Which of the ten basic functions in
our toolkit have all real numbers for
their range?

Step-by-step explanation:

it was not clear if an average change rate would be sufficient, or if you needed an immediate change rate (as I also don't know if you covered derivatives already or not).

so, it would be helpful, if you could put a message to an answer that was not giving you what you need.

so, here now an answer for an immediate change rate (hopefully that is what you need) :

we have a right-angled triangle.

the direct line of sight (the direct distance between police and red car) is the Hypotenuse (the baseline opposite of the 90° angle).

the 50 ft and 180 ft are the legs.

Pythagoras gives us the length of the Hypotenuse :

Hypotenuse² = 50² + 180² = 2500 + 32400 = 34,900

Hypotenuse = sqrt(34900) = 186.8154169... ft

in general terms let's say x is the distance of the cop to the road, y is the distance on the road to the crossing point with the distance cop to road, and z is the line of sight distance between the red car and the cop (the Hypotenuse).

x² + y² = z²

now, the first derivative of distance is the change of distance = speed.

then dy/dt (= y') is how fast the car is traveling down the road. dx/dt (= x') is how fast the cop is traveling toward the road. and dz/dt (= z') is how fast the distance between the cop and the car is changing.

now, we take the derivative of our equation

x² + y² = z² with respect to time, variable by variable :

d(x² + y² = z²)/dt =

dx²/dx × dx/dt + dy²/dy × dy/dt = dz²/dz × dz/dt

that gives us the equation

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

x(dx/dt) + y(dy/dt) = z(dz/dt)

from the problem we know x (50 ft), y (180 ft), dz/dt (85 ft/s). we calculated z (the Hypotenuse = sqrt(34900), and since the cop is not moving, we know dx/dt = 0.

and we get

50ft×0ft/s + 180ft×(y') = sqrt(34900)ft×(85)ft/s

we solve for y' (the speed of the car on the road)

y' = sqrt(34900)×85/180 = 88.21839132... ft/s

≈ 88.22 ft/s

and now here the difference for an average change rate over the unrevealed of 1 second :

the radar measured the change of the distance (Hypotenuse) from 1 second ago to now.

so, 1 second ago, the distance was

186.8154169... + 85 = 271.8154169... ft

the 50 ft leg stays the same, but the 180 ft leg was (again via Pythagoras)

271.8154169...² = 50² + leg²

leg² = 271.8154169...² - 50² = 71,383.62088...

leg = 267.1771339... ft

so, the red car traveled

267.1771339... - 180 = 87.1771339... ft/s

as you can see, it is close, but there has to be a difference, as the average change rate is only an approximation to the immediate change rate.

The function f(n) = n - 3 maps distinct integers to distinct integers, and thus is injective.

(a) The function f(n) = n - 3 is one-to-one (injective). To prove this, suppose that f(a) = f(b) for some integers a and b. Then, we have a - 3 = b - 3, which implies a = b. Therefore, the function f(n) = n - 3 maps distinct integers to distinct integers, and thus is injective.

(b) The function f(n) = n^2 - 1 is not one-to-one (not injective). To see this, note that f(1) = f(-1) = 0, so different inputs map to the same output. In general, for any positive integer k, we have f(k) = f(-k), since (k^2 - 1) = ((-k)^2 - 1). Therefore, the function f(n) = n^2 - 1 is not injective.

(c) The function f(n) = n^5 is one-to-one (injective). To prove this, suppose that f(a) = f(b) for some integers a and b. Then, we have a^5 = b^5, which implies a = b (since the fifth root of a non-zero real number is unique). Therefore, the function f(n) = n^5 maps distinct integers to distinct integers, and thus is injective.

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The area of the right triangle given in this problem is given as follows:

21 units squared -> Option A.

