share 80 in the ratio 1:4:5

According to the formula,  the area of rhombus JKI M is approximately 315.3 square centimeters.

What is area of rhombus formula?

The formula for the area of a rhombus is half the product of its diagonals. That is,

Area of rhombus = (diagonal 1 x diagonal 2)/2

where diagonal 1 and diagonal 2 are the lengths of the two diagonals of the rhombus.

Let D be the intersection of diagonals JK and IM.

Since JK and IM are perpendicular bisectors of each other, D is the midpoint of both diagonals. Let AD = x and BD = y. Then, we have:

[tex]$$\begin{aligned} x + y &= \frac{1}{2} JM = \frac{1}{2}(KL + KM) = \frac{1}{2}(2 \cdot 12.7 + 25.1) = 25.25 \ y - x &= \frac{1}{2} KL = \frac{1}{2} \cdot 12.7 = 6.35 \end{aligned}$$[/tex]

Solving for x and y, we get:

x = [tex]\frac{25.25 - 6.35}{2}[/tex]= 9.95cm

y = [tex]\frac{25.25 + 6.35}{2}[/tex] = 15.8cm

Therefore, the diagonals of rhombus JKI M have lengths 2x = 19.9 cm and 2y = 31.6 cm, respectively. The area of the rhombus is half the product of the diagonals, so we have:

[tex]$$\begin{aligned} A &= \frac{1}{2} \cdot 19.9 \cdot 31.6 \ &= 315.32 , \text{cm}^2 \end{aligned}$$[/tex]

Rounding to the nearest tenth, we get:

[tex]$$A \approx 315.3 , \text{cm}^2$$[/tex]

Therefore, the area of rhombus JKI M is approximately 315.3 square centimeters.

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According to the given information, Mack should make 4 necklaces and 8 wristbands to maximize his profit.

What are linear programming problems?

LPP stands for Linear Programming Problems. Linear programming is a mathematical optimization technique used to maximize or minimize a linear objective function subject to a set of linear constraints. An LPP involves identifying a set of decision variables, an objective function that depends linearly on these variables, and a set of constraints that specify linear relationships between the variables. The objective is to find the values of the decision variables that optimize the objective function while satisfying all the constraints.

To solve this problem, we need to use a system of linear inequalities to represent all the constraints. Let's start by defining the variables:

x = number of necklaces

y = number of wristbands

Now we can write the constraints:

Time constraint: Mack has 360 minutes to make the necklaces and wristbands.

40x + 25y ≤ 360

Production constraint: Mack wants to make no more than 12 items.

x + y ≤ 12

Non-negative constraint: Mack cannot make a negative number of necklaces or wristbands.

x ≥ 0, y ≥ 0

To maximize Mack's profit, we need to define the objective function as the total profit:

P = 3x + 2y

Now we can graph these constraints and find the feasible region:

40x + 25y ≤ 360

x + y ≤ 12

x ≥ 0, y ≥ 0

As shown in attachment 1.

The feasible region is the shaded polygon bounded by the lines: x=0, y=0, x + y ≤ 12, and 40x+25y=360.

To find the optimal solution, we need to evaluate the objective function at each corner point of the feasible region:

Corner point A: (0,0)

P = 3(0) + 2(0) = $0

Corner point B: (0,12)

P = 3(0) + 2(12) = $24

Corner point C: (4,8)

P = 3(4) + 2(8) = $28

Corner point D: (9,0)

P = 3(9) + 2(0) = $27

The maximum profit is $28, which occurs when Mack makes 4necklaces and 8 wristbands. Therefore, Mack should make 4 necklaces and 8 wristbands to maximize his profit.

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For the given inequality x < -7 we draw a straight line at the point -7. The inequality is not strict, hence the line is dotted.

A mathematical statement called an inequality compares two numbers or expressions, typically by employing a symbol. Inequalities show that two quantities are not equal and that one is more than or less than the other.

Number lines can be used to visualise inequality conditions. Depending on whether the endpoint is part of the solution set or not, numbers that meet the inequality are shaded in or denoted by an open or closed circle. A number line would, for instance, show the inequality x > 2 with an open circle at 2 and shading to the right to show all values higher than 2.

In many areas of mathematics as well as in other disciplines like economics and social sciences, inequality is used.

