Determine the area of the shaded regions in the figures below. Write the final polynomial in standard

The pοint that satisfies bοth inequalities is the pοint inside this triangular regiοn.

An inequality is a mathematical statement that cοmpares twο values οr expressiοns and indicates whether they are equal οr nοt, οr which οne is greater οr smaller.

Since the shading is nοt included, we will need tο use the lines themselves tο determine the cοrrect regiοn οf the cοοrdinate plane.

The first inequality y > (3/2)x - 5 has a slοpe οf 3/2 and a y-intercept οf -5. This means the line will have a pοsitive slοpe and will be lοcated belοw the pοint (0,-5).

The secοnd inequality y < (-1/6)x - 6 has a negative slοpe οf -1/6 and a y-intercept οf -6. This means the line will have a negative slοpe and will be lοcated abοve the pοint (0,-6).

Tο find the pοint that satisfies BOTH inequalities, we need tο lοοk fοr the regiοn οf the cοοrdinate plane that is belοw the line y = (3/2)x - 5 AND abοve the line y = (-1/6)x - 6. This regiοn is the triangular-shaped area that is bοunded by the twο lines and the x-axis.

The pοint that satisfies bοth inequalities is the pοint inside this triangular regiοn.

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a. S = {(1, -1), (2, 1)}Let's begin by calculating the determinant of the matrix composed of the vectors of S, and checking if it is equal to 0. Because the two vectors are not colinear, they should span R2.|1 -1||2 1| determinant is not 0, therefore S spans R2. No geometric description is required for this example.

b. S = {(1, 1)} The set S contains one vector. A set containing only one vector cannot span a plane because it only spans a line. Therefore, S does not span R2. Geometric description: S spans a line that passes through the origin (0, 0) and the point (1, 1).c. S = {(0, 2), (1, 4)} Let's again begin by calculating the determinant of the matrix composed of the vectors of S, and checking if it is equal to 0.|0 2||1 4| determinant is 0, thus S does not span R2. In this scenario, S only spans the line that contains both vectors, which is the line with the equation y = 2x.

Geometric description: S spans a line that passes through the origin (0, 0) and the point (1, 2).

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The following characteristics of racional decimal number abstraction apply: Conmutative property: The order of the remaining rational decimal numbers has no bearing on the operation's outcome,

Proprietary property: The racional decimal numbers may remain in various groups without affecting the operation's ultimate outcome, i.e., (a - b) - c = a - (b - c). Distributive property: Subtracting one racional decimal number from a sum of racional decimal numbers equals the sum of the subtractions of each one of them, or a - (b + c) = a - b - c. Neutral element: If a racional decimal number is left at zero, the outcome is the same number, i.e., a - 0 = a. Estas propiedades son útiles para simplificar y realizar cálculos más complejos con números racionales decimales.

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Answer:

P= 40 ft

Step-by-step explanation:

Perimeter is the sum of all the lengths

So,

Perimeter= 6+4+15+10ft

= 35ft

Nearest ten can be 40ft or 30ft

If you succeed In understanding then kindly mark my answer the brainliest. Thank you :)

Answer:

4(h+3) = 20

Step-by-step explanation:

Para empesar, disculpa si mi español no es perfecto, pero igual me encataria a ayudarte.

Pues, se sabe que estas temporadas de practica vienen en groupitos de horas a la ves. Dijo que cada dia, ella practica por alguans horas, las cuales suman a 20 en total. Como la problema nos dice que ella practica 4 veces a la semana, tienemos 4 de estos groupitos de horas. Por eso, la respuesa es 4(h+3) = 20, porque ella va por estas 4 temporadas de practicar 3 horas en la manana y quien sabe cuantos en la tarde. Addicionalmente, este"quien sabe" numero de horas se representa con h.

9 hikers are the bare minimum that might have covered the range of 6 to 17 miles because that distance falls inside the typical interval of 10 x 15 miles.

Rearrange the function using fundamental algebraic concepts to determine the value of x when the derivative equals 0.