To calculate the area of a triangle, you can use the formula presented as follows:

Area = (1/2) x base x height

In which the parameters are given as follows:

For a right triangle, we can consider one side to be the base and the other side to be the height, hence the parameters are given as follows:

Hence the area of the triangle is given as follows:

A = 0.5 x 7 x 6 = 21 units squared.

The complete problem is defined as follows:

"Calculate the area of the given triangle".

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Step-by-step explanation:

Might be a little easier to visualize if yo re-arrange it into y = mx + b form:

4x+ 5y = 20

5y = -4x + 20

y = - 4/5 x + 4         y-axis intercept at y = 4

                                x axis intercept is found by:

                                     0 = -4/5 x + 4

                                             - 4 = -4/5 x

                                                 x = 5

So===> plot the two intercept points ( 0,4)  and  ( 5,0)  and connect the dots

Answer:

Step-by-step explanation:

To find the total length of both pieces of fabric in centimeters, we need to add the length of the first piece of fabric (142 cm) and the length of the second piece of fabric (2 meters).

However, we need to make sure that the units are consistent before we add the lengths. We can convert the length of the second piece of fabric from meters to centimeters by multiplying by 100. Therefore, the total length in centimeters is:

142 cm + 2 meters * 100 cm/meter = 142 cm + 200 cm = 342 cm

The option that correctly gives the answer is "Multiply 2 by 100, then add 142" (Option C).

Answer:

y = 0.8x

Step-by-step explanation:

machine fills 24 jars in 30 seconds , then

fills 1 jar in [tex]\frac{24}{30}[/tex] = 0.8 seconds

thus number of jars filled in x seconds is

y = 0.8x

The median is the middle value of a data set when the values are arranged in order from lowest to highest (or highest to lowest). If there is an even number of values, then the median is the average of the two middle values. The median divides the data set into two halves, with half of the values being below the median and half of the values being above the median.

The first quartile, denoted as Q1, is the value that separates the lowest 25% of the data from the rest of the data. The second quartile, denoted as Q2, is the median of the data set. The third quartile, denoted as Q3, is the value that separates the lowest 75% of the data from the rest of the data.

The data-set has in this problem has two-halves of five elements, divided by the number 21, hence the median and the quartiles are given as follows:

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Answer:

m = 1

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (-1,0) (0,1)

We see the y increase by 1 and the x increase by 1, so the slope is

m = 1

The true statements are:

The graph of the function is a parabola.

The graph contains the point (0, 0).

The graph does not contain the point (20, -8).

The quadratic function is f(x) = x^2 - 5x + 12. Here are the statements that are true:

The value of f(-10) = 82:

To find f(-10), we substitute -10 for x in the function:

[tex]$$f(-10) = (-10)^2 - 5(-10) + 12 = 100 + 50 + 12 = 162$$[/tex]

Therefore, the statement "The value of f(-10) = 82" is false.

The graph of the function is a parabola:

Since the highest power of x in the function is 2, the graph of the function will be a parabola. Therefore, the statement "The graph of the function is a parabola" is true.

The graph of the function opens down:

The coefficient of [tex]$x^2$[/tex] in the function is positive (+1), which means the parabola opens upwards. Therefore, the statement "The graph of the function opens down" is false.

The graph contains the point (20, –8):

To see whether the point (20, -8) is on the graph of the function, we substitute x=20 into the function:

[tex]$$f(20) = (20)^2 - 5(20) + 12 = 400 - 100 + 12 = 312$$[/tex]

Since the y-coordinate of the point (20, -8) is not equal to 312, the statement "The graph contains the point (20, –8)" is false.

The graph contains the point (0, 0):

To see whether the point (0, 0) is on the graph of the function, we substitute x=0 into the function:

[tex]$$f(0) = (0)^2 - 5(0) + 12 = 12$$[/tex]

Since the y-coordinate of the point (0, 0) is equal to 12, the statement "The graph contains the point (0, 0)" is true.

Therefore, the true statements are:

The graph of the function is a parabola.