For the given inequality x < -7 we draw a straight line at the point -7.

The inequality is not strict, hence the line is dotted.

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As a result, the simplified expressiοn is (1/4)m(-3)p. by subtracting the apprοpriate expοnents frοm 3 and then dividing it by 12.

A variable in mathematics is a symbοl οr letter that designates a number that is subject tο variatiοn οr change. Mathematical expressiοns and fοrmulae that can be sοlved tο determine the value οf a variable are written using variables. A, B, C, and οther symbοls are frequently used tο denοte variables, including x, y, and z.

Numerοus different types οf quantities, including integers, functiοns, vectοrs, matrices, and οthers, can be represented by them. X and Y are factοrs in the equatiοn y = 2x + 1, fοr instance. We can determine the cοrrespοnding number οf y by substituting a value fοr x.

given

By dividing 3 by 12 and taking away the cοrrespοnding expοnents οf r and p, we can first simplify the numeratοr οf the expressiοn.

[tex](3m^{(-3)}r^{(4)}p^{(2)})/(12r^{(4)}) = (1/4)m^{(-3)}r^{(4-4)}p^{(2)}[/tex]

Even mοre simply put, we have:

[tex](1/4)m^{(-3){p^{(2)}[/tex]

As a result, the simplified expressiοn is (1/4)m(-3)p. by subtracting the apprοpriate expοnents frοm 3 and then dividing it by 12.

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The accurate definition of a type II error is failing to reject a true null hypothesis.

Type II error is known as a statistical term that happens when a null hypothesis is not rejected when it should have been rejected. Type II error can occur in a study when the researcher has failed to detect a real difference between the research subject group and the comparison group. It's often called the "false negative" because it incorrectly concludes that there is no difference when there actually is a difference.

Type I Error - It is known as a type I error when a researcher rejects a null hypothesis when it is true. Type I errors are often called "false positives."

Type II Error - Type II error is known as a statistical term that happens when a null hypothesis is not rejected when it should have been rejected. Type II error can occur in a study when the researcher has failed to detect a real difference between the research subject group and the comparison group. It's often called the "false negative" because it incorrectly concludes that there is no difference when there actually is a difference.

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Answer:

-6.99 , -4 , -6.9 , -2 , -6.999 , -6 , 1

Step-by-step explanation:

Any number to the right of -7 is greater than -7.

Answer:

If a = -1/2 is a root of the quadratic equation 8x² - bx - 3, then we know that when x = -1/2, the equation is equal to 0. We can use this information to solve for b.

Substituting x = -1/2 into the equation, we get:

8(-1/2)² - b(-1/2) - 3 = 0

Simplifying and solving for b, we get:

2 - (b/2) - 3 = 0

b/2 = -1

b = -2

Therefore, b = -2 is the value we are looking for.

To find the other root, we can use the fact that the product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient. In this case, the constant term is -3 and the leading coefficient is 8. Therefore, the product of the roots is:

(-1/2) times the other root = -3/8

Solving for the other root, we get:

(-1/2) times the other root = -3/8

other root = (-3/8) / (-1/2)

other root = (3/8) * 2

other root = 3/4

Therefore, the other root is 3/4.

Finally, to find (1/a - 1/b)², we can substitute a = -1/2 and b = -2 into the expression:

(1/a - 1/b)² = (1/(-1/2) - 1/(-2))²

= (-2 - 1/2)²

= (-5/2)²

= 25/4

Therefore, (1/a - 1/b)² is equal to 25/4.

Answer:

[tex]b=2[/tex]

[tex]\textsf{Other root} = \dfrac{3}{4}[/tex]

[tex]\left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2=\dfrac{25}{4}[/tex]

Step-by-step explanation:

Roots are also called x-intercepts or zeros.  They are the x-values of the points at which the function crosses the x-axis, so the values of x when f(x) = 0.

If x = α is a root of a polynomial f(x), then f(α) = 0.