This response gives the x-coordinate of the function's vertex, which is where the maximum or minimum will occur.

To determine the minimum or maximum, rewrite the solution into the original function.

The greatest and smallest values of a function, either within a specific range (the local or relative extrema) or throughout the entire domain, are collectively referred to as extrema (PL: extrema) in mathematical analysis.

b)  the maximum number of hikers who could have walked between 6 miles and 17 miles is 19.

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If the cost price of the goods is R200 and they are sold at a mark-up of 100%, then the selling price is equal to the cost price plus the mark-up, or:

Selling price = Cost price + Mark-up

Mark-up = 100% x Cost price

= 100% x R200

= R200

So the mark-up is R200.

Selling price = Cost price + Mark-up

= R200 + R200

= R400

Therefore, the selling price of the goods is R400.

Answer: 23 degrees

Step-by-step explanation:

Assuming that 117 is the entire angle we can find that:

94+x = 117

Subtract 94 from both sides:

x = 117-94

x = 23 degrees

Answer:

  3. 79.9 mm²

  4. 6.4 in

  5. 177.5 mi²

  6. 60.3°

Step-by-step explanation:

Given various quadrilaterals and their dimensions, you want to find missing dimensions.

In all cases, one or more area formulas and trig relations are involved. The trig relations are summarized by the mnemonic SOH CAH TOA. The relevant relation for these problems is ...

  Tan = Opposite/Adjacent

It is also useful to know that 1/tan(x) = tan(90°-x).

The formula for the area of a trapezoid is ...

  A = 1/2(b1 +b2)h

The relevant formula here for the area of a parallelogram is ...

  A = bs·sin(α) . . . . . where α is the angle between sides of length b and s

Using the area formula above, we find the area to be ...

  A = (21 mm)(9 mm)·sin(155°) ≈ 79.9 mm²

The area is about 79.9 square mm.

The given figure shows two unknowns. We can write equations for these using the area formula and using a trig relation.

If we draw a vertical line through the vertex of the marked angle, the base of the triangle to the right of it is (b2-4). The acute angle at the top of that right triangle is (121°-90°) = 31°. The tangent relation tells us ...

  tan(31°) = (b2 -4)/h   ⇒   h = (b2 -4)/tan(31°)

Using the area formula we have ...

  A = 1/2(b2 +4)h

and substituting for A and h, we get ...

  20.8 = 1/2(b2 +4)(b2 -4)/tan(31°)

  2·tan(31°)·20.8 = (b2 +4)(b2 -4) = (b2)² -16 . . . . . multiply by 2tan(31°)

  (b2)² = 2·tan(31°)·20.8 +16 . . . . . . . add 16

  b2 = √(2·tan(31°)·20.8 +16) ≈ 6.4 . . . . . . take the square root

Base 2 of the trapezoid is about 6.4 inches.

To find the area, we need to know the height of the trapezoid. To find the height we can solve a triangle problem.

If we draw a diagonal line parallel to the right side through the left end of the top base, we divide the figure into a triangle on the left and a parallelogram on the right. The triangle has a base width of 10 mi, and base angles of 60° and 75°.

Drawing a vertical line through the top vertex of this triangle divides it into two right triangles of height h. The top angle is divided into two angles, one being 90°-60° = 30°, and the other being 90°-75° = 15°. The bases of these right triangles are now ...

and their sum is 10 mi.

The height h can now be found to be ...

  h·tan(30°) +h·tan(15°) = 10

  h = 10/(tan(30°) +tan(15°))

Back to our formula for the area of the trapezoid, we find it to be ...

  A = 1/2(b1 +b2)h = 1/2(20 +10)(10/(tan(30°) +tan(15°)) ≈ 177.5

The area of the trapezoid is about 177.5 square miles.

The final formula we used for problem 5 can be used for problem 6 by changing the dimensions appropriately.

  A = 1/2(b1 +b2)(b1 -b2)/(tan(90-x) +tan(90-x))

  112 = 1/2(20+12)(20-12)/(2·tan(90-x)) = (20² -12²)·tan(x)/4

  tan(x) = 4·112/(20² -12²)

  x = arctan(4·112/(20² -12²)) = arctan(7/4) ≈ 60.3°

Angle x° in the trapezoid is about 60.3°.