The graph contains the point (0, 0).

The graph does not contain the point (20, -8).

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(a) θ is a biased estimator for θ.

(b) (α+1)Y/α is an unbiased estimator of θ.

(c) MSE(θ) = αθ^2/[(α+1)^2(α+2)]

(a) To show that θ is a biased estimator for θ, we need to show that E(θ) ≠ θ.

Using the definition of the maximum function, we have

P(θ ≤ y) = P(Y1 ≤ y, Y2 ≤ y, ..., Yn ≤ y) = (F(y))^n

where F(y) is the cumulative distribution function of Y.

Differentiating both sides with respect to y, we get:

f(θ) = n(F(θ))^(n-1)f(θ)

Simplifying, we get

F(θ) = (1/n)^(1/(n-1))

Using this result, we can find the expected value of θ

E(θ) = ∫₀^∞ θf(θ)dθ = ∫₀^θ θαθ^α-1dθ = αθ/(α+1)

Thus, E(θ) ≠ θ, which means that θ is a biased estimator for θ.

(b) To find a multiple of θ that is an unbiased estimator of θ, we can use the method of moments.

We know that the population mean of Y is

μ = ∫₀^θ yf(y)dy = αθ/(α+1)

The sample mean is

Y = (Y1+Y2+...+Yn)/n

Equating these two expressions and solving for θ, we get

θ = (α+1)Y/α

Thus, (α+1)Y/α is an unbiased estimator of θ.

(c) The mean squared error (MSE) of θ can be written as

MSE(θ) = E[(θ - θ)^2]

Expanding the square and using the linearity of expectation, we have

MSE(θ) = E[θ^2] - 2θE[θ] + E[θ]^2

We already know that E[θ] = αθ/(α+1).

To find E[θ^2], we can use the fact that θ = max(Y1,Y2,...Yn)

P(θ ≤ y) = P(Y1 ≤ y, Y2 ≤ y, ..., Yn ≤ y) = (F(y))^n

Differentiating both sides with respect to y, we get

f(θ) = n(F(θ))^(n-1)f(θ)

Using this result, we can find E[θ^2]

E[θ^2] = ∫₀^∞ θ^2f(θ)dθ = ∫₀^θ θ^2αθ^α-1dθ = αθ^2/(α+2)

Substituting these expressions into the MSE formula, we get

MSE(θ) = αθ^2/(α+2) - 2θ(αθ/(α+1)) + (αθ/(α+1))^2

Simplifying, we get

MSE(θ) = αθ^2/[(α+1)^2(α+2)]

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Answer:

z=3.75

Step-by-step explanation:

Answer:

j = - 9/14

Step-by-step explanation:

-4(7j+2) = 10

Distributing First

-28j - 8 = 10

Try to get the variable on one side.

-28j = 18

Divide both sides by -28

j = -18/28 = - 9/14

Answer:

To calculate the scale of the map, we can use the following formula:

Scale = Actual distance / Map distance

(a) Fractional scale:

The actual distance between Point C and Point D is 500 miles, and the distance on the map is 5 inches. Therefore, the fractional scale can be calculated as:

Scale = 500 miles / 5 inches

Scale = 100 miles per inch

So the fractional scale of the map is 1 inch = 100 miles.

(b) Written scale:

To express the scale in written form, we can use the ratio of inches to miles. Since 1 inch represents 100 miles, we can write the scale as:

1 inch represents 100 miles

Alternatively, we can simplify the scale to a more common ratio by dividing both sides by 100:

1/100 inch represents 1 mile

Therefore, the written scale of the map is 1/100 inch = 1 mile.

Answer:

1. a square plus b square is c square where c is the hypotenuse.

so 1. 7^2 + 6^2 = c^2

49+36=c^2

85=c^2

the square root of 85 is c

9.22 is c

all the other questions are blurry :OO

Step-by-step explanation:

if we make quarterly deposits and invest them at an nominal annual interest rate of 8% compounded quarterly for 40 years, we will have $143,004.54 at the end of the 40 years.