Therefore, given that a = -1/2 is a root of the quadratic equation 8x² - bx - 3, substitute x = -1/2 into the equation and set it to zero:

[tex]\implies 8\left(-\dfrac{1}{2}\right)^2-b\left(-\dfrac{1}{2}\right)-3=0[/tex]

Solve for b:

[tex]\implies 8\left(\dfrac{1}{4}\right)+\dfrac{1}{2}b-3=0[/tex]

[tex]\implies \dfrac{8}{4}+\dfrac{1}{2}b-3=0[/tex]

[tex]\implies 2+\dfrac{1}{2}b-3=0[/tex]

[tex]\implies \dfrac{1}{2}b-1=0[/tex]

[tex]\implies \dfrac{1}{2}b=1[/tex]

[tex]\implies b=2[/tex]

Therefore, the quadratic equation is:

[tex]\boxed{8x^2-2x-3}[/tex]

The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.  

The constant term of the quadratic equation is -3 and the leading coefficient is 8.  Let the other root be "r". Therefore:

[tex]\implies a \cdot r=\dfrac{-3}{8}[/tex]

Substitute the known value of a = -1/2 and solve for r:

[tex]\implies -\dfrac{1}{2} \cdot r=\dfrac{-3}{8}[/tex]

[tex]\implies r=\dfrac{3}{4}[/tex]

Therefore, the other root of the quadratic equation is 3/4.

To find the value of (1/a - 1/b)²​, substitute the given value of a and the found value of b into the equation and solve:

[tex]\implies \left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2[/tex]

[tex]\implies \left(\dfrac{1}{-\frac{1}{2}}-\dfrac{1}{2}\right)^2[/tex]

[tex]\implies \left(-2-\dfrac{1}{2}\right)^2[/tex]

[tex]\implies \left(-\dfrac{5}{2}\right)^2[/tex]

[tex]\implies \dfrac{25}{4}[/tex]

Therefore , the solution of the given problem of triangle comes out to be   m∠B = 29.5 degrees , m∠C = 132.25 degrees and m∠D = 18.25 degrees.

Because a triangle has two or so more extra parts, it is a polygon. It has a straightforward rectangular shape. Only two of a triangle's three sides—A and B—can differentiate it from a regular triangle. Euclidean geometry produces a single area rather than a cube when boundaries are still not perfectly collinear. Triangles are defined by their three sides and three angles. Angles are formed when a quadrilateral's three sides meet. There are 180 degrees of sides on a triangle.

Here,

Angles B and D are congruent because triangle BCD is isosceles with basis BD. As a result, we can equalise their measurements and find x:

=> m∠B = m∠D

=> (5x + 4) = (x + 15)

=> 4x = 11

=> x = 11/4

Knowing x allows us to determine the size of each angle.

=> m∠B = 5x + 4 = 5(11/4) + 4 = 29.5 degrees

=> m∠D = x + 15 = (11/4) + 15 = 18.25 degrees

Angles B and D being congruent, we can determine what mC is as follows:

=> m∠C = 180 - m∠B - m∠D = 180 - 29.5 - 18.25 = 132.25 degrees

As a result, the triangle's angles are each measured in degrees as follows:

=> m∠B = 29.5 degrees

=>  m∠C = 132.25 degrees

=>  m∠D = 18.25 degrees

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Answer:

A. area of larger rectangle: 18x^2

area of smaller rectangle: 8x^2

B. area of shaded region: 10x^2

Step-by-step explanation:

Larger rectangle: A = lw = (6x)(3x) = 18x^2

Smaller rectangle: A = lw = (4x)(2x) = 8x^2

The shaded region = larger rectangle - smaller rectangle

=> 18x^2 - 8x^2 = 10x^2

Answer:

13.2

Step-by-step explanation:

using Pythagorean theorem, create the equation for the unknown side, x.

x^2+9^2=16^2

subtract 9^2

x^2=16^2-9^2

Use difference of squares.

x^2=(16-9)*(16+9)

Solve

x^2=7*25

x^2=175

Take the square root of both sides

x=sqrt175

x=13.2

Answer: x = 1 ± √ 13 / 6

Step-by-step explanation:

For ax^2 + bx + c= 0, the values of x which are the solutions to the equation are given by: x = − b ± √ b^2 − (4ac) / 2 ⋅ a

Substituting:

3 for a

−1 for b

−1 for c gives: x = − ( −1 ) ± √ ( − 1 )^2 − (4 ⋅ 3 ⋅ − 1) / 2 ⋅ 3

x = 1 ± √ 1 − ( − 12 ) / 6

x = 1 ± √ 1 + 12 / 6

x = 1 ± √ 13 / 6

Hope this helps!