__

Additional comment

There is no set "step by step" for solving problems like these. In general, you work from what you know toward what you don't know. You make use of area and trig relations as required to create equations you can solve for the missing values. There are generally a number of ways you can go at these.

A nice scientific calculator has been used in the attachment for showing the calculations. A graphing calculator can be useful for solving any system of equations you might write.

The second attachment shows a graphing calculator solution to problem 5, where we let y = area, and x = the portion of the bottom base that is to the left of the top base. Area/15 represents the height of the trapezoid. This solution also gives an area of 177.5 square miles.

Answer:

C = 21,309.4

Step-by-step explanation:

Diameter of moon is miles is,

d = 2159.8 miles

We have,

The circumference of the moon is, 6783 miles

Since, We know that,

the circumference of circle is,

C = 2πr

Substitute given values,

6783 miles = 2 × 3.14 × r

6783 = 6.28 × r

r = 6783 / 6.28

r = 1079.9 miles

Therefore, Diameter of moon is miles is,

d = 2 x r

d = 2 x 1079.9

d = 2159.8 miles

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Answer:

142 sq cm

Step-by-step explanation:

A= 2(lh + wh + lw)

2(7*3+5*3+7*5)

2(21+15+35)

2(71)

A= 142 sq cm

We cannot prove that the quadrilateral is a parallelogram with only the given information that AB = DC.

What is quadrilateral and parallelogram ?

A quadrilateral is a four-sided polygon, which means it is a closed shape with four straight sides. Some examples of quadrilaterals include rectangles, squares, trapezoids, and rhombuses.

A parallelogram is a special type of quadrilateral where both pairs of opposite sides are parallel. This means that the opposite sides never intersect, and they have the same slope. Additionally, the opposite sides of a parallelogram are congruent (i.e., have the same length), and the opposite angles are also congruent. Some examples of parallelograms include rectangles, squares, and rhombuses.

To prove that a quadrilateral is a parallelogram, we need to show that both pairs of opposite sides are parallel. Knowing that AB = DC only gives us information about the lengths of the sides, but it doesn't tell us anything about their orientation or whether they are parallel.

We would need additional information, such as the measures of angles or the lengths of other sides, to determine whether the quadrilateral is a parallelogram.

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The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it vertically by the factor of 1/2.

The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will stretch it vertically by factor of 2.

Thus, Option B and Option D are correct.

A function is a relationship or expression involving one or more variables.  It has a set of input and outputs.  

A. F(x)= 2 square x has the same domain and range as f(x)= square x.

B. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it vertically by the factor of 1/2.

D. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will stretch it vertically by factor of 2.

Option A is false because multiplying the function by 2 will change the range of the function to include all non-negative real numbers (since the square of any number is non-negative).

Option B is true because multiplying the function by 2 will vertically shrink the graph by a factor of 1/2 (since the output values will be half the size of the original function).

Option C is false because multiplying the function by 2 will not affect the horizontal scale of the graph.

Option D is true because multiplying the function by 2 will vertically stretch the graph by a factor of 2 (since the output values will be twice the size of the original function).

Therefore, Option B and Option D are correct.

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The value of the double integral using the easier order, ydA bounded by y = x − 6; x = y² is 125/12.

The double integral, indicated by ', is mostly used to calculate the surface area of a two-dimensional figure. By using double integration, we may quickly determine the area of a rectangular region. If we understand simple integration, we can easily tackle double integration difficulties. Hence, first and foremost, we will go over some fundamental integration guidelines.