The first step in solving this problem is to calculate the amount of each quarterly deposit. We know that the first deposit is $400, and each subsequent deposit decreases by 1% from the previous deposit. This means that each deposit is 99% of the previous deposit. To calculate the size of each deposit, we can use the following formula:

deposit_ n = deposit_(n-1) * 0.99

Using this formula, we can calculate the size of each quarterly deposit as follows:

deposit_1 = $400

deposit_2 = deposit_1 * 0.99 = $396.00

deposit_3 = deposit_2 * 0.99 = $392.04

deposit_4 = deposit_3 * 0.99 = $388.12

...

We can continue this pattern for 40 years (160 quarters) to find the size of each quarterly deposit.

Next, we need to calculate the future value of these deposits using an nominal annual interest rate of 8% compounded quarterly. We can use the formula for compound interest to calculate the future value:

[tex]FV = PV * (1 + r/n)^(n*t)[/tex]

where FV is the future value, PV is the present value (which is zero since we are starting with deposits), r is the nominal annual interest rate (8%), n is the number of times the interest is compounded per year (4 since we are compounding quarterly), and t is the number of years (40).

We can substitute the values into the formula and solve for FV:

[tex]FV = $400 * (1 + 0.08/4)^(440) + $396.00 * (1 + 0.08/4)^(439) + $392.04 * (1 + 0.08/4)^(4*38) + ... + $1.64 * (1 + 0.08/4)^4[/tex]

After solving this equation, we get a future value of $143,004.54, rounded to two decimal places. This means that if we make quarterly deposits and invest them at an nominal annual interest rate of 8% compounded quarterly for 40 years, we will have $143,004.54 at the end of the 40 years.

This calculation highlights the power of compound interest over long periods of time. By making regular contributions and earning interest on those contributions, our investment grows exponentially over time. It also shows the importance of starting early and consistently contributing to an investment over time in order to achieve long-term financial goals.

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The amount a student received in merit scholarships was $3,456 ($478 per student). The cost of full tuition was $4,200. This means that the difference between the amount of the scholarship and the cost of tuition was $744.

Amount is a numerical value that represents a quantity of something. It is used to measure the size, amount, or degree of something, often in terms of money, time, or distance. Amounts are usually expressed in a specific unit, such as dollars, minutes, or kilometers. Amounts can also refer to the total number of something, such as the amount of people in a room or the amount of items in a box. Amounts can also be used to describe a portion or percentage of something, such as the amount of a discount or the amount of interest earned.  

To find the percentage of students who did not receive enough to cover full tuition, we need to divide the difference ($744) by the amount of the scholarship ($3,456). This gives us a percentage of 21.5%.

Rounded to the nearest whole percent, the answer is 22%. This means that 22% of students who received a merit scholarship did not receive enough to cover full tuition.

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Answer:

2 hours and 56 minutes.

Step-by-step explanation:

To divide 14 hours and 40 minutes by 5, we need to convert everything to minutes first.

14 hours is equal to 14 x 60 = 840 minutes.

So, 14 hours and 40 minutes are equal to 840 + 40 = 880 minutes.

Dividing 880 minutes by 5 gives us:

880 ÷ 5 = 176 minutes

Now, we need to convert the answer back to hours and minutes.

There are 60 minutes in 1 hour, so we can find how many hours are in 176 minutes by dividing by 60:

176 ÷ 60 = 2 with a remainder of 56.

So, the answer is 2 hours and 56 minutes.

Answer:

410 Black and Green shirts

Step-by-step explanation:

150 shirts were pink

135 shirts were green

so knowing this we can set up the equation 2x+85=465

there were 275 black shirts in the box

and 190 yellow shirts

therefore there were 410 Black and Green shirts in the box

Therefore, we can use the normal distribution with mean 27.9 and standard deviation 1.67 to approximate the binomial distribution with n = 31 and p = 0.9.

