10. Given that xy = a², show that + a+x a+y​

The answer of the given question based on to determine that the quadrilateral must be a parallelogram the answer is given below.

A parallelogram is  type of quadrilateral ( polygon with four sides) with two pairs of parallel sides. This means that the opposite sides of parallelogram are parallel and congruent.

Yes, we can determine that the given quadrilateral is a parallelogram based on the information given.

We are told that opposite sides of the quadrilateral are congruent (that is, AB is congruent to CD, and BC is congruent to DA), which is a property of parallelograms. We are also told that opposite angles of the quadrilateral are congruent (that is, angle A is congruent to angle C, and angle B is congruent to angle D), which is another property of parallelograms.

Therefore, based on these two pieces of information, we can conclude that the quadrilateral must be a parallelogram. This is because these properties are unique to parallelograms and do not apply to other types of quadrilaterals, such as rectangles, rhombuses, or trapezoids.

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So, by resolving the given, we obtain the : As a result, subtracting As a percentage result, subtracting $332.03 from $1,000 results in a loss of 66.797%.$332.03 from $1,000 results in a loss of 66.797%.

A ratio or value stated as a percentage of 100 is referred to as a percentage in mathematics. The abbreviations "pct.," "pct," and "pc" are also sporadically used. Nonetheless, the percentage symbol "%" is frequently used to denote it. The percentage amount is dimensionally empty. When the denominator is 100, percentages are essentially fractions. To show that a number is a percentage, a percent symbol (%) should be used next to it. For instance, you receive a 75% grade if you correctly answer 75 out of 25 questions (75/100) on a test. For percentage calculations, multiply the outcome by 100 after dividing the amount by the total. The percentage is calculated by multiplying (value/total) by 100%.  

You may use the following calculation to get the percentage change between $1,000 and $332.03:

(|new value - old value| / old value|) x 100% represents the percentage change.

In this instance, the new value is $332.03 while the previous value was $1,000.

percentage change = (|332.03 - 1000| / 1000| x 100% = f 66.797%.

As a result, subtracting $332.03 from $1,000 results in a loss of 66.797%.

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Total cost = 20x + 200

Predicted revenue = 100 + 30x

profit = revenue – cost

= (100 + 30x) - (20x + 200)

= 100 + 30x - 20x - 200

Profit = 10x - 100

Answer:

The guitar Francisca wants is on sale at two different stores. The original price of the guitar at both stores is $160. At which store is the guitar less expensive? How much less expensive?

The value of the expression 6 3/4 - 2 3/20 is -91/12.

An improper fraction is a fraction that cannot be further simplified and has a numerator higher than or equal to the denominator. For instance, 13/5 is a bad fraction. Let's find the mixed fraction equivalent of this incorrect fraction.

Divide the denominator by the numerator in step 1. Step 2: Track down the rest. In step 3, arrange the numbers as follows: quotient, followed by a portion of the remainder or divisor.

The given expression are:

6 3/4 - 2 3/20

Convert the mixed fraction into improper fraction:

27/4 - 43/3

Take the LCM of the two:

81/12 - 172/12

-91/12

Hence, the value of the expression 6 3/4 - 2 3/20 is -91/12.

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The speed at which the water level is rising when the water is 20 cm deep is 0.04285 m/min

Using differentiation:

dV/dt = 0.3

find dh/dt

the width of the water level is dependent on the height

at height 0, width of 20

height 60, width of 80

when height increases by 1, width increases by 1

dh/dt = dw/dt

V = 0.5*(0.2 + (0.2+h))*h*10

V = 5*(0.4 + h)*h

V = 5h^2 + 2h

dV/dt = 5*2h(dh/dt) + 2(dh/dt)

when h = 0.5

0.1 = 5*2*0.5*(dh/dt) + 2(dh/dt)

0.1 = 5(dh/dt) + 2(dh/dt)

7(dh/dt) = 0.3

dh/dt = 0.04285 m/min

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We should be cautious about making predictions that are far outside the range of the data used to create the line of best fit.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

Based on the line of best fit shown in the scatterplot, we can see that there is a positive linear relationship between the high temperature and the number of hot cocoas sold.

The equation of the line of best fit appears to be y = 23.25x - 705.5, where y represents the number of hot cocoas sold and x represents the high temperature.

