starts that angle in the same segment are equal. In triangle EFG, angle F and G are on the same segment, this means that angle F and G are equal.

Represent angle F as x

therefore 144+2x = 180° ( sum of angle in a triangle)

2x = 180-144

2x = 36

x = 36/2 = 18°

Therefore the measure of angle GFH is 18°

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we use linear equation in one variable to solve the problem. Kiran swam 10 laps in the pool.

Let's represent the number of laps Kiran swam as "z".

We know that Clare swam 18 laps, which is 9/5 times as many laps as Kiran. We can represent this relationship with the following equation:

18 = (9/5)z

To solve for z, we can isolate it by multiplying both sides of the equation by the reciprocal of 9/5, which is 5/9:

18 * (5/9) = (9/5)z * (5/9)

10 = z

Therefore, Kiran swam 10 laps in the pool.

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Sarah then purchased 9 coach seats as by increasing the first equation by 190 and deducting it from the second equation.

An equation is a logical statement that utilises the equal sign to demonstrate the equality of two expressions. Factors, constants, and mathematical like addition, reduction, multiply, division, and exponentiation can all be found in it. Equations are utilised to find solutions for problems in both mathematics and the real world.

given

Let's use the letters "c" for the quantity of coach tickets and "f" for the quantity of first-class tickets. We are aware that there were 12 travellers in all, so

c + f + 1 = 12

We also know that the entire cost of the airfare was $4650, with coach tickets costing $190 and first-class tickets costing $980. With this knowledge, we can construct the equation shown below:

[tex]190c + 980f = 4650 - 980[/tex]

When we simplify this equation, we obtain:

[tex]190c + 980f = 3670[/tex]

Elimination can now be used to find either "c" or "f." By increasing the first equation by 190 and deducting it from the second equation, let's get rid of "c":

[tex]190c + 190f + 190 = 2280[/tex]

-190c - 980f = -3670

-790f = -1390

f = 1.76

We can round "f" up to 2 because we cannot have a fractional number of persons.

c + 2 + 1 = 12

c = 9

Sarah then purchased 9 coach seats as by increasing the first equation by 190 and deducting it from the second equation.

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Just use the pythagorean theorem to solve the hypotenuse!

(3^2)+(2^2)=x^2

9+4=13^2

[tex]\sqrt{13}[/tex] = [tex]\sqrt{x}[/tex]

[tex]13^{2}[/tex] km

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The ball that weighs 30g is ball A.

The degree to which a measured value resembles the true or recognised value is known as accuracy. On the other hand, precision describes how closely two measurements of the same quantity agree. In other words, precision is a measure of consistency, whereas accuracy is a measure of correctness. The values obtained are consistent but not always close to the genuine value when a measurement is precise but not exact. On the other hand, a measurement might be accurate without being exact, which means that the result is close to the correct value but produces erratic or dispersed results when repeated.

From the given weights corresponding to different balls, 30 g corresponds to ball A.

Hence, the ball that weighs 30g is ball A.

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The Cube root of the resulting number is 8.

The smallest number that 40000 can be divided by so that the result is a perfect cube, we need to factorize 40000 into its prime factors:

[tex]40000 = 2^6 \times 5^4[/tex]

To make this a perfect cube, we need to ensure that the powers of each prime factor are multiples of 3.

The smallest number we can divide 40000 by so that the result is a perfect cube is:

[tex]40000 = 2^6 \times 5^4[/tex]

Now we can find the cube root of the resulting number:

[tex]3\sqrt (40000 \div 100) = 3\sqrt400 = 8.[/tex]

Factories 40000 into its prime components in order to determine.

The least number that the result may be divided by while still producing a perfect cube.

The powers of each prime factor must be multiples of three in order for this to be a perfect cube.

The least number that 40000 may be divided by to produce a perfect cube is:

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Without knowing which specific function you're referring to, the answer to this question may depend on the type of function and the nature of the iterative process applied to it. In some cases, the function value may converge towards a limiting value as the number of iterations increases, while in other cases it may oscillate or diverge.

For example, in the case of the fixed-point iteration method used to find the root of a function, the value of the function typically converges towards the root as the number of iterations increases. More specifically, if we have a function f(x) and a starting guess x0 for its root, we can use the iterative formula x(+1)=g(x()), where g(x) is some function that we set based on f(x), to generate a sequence of increasingly accurate approximations to the root. As the number of iterations increases, this sequence of approximations typically converges towards the root of the function, unless some conditions are not met (e.g., the method is not well-suited for some functions, or the iteration formula is not properly set.)

In the case of other types of iterative methods or other functions, however, the behavior of the function value as the number of iterations increases may differ. For instance, in some cases, the function value may oscillate between two or more values or diverge to infinity as the number of iterations increases.

Therefore, the specific behavior of the function value as the number of iterations increases may depend on the specific function being evaluated and the iterative method used.

The 'footprint' of CO2 emissions for a person in 1830 would be 818,199 tons of CO2 emissions per person.

To find the 'footprint' of CO2 emissions for a person in 1830, we need to substitute the value of x = 1830 - 1800 = 30 into the given function C(x) = 0.0365 (1.758)^x.