The binomial distribution is characterized by two parameters: the number of trials, denoted by n, and the probability of success on each trial, denoted by p. The probability of obtaining exactly k successes in n trials is given by the binomial probability mass function:

P(k) = (n choose k) * [tex]p^{k}[/tex] * [tex](1-p)^{(n-k)}[/tex],

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n distinct items.

Given by the question.

Yes, the binomial distribution with n = 31 and p = 0.9 can be approximated by a normal distribution.

The conditions for a binomial distribution to be approximated by a normal distribution are as follows:

n*p >= 10

n*(1-p) >= 10

In this case, n = 31 and p = 0.9, so:

np = 310.9 = 27.9 >= 10

n*(1-p) = 31*0.1 = 3.1 >= 10

Condition 1 is satisfied, but condition 2 is not. Therefore, it is recommended to use a correction factor to improve the approximation.

The correction factor is given by:

[tex]\sqrt[2]{np(1-p)}[/tex]

Substituting the values, we get:

[tex]\sqrt[2]{310.90.1}[/tex]= 1.67

The corrected values for mean and standard deviation are:

mean = np = 310.9 = 27.9

standard deviation = [tex]\sqrt[2]{np(1-p)}[/tex] = 1.67

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The value of (f • p)(-5) is 1885 when functions are given as f(s) = 3s + 2 and p(s) = s³+ 4s.

In mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. It is often represented by an equation or formula, and can be visualized as a graph. Functions are widely used in various areas of mathematics, science, engineering, and other fields to model real-world phenomena and solve problems.

Here,

f(s) = 3s + 2

p(s) = s³+ 4s

To find (f • p)(-5), we need to first find f(-5) and p(-5), and then multiply them together. To find f(-5), we substitute -5 into the function f(s) and simplify:

f(-5) = 3(-5) + 2

= -13

To find p(-5), we substitute -5 into the function p(s) and simplify:

p(-5) = (-5)³ + 4(-5)

= -125 - 20

= -145

Now we can multiply f(-5) and p(-5) together to find (f • p)(-5):

(f • p)(-5) = f(-5) * p(-5)

= (-13) * (-145)

= 1885

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The value stored in 0x10000004 on a big-endian machine is given by 0x00000011.

The word "endianness" refers to the arrangement of bytes as they are stored in computer memory. Endianness is classified as big or little depending on which value is stored first.

The "big end" (the most important item in the sequence) is put first and at the lowest storage address in a big-endian order. The "small end" (the least important item in the sequence) is put first in a little-endian order.

(1)In Big-endian Machine, first byte of multi-byte data will be stored first(at lower memory address)

Address                 Data

0x10000000            0x11

0x10000001            0x22

0x10000002            0x33

0x10000003            0x44

---------------------------------------------------------------------

lbu $t0, 0($t1)

Load unsigned byte in Register $t0 at address 0x10000000

Here byte at address 0x10000000 is 0x11

$t0 = 0x00000011

----------------------------------------------------

sw $t0, 0($t2)

Store a word(4 bytes) from Register $t0 to memory address 0x10000004

value stored in 0x10000004 is 0x00000011

--------------------------------------------------------------------------------

(2)In Little-endian Machine, last byte of multi-byte data will be stored first(at lower memory address)

Address                 Data

0x10000000            0x44

0x10000001            0x33

0x10000002            0x22

0x10000003            0x11

---------------------------------------------------------------------

lbu $t0, 0($t1)

Load unsigned byte in Register $t0 at address 0x10000000

Here byte at address 0x10000000 is 0x44

$t0 = 0x00000044

----------------------------------------------------

sw $t0, 0($t2)

Store a word(4 bytes) from Register $t0 to memory address 0x10000004

value stored in 0x10000004 is 0x00000044.