To make a prediction for a high temperature of 54, we substitute x = 54 into the equation and solve for y:

y = 23.25x - 705.5

y = 23.25(54) - 705.5

y = 694.5 - 705.5

y = -11

According to this prediction, if the high temperature is 54, Josue would sell approximately -11 hot cocoas. However, this result doesn't make sense, as we can't sell a negative number of hot cocoas.

It's possible that the line of best fit is not an accurate model for the data at very high or very low temperatures, or that there are other factors besides temperature that affect hot cocoa sales.

Therefore, we should be cautious about making predictions that are far outside the range of the data used to create the line of best fit.

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To solve the system of equations, we can substitute the expression for y in the first equation with the expression for y in the second equation:

   x^2 + 4 = 2x + 1

Now we can rearrange the equation by bringing all the terms to one side:

   x^2 - 2x + 3 = 0

We can solve this quadratic equation by using the quadratic formula:

   x = [-(-2) ± sqrt((-2)^2 - 4(-1)(3))] / 2(-1)

   x = [2 ± sqrt(16)] / -2

   x = -1 or x = 3

Now we can find the corresponding values of y for each solution:

   When x = -1, y = 2(-1) + 1 = -1

   When x = 3, y = -3^2 + 4 = -5

Therefore, the solution to the system of equations is (-1, -1) and (3, -5).

To check, we can substitute each solution into the original equations:

   When x = -1 and y = -1:

       y = -x^2 + 4 becomes -1 = -(-1)^2 + 4, which is true

       y = 2x + 1 becomes -1 = 2(-1) + 1, which is true

   When x = 3 and y = -5:

       y = -x^2 + 4 becomes -5 = -(3)^2 + 4, which is true

       y = 2x + 1 becomes -5 = 2(3) + 1, which is true

Therefore, the solutions are valid.

The inverse of a function can be found by switching the roles of x and y and solving for y:

   For f(x) = -x^2 + 4, we have:

       x = -y^2 + 4

       y^2 = 4 - x

       y = ± sqrt(4 - x)

       So, the inverse of f is f^-1(x) = ± sqrt(4 - x)

   For g(x) = x^2 - 6, we have:

       x = y^2 - 6

       y^2 = x + 6

       y = ± sqrt(x + 6)

       So, the inverse of g is g^-1(x) = ± sqrt(x + 6)

The domain and range of the original functions and their inverses can follow a pattern, depending on the specific functions. However, in general, the domain and range of a function and its inverse are switched. For example, if the domain of f(x) is all real numbers, then the range of f^-1(x) will be all real numbers. Similarly, if the range of f(x) is all positive numbers, then the domain of f^-1(x) will be all positive numbers.

The measurement of angle DAO in the given rhombus  is 40°

A rhombus is a quadrilateral with both pairs of opposite sides parallel and all sides the same length, i.e., an equilateral parallelogram. The word rhombus is sometimes used instead of rhombus.

Given that, ABCD is a rhombus and ADC = 50°. we need to find DAO.

Please refer to the figure attached

We know that, the diagonals of the rhombus bisects at right angle,

Therefore,

AOD is a right-angled triangle.

We know that, the sum of the acute angles of a right-angled triangle is 90°

Therefore,

∠ ADC + ∠ DAO = 90°

∠ DAO = 90° - 50°

∠ DAO = 40°

Hence, the measurement of angle DAO in the given rhombus  is 40°

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In   linear equation, you can get 4 cookies for $1.00 at the Math munch cafe.

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                             Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

If you can get 6 cookies for $1.50, then the cost of each cookie is:

$1.50 ÷ 6 = $0.25

This means that for $1.00, you can get:

$1.00 ÷ $0.25 per cookie = 4 cookies

Therefore, you can get 4 cookies for $1.00 at the Math munch cafe.

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The radius of the frame is approximately 1.27 inches

First, we can start by using the formula for the circumference of a circle:

C = 2πr

Where

We know that 75% of the circumference is 6 inches, so we can set up the following equation:

0.75C = 6

Simplifying this equation by dividing both sides by 0.75, we get:

C = 8

Now we can use this value of the circumference and the formula for the circumference to solve for the radius:

C = 2πr

8 = 2πr

Dividing both sides by 2π, we get:

r = 8/(2π)

Using a calculator, we can evaluate this expression to get:

r ≈ 1.27

Therefore, the radius of the frame is approximately 1.27 inches.

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When polygon ABCD is dilated by a factor of 2. Then,

Point A′ is at  (-2, -2) and point D′ is at (4, -2).