The values of p are -3 or 3 such that the rank of A is 3.[1,1,-1,0][4,4,-3,1][p,2,2,2][9,9,p,3] are the values of A.

Given, A = [[1,1,-1,0],[4,4,-3,1],[p,2,2,2],[9,9,p,3]] To find the value of p, such that the rank of A is 3.

Rank of a matrix is defined as the maximum number of linearly independent row vectors or column vectors. It is denoted by R(A).When a matrix is in echelon form, its rank is equal to the number of pivots.The rank of a matrix is equal to the maximum number of linearly independent rows or columns in the matrix.Here, A is a matrix,The rank of A is 3.Thus, we can say that there will be 3 linearly independent rows or columns in the matrix.

The augmented matrix [A|0] should have 1 pivot for each linearly independent rows.Therefore, [A|0] will have 3 pivots.Then, the last row of [A|0] should be a linear combination of the first three rows.To find the value of p,Let the matrix A is in echelon form [1,1,-1,0][0,0,1,1][0,0,0,0][0,0,0,0]Let's analyze the matrix A for rank 3If we swap R2 and R3,R2 <-> R3 [1,1,-1,0][p,2,2,2][4,4,-3,1][9,9,p,3]Then, the matrix in echelon form is [1,1,-1,0][0,0,1,1][0,0,0,0][0,0,0,0]We can see that the third row is not a linear combination of the first two rows. Therefore, the first three rows of A are linearly independent.Then, A has rank 3 if we can get rid of the fourth row using linear combinations of the first three rows.

9R1 + (-9)R2 + (-p)R3 = 0Thus, 9 - 9p - p² = 0p² + 9p - 9 = 0(p + 3)(p - 3) = 0p = -3 or 3So, the values of p are -3 or 3 such that the rank of A is 3.[1,1,-1,0][4,4,-3,1][p,2,2,2][9,9,p,3] are the values of A.

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Using probability, we can find that the probability of getting a triangle shape block when selected at random is 1/5.

The chance of an event can be calculated using the probability formula by only dividing the favourable number of possibilities by the total number of potential outcomes.

The likelihood of an event occurring can be anything between 0 and 1, as the favourable number of outcomes can never exceed the total number of outcomes. Therefore, the percentage of successful results cannot be zero.

In the question,

Malaya has 5 different shapes of blocks.

Malaya has 10 blocks of each shape.

So, the total no. of blocks = 5 × 10 = 50.

Now, Malaya has 10 blocks of triangular shape.

probability of getting a triangular shape is:

P = 10/50

= 1/5

Therefore, the probability of getting a triangle shape block when selected at random is 1/5.

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The complete question is:

Malaya has 5different shapes of blocks: rectangle, rhombus, square, trapezoid, and triangle. She has 10 of each shape. She selects a block at random. What is the probability of getting a triangle shape block when selected at random?

Answer:

To find the area of a parallelogram, we need to multiply the base by the height. In this case, we need to find the height of the parallelogram.

We can see that segment AL and segment RI are both perpendicular to segment AR. Therefore, we can use either of them to find the height.

Let's use segment AL as our height. To find the length of segment L​S, we can use the Pythagorean theorem:

L​S² = A​R² - R​I²

L​S² = 6.2² - 5.3²

L​S² = 16.21

L​S = √16.21

L​S ≈ 4.02 feet

Now we can find the area of the parallelogram:

Area = base x height

Area = AL x LS

Area = 26.5 x 4.02

Area ≈ 106.23 square feet

Therefore, the area of parallelogram RALS is approximately 106.23 square feet.

Step-by-step explanation:

Answer:

8

Step-by-step explanation:

Not sure this is correct

substitute y in the second equation

X+2(-1/4x)=4

X-2/4x=4

4/4x-2/4x=2/4x

2/4x=4

4 /2/4

4•4/2

16/2

8

The value of the probability that Y is less than or equal to 9 is approximately 0.893

Given that the moment generating function:

M(t) = (pe⁺ + q)ⁿ

And also

q = 1 - p

When M(t) = (0.8e⁺ + 0.2)¹⁰ and M(t) = (pe⁺ + q)ⁿ are compared, we have

n = 10

p = 0.8

q = 0.2

To find P(Y ≤ 9), we can use the cumulative distribution function (CDF) for the binomial distribution:

[tex]F(k) = P(Y \le k) = \sum\limits^k_{i=0}\left[\begin{array}{c}n&i\end{array}\right] p^i q^{n-i}[/tex]

In this case, we want to find P(Y ≤ 9), so we can evaluate the CDF at k=9:

So, we have

[tex]P(Y \le 9) = \sum\limits^9_{i=0}\left[\begin{array}{c}10&i\end{array}\right] 0.8^i * 0.2^{n-i}[/tex]

Using a calculator to evaluate this sigma notation, we find that

P(Y ≤ 9) ≈ 0.89263

Approximate

P(Y ≤ 9) ≈ 0.893

Therefore, the probability that Y is less than or equal to 9 is approximately 0.893

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Complete question

If Y has a binomial distribution with n trials and probability of success p, the moment-generating function for Y is M(t) = (pe⁺ + q)ⁿ, where q = 1 − p.

If Y has moment-generating function M(t) = (0.8e⁺ + 0.2)¹⁰, what is P(Y ≤ 9)?

1. The surface area of the pool is given as follows: 2,202,000 square feet.

2. The amount of water that the pool holds is of: 10,000,000 cubic feet.

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.

The dimensions for this problem are given as follows:

l = 10000, w = 100, h = 10.

Hence the surface area is given as follows:

S = 2 x (10000 x 100 + 10000 x 10 + 100 x 10)

S = 2,202,000 square feet.

The volume of a rectangular prism is given by the multiplication of it's dimensions, hence:

V = 10000 x 100 x 10

V = 10,000,000 cubic feet.

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The volume of a liquid can also be compared to the volume of a solid, as liquids and solids both occupy space.

What is Volume ?

Volume is a measure of the amount of space occupied by an object or substance in three-dimensional space. It is the amount of space that a solid, liquid, or gas occupies.

Liquid volume is the amount of space occupied by a liquid.

The units of liquid volume are typically liters, milliliters, gallons, or fluid ounces.

Liquid volume can be measured using a graduated cylinder or other measuring tools.

The volume of a liquid can be affected by changes in temperature and pressure.

The volume of a liquid can be calculated by multiplying its height, width, and length.

The density of a liquid can also affect its volume, as denser liquids will occupy less space than less dense liquids.

The volume of a liquid can be converted to other units of measurement using conversion factors.

Therefore, The volume of a liquid can also be compared to the volume of a solid, as liquids and solids both occupy space.

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The graph of the piecewise function is shown in the image attached below.

In this problem we need to graph a piecewise function formed by two linear equations, a horizontal line and an oblique line. According to Euclidean geometry, a line can be formed from two distinct points set on Cartesian plane. The procedure is summarized below:

Lastly, the piecewise function is shown in the image attached below.

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10,-2

Step-by-step explanation: 10-2, -9,6, 5-8, 2,-4

Domain: 10,-9, 5, 2,

Range -8, 6,-4, -2

The sunflower's height is increasing at an average rate of 13.045 inches per week over the third week.

The average rate of change of a function over an interval is the slope of the secant line that passes through the two endpoints of the interval. Mathematically, if we have a function f(x) and an interval [a,b], the average rate of change of f(x) over [a,b] is given by:

average rate of change = (f(b) - f(a))/(b - a)

For our problem, the function is f(t) = 5(1.7)ˣ, and we need to find the average rate of change over different time intervals. Let's consider each interval separately:

The average rate of change over the [0,1] interval is:

average rate of change = (f(1) - f(0))/(1 - 0) = (5(1.7)¹ - 5(1.7)⁰)/(1 - 0) = 4.5

Therefore, the sunflower's height is increasing at an average rate of 4.5 inches per week over the first week.

The average rate of change over the [1,2] interval is:

average rate of change = (f(2) - f(1))/(2 - 1) = (5(1.7)² - 5(1.7)¹)/(2 - 1) = 7.65

Therefore, the sunflower's height is increasing at an average rate of 7.65 inches per week over the second week.

The average rate of change over the [2,3] interval is:

average rate of change = (f(3) - f(2))/(3 - 2) = (5(1.7)³ - 5(1.7)²)/(3 - 2) = 13.045

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4x + 10 = 30To solve for x, we can start by subtracting 10 from both sides:4x + 10 - 10 = 30 - 10

4x = 20Then, we can divide both sides by 4 to isolate x:4x/4 = 20/4, x = 5

Therefore, the solution to this equation is x = 5.