Given, the double integral ∫∫yA and the region y = x-6 and x = y²

y = x-6

x = y²

y² = y +6

y² - y - 6 = 0

y² - 3y +2y - 6 = 0

(y-3) (y+2) = 0

y = 3 and y = -2

[tex]\int\int\limits_\triangle {y} \, dA\\ \\[/tex]

= [tex]\int\limits^3_2 {y(y+6-y^2)} \, dx \\\\\int\limits^3_2 {(y^2+6y-y^3)} \, dx \\\\(\frac{y^3}{3} + 3y^2-\frac{y^4}{4} )_-_2^3\\\\\frac{63}{4} -\frac{16}{3} \\\\\frac{125}{12}[/tex]

The value for the double integral is 125/12.

Integration is an important aspect of calculus, and there are many different forms of integrations, such as basic integration, double integration, and triple integration. We often utilise integral calculus to determine the area and volume on a very big scale that simple formulae or calculations cannot.

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Answer:

p = 0, p = -7/3

Step-by-step explanation:

We are given the following equation:
3p² + 7p = 0

We want to solve the equation by factoring.

To factor, we want to look for a common term that we can pull out.

You may notice that both terms have 'p' in common, so we can pull out p from both terms.

This will then make the equation:

p(3p + 7) = 0

Now, we can use zero product property to solve the equation.

p = 0

3p + 7 = 0

Subtract.

3p = -7

Divide.

p = -7/3

Our answers are p = 0 and p = -7/3

The triangles ABC and DEF are similar triangles, but DEF is twice as big as ABC.

What does it signify when two triangles are similar?

Congruent triangles are triangles that share similarity in shape but not necessarily in size. All equilateral triangles and squares of any side length serve as illustrations of related objects.

                        Or to put it another way, the corresponding angles and sides of two triangles that are similar to one another will be congruent and proportionate, respectively.

How do the sizes of the circles compare?

Given the triangles ABC and DEF

From the figure, we have

AB = 1

DE = 2

This means that the triangle DEF is twice the size of the triangle ABC

Are triangles ABC and DEF similar?

Yes, the triangles ABC and DEF are similar triangles

This is because the corresponding sides of  DEF is twice the corresponding sides of triangle ABC

How can you use the coordinates of A to find the coordinates of D?

Multipliying the coordinates of A by 2 gives coordinates of D.

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Answer:

5.333 hours

Step-by-step explanation:

We know

4 Pipes fill a tank in 16 hours.

How long will it take to fill the tank if 12 pipes of the same dimensions are used?

We Take

16 x 1/3 = 5.333 hours

So, it takes about 5.333 hours to fill the tank.

The value of (37)-2 is 1/1369, in its simplest form as an improper fraction.

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. In other words, it is a fraction that is larger than a whole number.

When an expression is written in the form of [tex]x^{(-n)[/tex], it means the reciprocal of [tex]x^n.[/tex] In this case, we have the expression[tex](37)^{(-2)[/tex] which means the reciprocal of 37².

The expression (37)-2 means 37 raised to the power of -2, or 1/(37²). To simplify this fraction, we can multiply the numerator and denominator by 1,296 (37²):

1/(37²) = 1 * 1 / (37 * 37)

= 1/1369

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The average number of hours of sleep for working college students is between 6.28 and 6.72 hours of sleep each night

According to the given data,

Sample size n = 101

Sample mean x = 6.5

Standard deviation s = 2.14

Level of confidence C = 96%

Using a 96% confidence level, the value of t* for 100 degrees of freedom is 2.081, as given in the question.

Now, the formula for the confidence interval is:x ± (t* × s/√n)Here, x = 6.5, s = 2.14, n = 101, and t* = 2.081

Substituting the values in the above formula, we get:

Lower limit = x - (t* × s/√n) = 6.5 - (2.081 × 2.14/√101) = 6.28

Upper limit = x + (t* × s/√n) = 6.5 + (2.081 × 2.14/√101) = 6.72

Therefore, the 96% confidence interval for the average number of hours of sleep for working college students is between 6.28 and 6.72 hours of sleep each night.

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The zero vector: The zero vector 0 = [0,0;0,0;0,0] is also a member of W. Thus, the third criterion is satisfied.