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Answer:

27.78%

Step-by-step explanation:

We know

Total of 18 students 5 students prefer country.

What is the probability?

5/18 = 27.78%

So, the answer is 27.78%

The perimeter of the garden in the drawing is 22 inches and the perimeter of the actual garden is 770 inches

The perimeter of the garden in the drawing is calculated as

Perimeter of garden in drawing = 2(length + width)

So, we have

Perimeter of garden in drawing = 2(7 inches + 4 inches)

Evaluate

Perimeter of garden in drawing = 22 inches

So the perimeter of the garden in the drawing is 22 inches.

For the perimeter of the actual garden, we have

Perimeter of actual garden = 22 inches * 35

Perimeter of actual garden = 770 inches

We can see that the perimeter of the actual garden is 35 times larger than the perimeter of the garden in the drawing.

This makes sense since the length and width of the actual garden are 35 times larger than the length and width in the drawing, so the perimeter (which is the sum of the lengths of all four sides) would also be 35 times larger.

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Answer: 6,160

Step-by-step explanation:

Immediately, without even doing any math, the only logical answer would be 6160. This is because the current tuition is 5,500 and it is increasing so the answer cannot be lower.

However, mathematically you can prove this by turning 12% into a decimal and multiplying it by 5,500. 12% could be converted to .12 and because it is increasing you must add 1, or 100%, since that is what it started with. 5,500 x 1.12 = 6,160.

The value of the angles in the triangle are:

∠A = 60°, ∠B = 80° and ∠C = 40°

We  are given that BD = DC

Thus, ∠DBC = ∠BCD ---- 1  (angle in isosceles triangle)

We also have ∠BDC = 100°

In ΔBDC

∠BDC + ∠DBC + ∠BCD = 180°  (sum of angles of triangle is 180°)

Using 1:

∠BDC + 2∠DBC = 180°  

100° + 2∠DBC = 180°  

2∠DBC = 180 - 100

2∠DBC = 80

∠DBC = 80/2

∠DBC = 40°

∠DBC = ∠BCD = ∠2 = 40°

Thus, ∠C = 40°

We are given that  m∠1 = m∠2

Thus, ∠1 = ∠2 = 40°

Now,  ∠BDC +  ∠BDA = 180° (Linear pair)

100° + ∠BDA = 180°

∠BDA = 180 - 100

∠BDA = 80°

In ΔABD

∠ABD + ∠BDA +  ∠BAD = 180° (sum of angles of triangle is 180°)

∠1 + ∠BDA +  ∠BAD = 180°

40° +  80° +  ∠BAD = 180°

120° +  ∠BAD = 180°

∠BAD = 60°

So,  ∠A = 60°

∠B =  ∠1  + ∠2 = 40° + 40° = 80°

Therefore,  ∠A = 60°, ∠B = 80° and ∠C = 40°

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Complete Question

m∠1=m∠2

D∈

AC

, BD = DC

m∠BDC = 100°

Find: m∠A, m∠B, m∠C

Answer:

  6 cm

Step-by-step explanation:

You want the height of a trapezium with bases 17 cm, 35 cm and area 156 cm².

The formula for the are of a trapezium is ...

  A = 1/2(b1 +b2)h

Filling in the given values, we have ...

  156 = 1/2(17 +35)h = 26h

  6 = h . . . . . . . . . . divide by 26

The height of the trapezium is 6 cm.

Answer:

6cm

Step-by-step explanation:

To find:-

Answer:-

We are here given that the area of the trapezium is 156cm² and two of the parallel sides are 17cm and 35cm .We are interested in finding out the height of the trapezium.

The area of the trapezium is given by the formula,

[tex]:\implies \sf Area =\dfrac{1}{2}\times (s_1+s_1)\times h \\[/tex]

where s1 and s2 are the || sides of the trapezium and h is the height of the trapezium.