Plugging in x = 30 into the function, we get:

C(30) = 0.0365 * (1.758)^30

Substituting this value back into the function, we get:

C(30) = 0.0365 * 22416413.1381

C(30) = 818199.079541

C(30) ≈ 818,199.08

Answered question "Scientists studying the 'footprint' of carbon dioxide (CO2) emissions attributed to the average person for each decade from 1800 to 1910 used the function C(x) = 0.0365 (1.758)*, where x is the number of decades since 1800 and C is the number of tons of CO2 emissions per person. What is the 'footprint' of CO2 emissions for a person in 1830??"

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Answer:

x = -5, x = 1

As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).

Step-by-step explanation:

The x-intercepts are the x-values of the points at which the curve crosses the x-axis, so when y = 0.

From inspection of the given graph, we can see that the parabola crosses the x-axis at x = -5 and x = 1.

Therefore, the x-intercepts of the parabola are:

As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).

The 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is approximately [-6.62, 22.02].

The critical value that should be used in constructing the confidence interval.

Since we are looking for a 90% confidence interval, we need to find the critical value associated with a 5% level of significance in a two-tailed test.

Using a t-distribution with (n1-1) + (n2-1) degrees of freedom and a significance level of 0.05, we find the critical value to be:

t-critical = 1.717 (using a t-distribution table or a calculator)

Step 2 of 3:

Next, we need to find the standard error of the sampling distribution to be used in constructing the confidence interval.

Since the population variances are not equal, we need to use the Welch-Satterthwaite equation to calculate the standard error:

SE = sqrt[([tex]s1^2[/tex]/n1) + ([tex]s2^2[/tex]/n2)]

where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Substituting the given values, we get:

SE = sqrt[([tex]19.25^2[/tex]/17) + ([tex]26.25^2[/tex]/12)]

SE ≈ 8.35

Step 3 of 3:

To construct the 90% confidence interval, we can use the formula:

(mean1 - mean2) ± t-critical * SE

where mean1 and mean2 are the sample means, and t-critical and SE are the values calculated in steps 1 and 2.

Substituting the given values, we get:

= (84.80 - 77.10) ± 1.717 x 8.35

= 7.70 ± 14.32

Therefore,

The 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is (approx) [-6.62, 22.02].

We can be 90% confident that the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases falls within this interval.

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the simplified  form  of expression is: -(x² + 2x - 2)/((x+2)*(x+4))

what is expression  ?

In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that represents a mathematical quantity or relationship. Expressions can be simple or complex

In the given question,

To evaluate the expression 1/(x+2) - (x+1)/(x+4), we need to find a common denominator for the two terms. The least common multiple of (x+2) and (x+4) is (x+2)(x+4).

So, we can rewrite the expression as:

(1*(x+4) - (x+1)(x+2))/((x+2)(x+4))

Expanding the brackets, we get:

(x+4 - x² - 3x - 2)/((x+2)*(x+4))

Simplifying the numerator, we get:

(-x² - 2x + 2)/((x+2)*(x+4))

Therefore, the simplified expression is:

-(x² + 2x - 2)/((x+2)*(x+4))

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The chance that the number of heads will be in the range 4850 to 5150 is approximately 0.9973, or about 99.73%.

The number of heads in 10,000 coin tosses follows a binomial distribution with parameters n = 10,000 (the number of trials) and p = 0.5 (the probability of heads on a single toss).

We can approximate this binomial distribution using the normal distribution, with mean μ = np = 5000 and variance σ² = np(1-p) = 2500.

To find the probability that the number of heads is in the range 4850 to 5150, we can use the normal distribution and standardize the range using the z-score formula:

z = (x - μ) / σ

where x is the number of heads in the range we're interested in.

For the lower bound of 4850, we have:

[tex]z_lower = (4850 - 5000) / \sqrt{(2500)}[/tex]

= -3

For the upper bound of 5150, we have:

[tex]z_upper = (5150 - 5000) / \sqrt{(2500)} = 3[/tex]

Using a standard normal distribution table or calculator, we can find the probability of being within 3 standard deviations of the mean:

P([tex]z_lower[/tex]  < Z < [tex]z_upper[/tex] ) ≈ P(-3 < Z < 3)

= 0.9973.

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The equation x = 10 tan (32°) could be used to find x in ∆ABC.

A triangle is classified as a right triangle when it presents one of your angles equal to 90º.  The greatest side of a right triangle is called the hypotenuse. And, the other two sides are called legs.

The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem.

The Pythagorean Theorem says: (hypotenuse)²= (leg1)²+(leg2)² . And the main trigonometric ratios are: sin (x) , cos  (x) and tan  (x) , where:

[tex]sin(x)=\frac{opposite\ side}{hypotenuse} \\ \\ cos(x)=\frac{adjacent\ side}{hypotenuse}\\ \\ tan(x)=\frac{sin(x)}{cos(x)} =\frac{opposite\ side}{adjacent\ side}[/tex]

The question gives the value of the two sides and the value of an angle. From the trigonometric ratios presented before, you can write:

[tex]tan(32)=\frac{opposite\ side}{adjacent\ side}=\frac{x}{10} \\ \\ x=10\ tan (32\°)[/tex]

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