A reference point needed to scale a dilation of such a figure correctly is the centre of dilatation.

A fixed point there in plane around whereby all points are enlarged or contracted is the centre of a dilation. The centre, which may be found within, outside, or on a figure, is the only unchanging.

Thus,  A' will occur at :(2 × (-1), 2 × (-1)) = (-2, -2)

D' will occur at (2 × 2, 2 × (-1)) = (4, -2).

Thus, when polygon ABCD is dilated by a factor of 2. Then,

Point A′ is at  (-2, -2) and point D′ is at (4, -2).

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The complete question is-

In the figure, polygon ABCD is dilated by a factor of 2 to produce A′B′C′D′ with the origin as the center of dilation.

Point A′ is at

(2 , 1)

(-2, -2)

(1 , -2)

( -2 , 2)

and point D′ is at

(-2 , 2)

(4 , -2)

(4 , 2)

(-2 , 2)

The calculation shows that the yearly salary of a supervisor and two drivers is R114,000.

By summing the individual wages of each employee, it is possible to get the combined monthly salary of all the new employees:

Overall monthly compensation [tex]\mathrm{= (2 \times R2,000) + (3\times R4,000) +(1\times (R2,000 +R2,500)) +(2\times R2,500) }[/tex]

[tex]\mathrm{= R4000, R12000, R4,500, and\ R5,000, or \ R25500.}[/tex]

As a result, R25 500 is the total monthly wage for all of the additional employees.

4.1.2

Let x represent the messenger's monthly wage.

Hence the supervisor's monthly pay is x + R2 500, and the driver's monthly pay is R2 500 as well.

The following formula can be used to determine the supervisor's and the two drivers' annual salaries:

Supervisor's annual compensation is equal to (x + R2 500) x 12.

Two drivers' annual salaries come out to (R2 500 x 2) x 12 = R60 000.

Overall annual pay for the supervisor and two drivers is

[tex]\mathrm{= x + R2,500) \times 12 + R60,000}[/tex]

[tex]\mathrm{= 12x + R30,000 + R60,000 \ and \ 12x + R90,000}[/tex]

Assuming that the supervisor and two drivers each earn R114 000 annually, we may construct the following equation:

12x + R90,000 = R114,000

Calculating x:

12x = R24,000 = R2,000

As a result, the messenger's monthly wage is R2000.

Returning this amount to the equation in place of the supervisor's and the two drivers' yearly salaries:

12x + R90 000 = 12(R2 000) + R90 000 = R114 000

As a result, the computation indicates that a supervisor and two drivers each earn R114 000 year.

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Answer:

We can use the formula for the future value of an annuity to solve this problem. An annuity is a series of equal payments made at regular intervals. In this case, we are making a single lump-sum deposit now to grow into the desired future value.

The formula for the future value of a lump-sum deposit is:

FV = PV x (1 + r/n)^(n*t)

Where:

FV = future value

PV = present value (the amount to be deposited now)

r = annual interest rate (as a decimal)

n = number of times the interest is compounded per year

t = number of years

In this case, we want to find the present value (PV) required to grow into the future value (FV) of $800,000 in 25 years with an annual interest rate of 5% compounded monthly, so:

FV = $800,000

r = 5% = 0.05 (annual interest rate)

n = 12 (monthly compounding)

t = 25 (number of years)

Substituting these values into the formula, we get:

$800,000 = PV x (1 + 0.05/12)^(12*25)

Simplifying the exponent on the right-hand side, we get:

$800,000 = PV x (1.004167)^300

Dividing both sides by (1.004167)^300, we get:

PV = $800,000 / (1.004167)^300

Using a calculator or spreadsheet, we can evaluate this expression to find:

PV ≈ $267,227.92

Therefore, you would need to deposit approximately $267,227.92 now into the account to reach your retirement goal of $800,000 in 25 years with an annual interest rate of 5% compounded monthly.

So, the codomain of the function is the set of all real numbers: codomain = R.

In mathematics, a function is a rule that maps a set of inputs (domain) to a set of outputs (codomain) in such a way that each input is associated with exactly one output. A function can be represented using various notations, including equations, graphs, and tables. A function can be thought of as a machine that takes an input and produces an output. The input is usually represented by the variable x, and the output is represented by the variable y. The relationship between the input and output is defined by the function rule. Functions are used in many areas of mathematics, science, and engineering to describe various phenomena, such as motion, growth, and decay. They are also used to model and analyze data in statistics and economics, and to design and control systems in engineering and computer science.