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The probability of drawing a club, a red card, and the six of hearts in that order from a standard deck of 52 cards is  [tex]1/13,552.[/tex]

This is because the probability of drawing a club is 1/4, and the probability of drawing a red card is 1/2, and the probability of drawing the six of hearts is 1/52.
Since the cards are drawn with replacement, the total probability is the product of the individual probabilities, which is equal to [tex]1/4 * 1/2 * 1/52 = 1/13,552[/tex].
It is important to note that if the cards were not drawn with replacement, then the probability of drawing the three cards would be slightly different. The total probability would be equal to [tex]1/4 * 1/2 * 1/51 = 1/12,600.[/tex]
It is also important to note that since this is a probability question, the answer can be expressed as a decimal or percentage. In decimal form, the probability of drawing the three cards is 0.000074, and in percentage form, the probability of drawing the three cards is 0.0074%.

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The sum of the real and imaginary parts of $c$ is$$\operatorname{Re}(c) + \operatorname{Im}(c) = \frac{\operatorname{Re}(2c)}{2} + \frac{\operatorname{Im}(2c)}{2}$$$$= \frac{\operatorname{Re}(z_0+z_2)}{2} - \frac{\operatorname{Im}(z_0)}{2}(1-\cos(\theta/2)) - \frac{\operatorname{Re}(z_0)}{2}\sin(\theta/2)$$$$+ \frac{\operatorname{Im}(z_0+z_2)}{2} - \frac{\operatorname{Re}(z_0)}{2}(1-\cos(\theta/2)) + \frac{\operatorname{Im}(z_0)}{2}\sin(\theta/2).$$

The given problem can be solved using algebraic and geometric methods. We can use algebraic methods, such as the equations given in the problem, and we can use geometric methods by visualizing what the problem is asking. To start, let's translate the given problem into mathematical equations. Let $z_0$ be the original complex number. We want to rotate this point by 90 degrees counter-clockwise about some complex number $c$ to get $z_2$. Thus,$$z_2 = c + i(z_0 - c)$$$$=c + iz_0 - ic$$$$= (1-i)c + iz_0.$$We also know that this transformation will rotate the point $z_1 = (z_0 + z_2)/2$ by 45 degrees. Thus, using similar logic,$$z_1 = (1-i/2)c + iz_0/2.$$Now let's use the formula for rotating a point about the origin by $\theta$ degrees (where $\theta$ is measured in radians) to find a relationship between $z_1$ and $z_0$.$$z_1 = z_0 e^{i\theta/2}$$$$\implies (1-i/2)c + iz_0/2 = z_0 e^{i\theta/2}$$$$\implies (1-i/2)c = (e^{i\theta/2} - 1)z_0/2.$$We can solve for $c$ by dividing both sides by $1-i/2$.$$c = \frac{e^{i\theta/2} - 1}{1-i/2}\cdot\frac{z_0}{2}.$$We can now use the information given in the problem to solve for the sum of the real and imaginary parts of $c$. We know that rotating $z_0$ by 90 degrees counter-clockwise will result in the complex number $z_2$. Visually, this means that $c$ is located at the midpoint between $z_0$ and $z_2$ on the line that is perpendicular to the line segment connecting $z_0$ and $z_2$. We can use this geometric interpretation to solve for $c$. The midpoint of the line segment connecting $z_0$ and $z_2$ is$$\frac{z_0+z_2}{2} = c + i\frac{z_0-c}{2}.$$Solving for $c$, we get$$c = \frac{z_0+z_2}{2} - \frac{i}{2}(z_0-c)$$$$\implies 2c = z_0+z_2 - i(z_0-c)$$$$\implies 2c = z_0+z_2 - i(z_0- (e^{i\theta/2} - 1)(z_0/2)/(1-i/2)).$$We can now find the real and imaginary parts of $c$ and add them together to get the desired answer. Let's first simplify the expression for $c$.$$2c = z_0+z_2 - i(z_0 - (e^{i\theta/2} - 1)\cdot(z_0/2)\cdot(1+i)/2)$$$$= z_0 + z_2 - i(z_0 - z_0(e^{i\theta/2} - 1)(1+i)/4)$$$$= z_0 + z_2 - i(z_0 - z_0e^{i\theta/2}(1+i)/4 + z_0(1-i)/4)$$$$= z_0 + z_2 - i(z_0(1-e^{i\theta/2})/4 + z_0(1-i)/4)$$$$= z_0 + z_2 - i(z_0/4(1-e^{i\theta/2} + 1 - i))$$$$= z_0 + z_2 - i(z_0/2(1-\cos(\theta/2) - i\sin(\theta/2)))$$$$= z_0 + z_2 - i(z_0(1-\cos(\theta/2)) + z_0\sin(\theta/2) - i(z_0\cos(\theta/2))/2.$$Now we can find the real and imaginary parts of $2c$ and divide by 2 to get the real and imaginary parts of $c$. We have$$\operatorname{Re}(2c) = \operatorname{Re}(z_0+z_2) - \operatorname{Im}(z_0)(1-\cos(\theta/2)) - \operatorname{Re}(z_0)\sin(\theta/2)$$$$\operatorname{Im}(2c) = \operatorname{Im}(z_0+z_2) - \operatorname{Re}(z_0)(1-\cos(\theta/2)) + \operatorname{Im}(z_0)\sin(\theta/2).$$Thus, the sum of the real and imaginary parts of $c$ is$$\operatorname{Re}(c) + \operatorname{Im}(c) = \frac{\operatorname{Re}(2c)}{2} + \frac{\operatorname{Im}(2c)}{2}$$$$= \frac{\operatorname{Re}(z_0+z_2)}{2} - \frac{\operatorname{Im}(z_0)}{2}(1-\cos(\theta/2)) - \frac{\operatorname{Re}(z_0)}{2}\sin(\theta/2)$$$$+ \frac{\operatorname{Im}(z_0+z_2)}{2} - \frac{\operatorname{Re}(z_0)}{2}(1-\cos(\theta/2)) + \frac{\operatorname{Im}(z_0)}{2}\sin(\theta/2).$$