4x - 8 = 20, To solve for x, we can start by adding 8 to both sides: 4x - 8 + 8 = 20 + 8, 4x = 28

Then, we can divide both sides by 4 to isolate x:

4x/4 = 28/4, x = 7 Therefore, the solution to this equation is x = 7.

5 + 2x = 65, To solve for x, we can start by subtracting 5 from both sides: 5 + 2x - 5 = 65 - 5, 2x = 60

Then, we can divide both sides by 2 to isolate x:

2x/2 = 60/2, x = 30 Therefore, the solution to this equation is x = 30. 9 + 4x = -5, To solve for x, we can start by subtracting 9 from both sides: 9 + 4x - 9 = -5 - 9, 4x = -14

Then, we can divide both sides by 4 to isolate x:

4x/4 = -14/4, x = -3.5, Therefore, the solution to this equation is x = -3.5. 14 + 6x = 2,To solve for x, we can start by subtracting 14 from both sides:14 + 6x - 14 = 2 - 14, 6x = -12Then, we can divide both sides by 6 to isolate x: 6x/6 = -12/6, x = -2

Therefore, the solution to this equation is x = -2.

2x - 3 = -2

To solve for x, we can start by adding 3 to both sides:

2x - 3 + 3 = -2 + 3

2x = 1

Then, we can divide both sides by 2 to isolate x:

2x/2 = 1/2

x = 1/2 or 0.5

Therefore, the solution to this equation is x = 0.5.

5 + 10x = -5

To solve for x, we can start by subtracting 5 from both sides:

5 + 10x - 5 = -5 - 5

10x = -10

Then, we can divide both sides by 10 to isolate x:

10x/10 = -10/10

x = -1

Therefore, the solution to this equation is x = -1. 10 = 7=x ,This equation is not solvable. It appears to be a typographical error, as it does not make sense to say that 10 is equal to both 7 and x at the same time.

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Answer:

please mark as brainliest

Answer:

[tex]The \: problem \: probably \\ \: assumes \: direct \: variation \: \\ \\

y \: = \: k \: x \\ \\

IF \: so, \: then \: plug \: in \: the \: \\ values \: and solve \: for \: k \\ \\

27 \: = \: k(8) \\ \\

k \: = \: \frac{27}{8}

\\ \\

y \: = \: ( \frac{27}{8} )x. \\ \\ Now \: let \: x \: = \: 11 \\ \\

y \: = \: ( \frac{27}{8} )11 = \\ \\ \frac{27(11)}{8} \\ \\ = 37.125 = \\ \\ \frac{371}{8y} \\ \\

y \: = \frac{371}{8}

[/tex]

The value of sin θ is -[tex]\frac{7\sqrt{130} }{130 }[/tex] and cos θ is [tex]\frac{9\sqrt{130} }{130 }[/tex] . The solution has been obtained by using trigonometry.

What is trigonometry?

The study of right-angled triangles, including their sides, angles, and connections, is referred to as trigonometry.

We are given that tan θ is -7/9. The minus sign is there because it lies in the fourth quadrant.

This means that the perpendicular is 7 and the base is 9.

Let the hypotenuse be x.

Now, by using Pythagoras theorem, we get

⇒ [tex]7^{2}[/tex] + [tex]9^{2}[/tex] = [tex]x^{2}[/tex]

⇒ 49 + 81 = [tex]x^{2}[/tex]

⇒ [tex]x^{2}[/tex] = 130

⇒ x = √130

By trigonometry,

⇒ Sin θ = -[tex]\frac{7}{\sqrt{130} }[/tex]

⇒ Sin θ = -[tex]\frac{7\sqrt{130} }{130 }[/tex]

Similarly,

⇒ Cos θ = [tex]\frac{9}{\sqrt{130} }[/tex]

⇒ Cos θ = [tex]\frac{9\sqrt{130} }{130 }[/tex]

Hence, the values for sin θ and cos θ have been obtained.