Here,

The codomain is the set of all possible output values of a function.

For the given function, f(x) = x² - 4, we can see that the output values are obtained by squaring the input value, subtracting 4, and therefore can be any real number.

To find the codomain of the function f(x) = x² - 4, we need to determine the range of the function, which is the set of all possible output values.

Let y = f(x) = x² - 4.

To find the range, we need to solve for y:

y = x² - 4

y + 4 = x²

Taking the square root of both sides, we get:

x = ±√(y + 4)

Therefore, the range of the function is the set of all real numbers greater than or equal to -4, since y + 4 must be non-negative for real values of y:

range = {y | y + 4 ≥ 0} = {y | y ≥ -4}

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The probability both will be peanut butter sandwiches will be 1/6.

Probability is an area of mathematics that deals with numerical representations of how probable an event is to occur or how likely a statement is to be true.

here, we have,

The probability of an occurrence is a number between 0 and 1, where 0 denotes the event's impossibility and 1 represents certainty.

The probability that John picks a peanut butter sandwich from the first bag is 6/12 = 1/2.

The probability that John picks a peanut butter sandwich from the second bag is 4/12 = 1/3.

The probability that John picks a peanut butter sandwich from both bags is (1/2) * (1/3) = 1/6.

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The slope of the line passing through (- 1, 3) and (1, - 2) is - 5/2.

The slope is the rate of change of the y-axis with respect to the x-axis.

The equation of a line in slope-intercept form is y = mx + b, where

slope = m and y-intercept = b.

We know the greater the absolute value of a slope is the steeper its graph or the rate of change is large.

From the given line we can identify two coordinate points,

(- 1, 3) and (1, - 2).

Now, We know slope(m) = (y₂ - y₁)/(x₂ - x₁).

slope(m) = (- 2 - 3)/(1 + 1).

slope(m) = - 5/2.

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The growth percentage for the given function is 92.2%.

What is the percentage?

A number can be represented as a fraction of 100 using a percentage. "%" is used to represent it.

For instance, when we state that 20% of a class is made up of guys, we mean that 20 of the class's 100 pupils are male. In other words, there are 20 out of 100 boys in the class, or 0.2.

We can multiply by 100 to change a fraction or decimal into a percentage. For instance, we can multiply the fraction 3/5 by 100 to achieve the following percentage conversion:

3/5 × 100% = 60%

Now,

The given function is [tex]y = 990^{(1.922)}[/tex]

We can compare this to the standard form of an exponential function y = abˣ, where "a" represents the initial value or starting point, "b" represents the growth or decay factor, and "x" represents the time or number of periods.

In this case, we have:

a = 990 (the initial value)

b = 1.922 (the growth factor)

To find the growth percentage, we can subtract the initial value from the final value and divide by the initial value:

(1.922 - 1) / 1 × 100% = 92.2%

Therefore, the growth percentage for the given function is 92.2%.

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You therefore paid $21.26 to travel to the beach.

One mile is 5,280 feet long. To determine how many steps this should require walking a mile, simply split 5,280 feet by the length of your typical stride. For instance, if your typical stride is 2 feet, walking a mile will need 2,640 steps.

a) We must divide the total amount of miles travelled by the quantity of gas consumed in order to compute mpg (miles per gallon). If 141 miles are covered by 3/8 of a tank, we may calculate the tank's total capacity as follows:

18 gallons = 8/8 tank (full tank)

So, 3/8 tank = (3/8) x 18 = 6.75 gallons

Therefore, the mpg is:

mpg = distance traveled ÷ gas used

mpg = 141 miles ÷ 6.75 gallons

mpg = 20.89 miles per gallon (rounded to two decimal places)

b) The entire amount of gas consumed must be multiplied by the gallon price in order to get the cost of gas. Gas is expensive at:

Cost of gas = gas used x price per gallon

Cost of gas = 6.75 gallons x $3.15/gallon

Cost of gas = $21.26

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It will take apprοximately 23.29 years fοr the accοunt tο reach $15,000, assuming the given initial investment and interest rate.

A lοgarithm is an inverse οperatiοn tο expοnentiatiοn. It is a mathematical functiοn that helps tο sοlve equatiοns where the variable is in the expοnent. The lοgarithm οf a number tο a certain base is the expοnent tο which the base must be raised tο get that number.