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The probability that student A or B can solve a problem chosen at random is 0.95.

Probability is calculated by dividing the number of favourable outcomes by the number of possible outcomes.

Random: An event is referred to as random when it is not possible to predict it with certainty. The probability that either student A or B will be able to solve a problem chosen at random can be calculated as follows:

P(A or B) = P(A) + P(B) - P(A and B) where: P(A) = probability of A solving a problem = 0.75, P(B) = probability of B solving a problem = 0.7, P(A and B) = probability of both A and B solving a problem. Since A and B are independent, the probability of both solving the problem is:

P(A and B) = P(A) x P(B) = 0.75 x 0.7 = 0.525

Now, using the above formula: P(A or B) = P(A) + P(B) - P(A and B) = 0.75 + 0.7 - 0.525 = 0.925

Therefore, the probability that student A or B can solve a problem chosen at random is 0.95 (or 95%).

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Answer:

x=33

Step-by-step explanation:

all angles in a triangle sum to 180 degrees

x+2x+(2x+15) = 180 <---- simplify this

5x+15 = 180

5x=165

x = 33

The Trapezoidal rule and Simpson's rule are two methods used to approximate the value of a definite integral. The Trapezoidal rule approximates the integral by dividing the region between the lower and upper limits of the integral into n trapezoids, each with a width h. The approximate value of the integral is then calculated as the sum of the areas of the trapezoids. The Simpson's rule is similar, except the region is divided into n/2 trapezoids and then the integral is approximated using the weighted sum of the area of the trapezoids.

For the given integral 4 x x2 1 0 dx, with n = 200, the Trapezoidal rule and Simpson's rule approximate the integral to be 7.4528 and 7.4485 respectively, rounded to four decimal places. The exact value of the integral is 7.4527. The difference between the exact and approximate values is very small, thus indicating that both the Trapezoidal rule and Simpson's rule are accurate approximations.

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According to a census, 3.3% of all births in a country are twins. In a month, there are 2,500 births. The census reports that 3.3% of all births result in twins, and the probability of having more than 90 twins in a month is "0.4351."