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The length of arc of the sector is 52.2 cm and the area of the sector is 260.8 cm²

Arc length is defined as the distance between the two points placed on the circumference of the circle and measured along the circumference. Arc length is the curved distance along the circumference of the circle.

area of an arc = tetha/360 × πr²

l = 299/360 × 3.14 × 10²

l = 93886/360

l = 260.8 cm² ( 1 dp)

The length of arc of the sector

=( tetha)/360 × 2πr

= 299/360 × 2 × 3.14 × 10

= 18777.2/360

= 52.2 cm

therefore the area of the sector is 260.8cm² and the length of the arc is 52.2 cm

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the distance between the points (10, 4) and (1, -8) is 15 units. We round this to the nearest tenth by looking at the first decimal place after the decimal point, which is 5. Since 5 is greater than or equal to 5, we round up the tenths place, giving us a final answer of 15.0 units.

To find the distance between two points in a coordinate plane, we can use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by:

d = √[(x2 - x1)² + (y2 - y1)²]    

Where (x1, y1) and (x2, y2) are the coordinates of the two points, and d is the distance between them.

In this case, the coordinates of the two points are (10, 4) and (1, -8). Substituting these values into the distance formula, we get:

d = √[(1 - 10)² + (-8 - 4)²]

= √[(-9)² + (-12)²]

= √(81 + 144)

= √225

= 15

Therefore, the distance between the points (10, 4) and (1, -8) is 15 units. We round this to the nearest tenth by looking at the first decimal place after the decimal point, which is 5. Since 5 is greater than or equal to 5, we round up the tenths place, giving us a final answer of 15.0 units.

In summary, to find the distance between two points in a coordinate plane, we can use the distance formula. In this case, we found that the distance between the points (10, 4) and (1, -8) is 15 units, rounded to the nearest tenth as 15.0 units.

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a)  The probability that there will be exactly four droughts in the region over the next decade is approximately 0.193.

b) The probability that it will be a severe drought given that exactly one drought actually occurred in 2 years is approximately 0.164.

c)  The probability that all three droughts that actually occurred in 5 years will be moderate is 0.016.

To determine the probability of there being exactly four droughts in the region over the next decade, the expected value of droughts over a decade must first be calculated. λ, the expected number of droughts over a period, can be calculated using the formula:λ = (number of droughts in the last 60 years)/(60 years)λ = (6+16)/(60)λ = 0.367

Therefore, the expected number of droughts in the region over the next decade is 0.367 x 10 = 3.67.Using the Poisson distribution formula, the probability of there being exactly four droughts in the region over the next decade can be calculated as:P(4) = (e^-3.67)(3.67^4)/(4!)P(4) ≈ 0.193

Therefore, the probability that there will be exactly four droughts in the region over the next decade is approximately 0.193.

Assuming that exactly one drought actually occurred in 2 years, the probability that it will be a severe drought can be calculated using Bayes' theorem:P(severe | 1) = P(1 | severe)P(severe) / P(1)First, P(1) must be calculated:P(1) = P(1 | severe)P(severe) + P(1 | moderate)P(moderate)P(1 | severe) = e^-λ(λ^1) / 1! = e^-0.367(0.367^1) / 1! ≈ 0.312P(1 | moderate) = e^-λ(λ^1) / 1! = e^-0.367(0.367^1) / 1! ≈ 0.592P(moderate) = 16 / 60 = 0.267P(severe) = 6 / 60 = 0.1P(1) ≈ 0.312(0.1) + 0.592(0.267) ≈ 0.279Next, P(severe | 1) can be calculated:P(severe | 1) = P(1 | severe)P(severe) / P(1)P(severe | 1) ≈ (0.312)(0.1) / 0.279 ≈ 0.164

Therefore, the probability that it will be a severe drought given that exactly one drought actually occurred in 2 years is approximately 0.164.

Assuming that exactly three droughts actually occurred in 5 years, the probability that all will be moderate droughts can be calculated using the binomial distribution formula:P(3 moderate) = (n choose k)(p^k)(1-p)^(n-k)where n = 3, k = 3, and p = 16 / 60 = 0.267(n choose k) = (n! / k!(n-k)!) = (3! / 3!(3-3)!) = 1P(3 moderate) = (1)(0.267^3)(1-0.267)^(3-3) = 0.016

Therefore, the probability that all three droughts that actually occurred in 5 years will be moderate is 0.016.

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