Accοrding tο given infοrmatiοn :

In the given expressiοn, 1,000 • e0.034t, the variable "t" represents the time in years, and "e" is a mathematical cοnstant called Euler's number (apprοximately 2.71828).

The entire expressiοn represents the amοunt οf mοney in an accοunt after "t" years, assuming an initial investment οf $1,000 and a cοntinuοus interest rate οf 3.4% per year (0.034 in decimal fοrm).

Tο find the number οf years it will take fοr the accοunt tο reach $15,000, we can set the expressiοn equal tο 15,000 and sοlve fοr "t".

1,000 • e0.034t = 15,000

Dividing bοth sides by 1,000 gives:

e0.034t = 15

Tο sοlve fοr "t", we can take the natural lοgarithm (ln) οf bοth sides:

ln(e0.034t) = ln(15)

0.034t • ln(e) = ln(15)

Since ln(e) = 1, we can simplify tο:

0.034t = ln(15)

Dividing bοth sides by 0.034 gives:

t = ln(15) / 0.034

Using a calculatοr, we get:

t ≈ 23.29

Therefοre, it will take apprοximately 23.29 years fοr the accοunt tο reach $15,000, assuming the given initial investment and interest rate.

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For the same quantity of blue, the two highlighted rows demonstrate this. Purple 1 then utilizes more red than Purple 2.

Purple 1 is thus a redder hue of purple than Purple 2.

Purple 2 is a more blue-based purple than Purple 1.

The utilization of two or more additional numbers that compares is known as the ratio.

The ratio of red to blue in purple 1 and purple 2 will be 1:2 and 1:3, respectively. The table is given below.

Purple #1              Purple #2

Red: Blue             Red: Blue

  1      2                    1      3

 2      4                   2      6

 3      6                   3      9

For the same quantity of blue, the two highlighted rows demonstrate this. Purple 1 then utilizes more red than Purple 2.

Purple 1 is thus a redder hue of purple than Purple 2.

Purple 2 is a more blue-based purple than Purple 1.

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Using simple interest, The interest will be earned will be $144391.48.

Using the simple interest technique, which always adds interest at the same rate to the initial principal amount, is a quick and easy way to calculate interest on money. We will receive interest on our investment from any bank where we deposit our money. Simple interest is one of the various interest rates that banks impose. Now, before delving more into the concept of fundamental curiosity,

you want to have $300,000 for retirement in 30 years. Your account earns 4% interest.

Monthly interest rate, r = 4%/12 = 0.003333333333

n = 30 * 12 = 360

We need to deposit $432.2458864292 each month

b) Interest amount = 300,000 - 432.2458864292 * 360

Hence the interest will be earned will be $144391.48.

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The circle in the middle represents the number of households with an Aggie or a Longhorn fan and it is 90.

Household refers to a single dwelling unit occupied by a family or other group of people who share common living arrangements. This includes a home, apartment, condominium, or other dwelling. Households can be single-family, multi-family, or even a group of unrelated people. They are typically headed by one or more adults, and may include children, elderly family members, and other dependents.

The total number of households surveyed is 1000 and is represented by the circle outside the two others. The circle on the left represents the number of households with an Aggie fan and it is 670. The circle on the right represents the number of households with a Longhorn fan and it is 450. The circle in the middle represents the number of households with an Aggie or a Longhorn fan and it is 90.

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The interest rate is 0.04 or 4% when the time is equal to 2.5 years, principal amount is $3000 and interest is $300.

Simple interest is a type of interest that is calculated only on the initial principal amount of a loan or investment. It does not take into account any interest earned on the interest that has accumulated over time, as is the case with compound interest.

To find the value of r, the formula for simple interest is I = P * r * t

where:

I = Interest

P = Principal

r = Interest rate

t = Time

By substituting the given values in equation, we get:

$300 =  $3000 * r * 2.5

r = $300/($3000*2.5)

r = 0.04

Therefore, the interest rate is 0.04 or 4%.

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The distance between the two other landmarks on the same map is approximately 23 inches.

What is a scale factor?

A scale factor is a measure of the proportional relationship between two similar figures. It is the ratio of any two corresponding lengths in the two figures.