We will solve this problem using the binomial distribution formula, which is as follows:P (X > 90) = 1 - P (X ≤ 90)where P represents the probability, X represents the number of twins born in a month, and X is a binomial random variable with a sample size of n = 2,500 and a probability of success (having twins) of p = 0.033. Using the TI-83 calculator, TI-83 Plus, or TI-84 calculator, the following steps can be followed:

Press the "2nd" button followed by the "VARS" button (DISTR) to access the distribution menu. Scroll down and select "binomcdf (" from the list of options (use the arrow keys to navigate). The binomcdf ( menu will appear on the screen. The first number in the parentheses is the number of trials, n, and the second number is the probability of success, p. We want to find the probability of having more than 90 twins, so we need to use the "compliment" option. Therefore, we will subtract the probability of having 90 twins or less from 1 (using the "1 -" key). Type in "binomcdf (2500,0.033,90)" and press the "ENTER" button on your calculator.

This will give you the probability of having 90 twins or fewer in a month. Subtract this value from 1 to obtain the probability of having more than 90 twins in a month, which is the answer to our question. P(X>90) = 1 - binomcdf (2500,0.033,90)P(X>90) = 1 - 0.5649P(X>90) = 0.4351Therefore, the probability of having more than 90 twins in a month is 0.4351.

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Answer:

[tex] |x - 300| \leqslant 10[/tex]

 In conclusion, the Poisson distribution was successfully applied by Student (1907) to the study of errors in counting yeast cells or blood corpuscles with a haemacytometer. It is possible to calculate an approximate 95% confidence interval for each estimated count, as well as to compare observed and expected counts.

The Poisson distribution was first applied to the study of errors made in counting yeast cells or blood corpuscles with a haemacytometer by Student (1907). The study involved the preparation of four different concentrations of a mixture of yeast cells, water, and gelatin spread on a glass. Counts were made on 400 squares and the data summarized in the following table.

An approximate 95% confidence interval for each estimate can be calculated using the Poisson distribution. For each of the four concentrations, the lower bound of the confidence interval is given by the formula x - 1.96*sqrt(x) and the upper bound is given by the formula x + 1.96*sqrt(x), where x is the observed count for that concentration.

It is also possible to compare the observed counts with the expected counts for each concentration. The expected count for each concentration is given by the formula λ = n*p, where n is the number of squares and p is the probability of an event occurring in a single square. The expected counts can be compared to the observed counts to determine whether they are in agreement with the Poisson distribution.

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(a) The probability that exactly two cars finish the race is 0.0512.

(b) The probability that at most two cars finish the race is 0.05792.

(c) The probability that at least three cars finish the race is 0.94208.

(a) To determine the probability that exactly two cars finish the race, we have to use binomial distribution. In this case, we have n = 5 trials, and p = 0.8 is the probability that a car finishes the race (1 - 0.2). Using the binomial distribution formula:

P(X = k) = (nCk)(p^k)(1 - p)^(n - k)

Where X is the number of cars that finish the race, we get:

P(X = 2) = (5C2)(0.8²)(0.2)³= (10)(0.64)(0.008)= 0.0512

Therefore, the probability that exactly two cars finish the race is 0.0512.

(b) To determine the probability that at most two cars finish the race, we have to calculate the probabilities of 0, 1, and 2 cars finishing the race and add them up.

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)= (5C0)(0.8⁰)(0.2)⁵ + (5C1)(0.8¹)(0.2)⁴ + (5C2)(0.8²)(0.2)³= 0.00032 + 0.0064 + 0.0512= 0.05792

Therefore, the probability that at most two cars finish the race is 0.05792.

(c) To determine the probability that at least three cars finish the race, we can calculate the probability of 0, 1, and 2 cars finishing the race and subtract it from 1, which gives us the probability of at least three cars finishing the race.

P(X ≥ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]= 1 - (0.00032 + 0.0064 + 0.0512)= 0.94208

Therefore, the probability that at least three cars finish the race is 0.94208.

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