We can set up a proportion to solve this problem:

Distance on map / Actual distance = Scale factor

Let x be the distance between the two other landmarks on the map. We can set up the following proportion:

2 feet 5 inches / 65 miles = x / 52 miles

To solve for x, we first convert 2 feet 5 inches to inches:

2 feet = 24 inches

5 inches = 5 inches

2 feet 5 inches = 24 inches + 5 inches = 29 inches

Substituting the values into the proportion, we get:

29 inches / 65 miles = x / 52 miles

To solve for x, we cross-multiply:

29 inches × 52 miles = 65 miles × x

Simplifying, we get:

x = 29 inches × 52 miles / 65 miles

x = 23.2 inches

Rounding to the nearest inch, we get x ≈ 23 inches.

Therefore, the distance between the two other landmarks on the same map is approximately 23 inches.

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This is happening because the standard deviation of the scores is 12.

Standard deviation is a measure of how spread out a set of data is. It is a measure of how far each data point is from the mean of the data set. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. Standard deviation is an important concept in statistics and is used to measure the variability of a data set. It is used to assess the risk associated with investments, as well as to determine the accuracy of an estimate.

(a) Approximately 68% of the students scored between blank and blank.
39 and 63
(b) Approximately 95% of the students scored between blank and blank.
27 and 75

This is happening because the standard deviation of the scores is 12. When the scores are normally distributed, roughly 68% of them will fall within 1 standard deviation of the mean, which is between 39 and 63. Similarly, 95% of the scores will fall within 2 standard deviations of the mean, which is between 27 and 75.

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There is approximately a 36.11% probability of rolling an odd sum or doubles when a pair of dice is rolled.

Probability is defined as the possibility of an event being equal to the ratio of the number of favorable outcomes and the total number of outcomes.

There are 3 odd numbers among the possible outcomes of each die (1, 3, 5), so there are 3 x 3 = 9 ways to obtain an odd sum:

(1,2), (1,4), (1,6), .... (6,1), (6,3), (6,5)

There are 6 possible outcomes for rolling the same number on both dice, so there are 6 ways to obtain doubles:

(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

We counted (1,1), (3,3), and (5,5) twice in the above two categories, so we need to subtract 2 from the total count.

Therefore, there are 9 + 6 - 2 = 13 ways to obtain an odd sum or doubles.

Therefore, the probability of rolling an odd sum or doubles is:

P(odd sum or doubles) = 13/36

Thus, there is about a 36.11% probability of rolling an odd sum or doubles when a pair of dice is rolled.

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a) The altitude of the plane is approximately 8.4 kilometers.

b) The pilot has approximately 11.2 minutes before he has to land.

What is the Pythagorean theorem?

The Pythagorean theorem is a fundamental theorem in mathematics that relates to the sides of a right triangle. It states that the square of the length of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the lengths of the other two sides (the legs).

In mathematical terms, if we have a right triangle with legs a and b and hypotenuse c, the Pythagorean theorem can be written as:

[tex]a^2 + b^2 = c^2[/tex]

This theorem is named after the ancient Greek mathematician Pythagoras, who was the first to prove it. The Pythagorean theorem has numerous applications in geometry, trigonometry, physics, and engineering, among other fields.

The Pythagorean theorem can also be used to determine whether a triangle is a right triangle or not. If the lengths of the three sides of a triangle satisfy the equation [tex]a^2 + b^2 = c^2,[/tex] then the triangle is a right triangle with sides a and b and hypotenuse c.

a) In the given image, we can see that the slant range, ground range, and altitude form a right triangle. We can use the Pythagorean theorem to find the altitude of the plane.

Pythagorean theorem: In a right triangle, the square of the hypotenuse (slant range in this case) is equal to the sum of the squares of the other two sides (ground range and altitude in this case).

Therefore, we can write:

[tex]Slant range^2 = Ground range^2 + Altitude^2[/tex]

Substituting the given values, we get:

[tex]14.5^2 = 10.15^2 + Altitude^2[/tex]

Simplifying and solving for altitude, we get:

Altitude ≈ 8.4 km (rounded to the nearest meter)

Therefore, the altitude of the plane is approximately 8.4 kilometers.

b) If the plane descends suddenly at a rate of 750 meters per minute, we can use the altitude of the plane (calculated in part a)) to find the time the pilot has before landing.

Time = Altitude/Descent rate

Substituting the values, we get:

Time = [tex]\frac{8.4 }{0.75}[/tex] ≈ 11.2 minutes

Therefore, the pilot has approximately 11.2 minutes before he has to land.

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