Now on substituting the respective values in the given formula, we have;

[tex]:\implies \sf 156cm^2 =\dfrac{1}{2} (17cm+35cm)\times h \\[/tex]

[tex]:\implies \sf 156cm^2(2) = 52cm (h) \\[/tex]

[tex]:\implies \sf h =\dfrac{156(2)}{52} cm\\[/tex]

[tex]:\implies \sf \pink{ height = 6 cm }\\[/tex]

Hence the height of the trapezium is 6cm .

Therefore, the distance between the 90 degree angle and the hypotenuse is approximately 0.829 units.

A triangle is a two-dimensional geometric shape that is formed by three straight line segments that connect to form three angles. It is one of the most basic shapes in geometry and has a wide range of applications in mathematics, science, engineering, and everyday life. Triangles can be classified by the length of their sides (equilateral, isosceles, or scalene) and by the size of their angles (acute, right, or obtuse). The study of triangles is an important part of geometry, and their properties and relationships are used in many areas of mathematics and science.

Here,

1. To find HF, we can use the angle bisector theorem, which states that if a line bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the adjacent sides. Let's denote the length of HF as x. Then, by the angle bisector theorem, we have:

JF/FH = JG/HG

Substituting the given values, we get:

15/x = 18/21

Simplifying and solving for x, we get:

x = 15 * 21 / 18

x = 17.5

Therefore, HF is 17.5 cm.

2. Let's denote the length of the hypotenuse as h and the length of the leg opposite the 18-unit perpendicular as a. We can then use the Pythagorean theorem to write:

h² = a²  + 18²

We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:

h = x + 6

Substituting this expression into the first equation, we get:

(x + 6)² = a² + 18²

We are also told that the leg adjacent to the angle opposite the 4-unit segment is divided into segments of length 4 and (a - 4), so we can write:

a = 4 + (a - 4)

Simplifying this equation, we get:

a = a

Now we can substitute this expression for a into the previous equation and solve for x:

(x + 6)² = (4 + (a - 4))² + 18²

Expanding and simplifying, we get:

x² + 12x - 36 = 0

Using the quadratic formula, we get:

x = (-12 ± √(12² - 4(1)(-36))) / (2(1))

x = (-12 ± √(288)) / 2

x = -6 ± 6√(2)

Since the length of a segment cannot be negative, we take the positive root:

x = -6 + 6sqrt(2)

x ≈ 1.46

Therefore, the value of x is approximately 1.46 units.

3. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the 9-unit perpendicular as b. We can then use the Pythagorean theorem to write:

h² = b² + 9²

We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:

h = x + 6

Substituting this expression into the first equation, we get:

(x + 6)² = b² + 9²

Expanding and simplifying, we get:

x² + 12x - b² = 27

We also know that the length of the leg opposite the 9-unit perpendicular is:

a = √(h² - 9²)

= √((x + 6)² - 9²)

= √(x² + 12x + 27)

Now we can use the fact that the tangent of the angle opposite the 9-unit perpendicular is equal to the ratio of the lengths of the opposite and adjacent sides:

tan(θ) = a / b

Substituting the expressions for a and b, we get:

tan(θ) = √(x² + 12x + 27) / (x + 6)

We also know that the tangent of the angle theta is equal to the ratio of the length of the opposite side to the length of the adjacent side:

tan(θ) = 9 / b

Substituting the expression for b, we get:

tan(θ) = 9 / √(h² - 9²)

Substituting the expression for h, we get:

tan(θ) = 9 / √((x + 6)² - 9²)

Since the tangent function is the same for equal angles, we can set these two expressions for the tangent equal to each other:

√(x² + 12x + 27) / (x + 6) = 9 / √((x + 6)² - 9²)

Squaring both sides, we get:

(x² + 12x + 27) / (x + 6)² = 81 / ((x + 6)² - 81)

Cross-multiplying and simplifying, we get:

x⁴ + 36x³ + 297x² - 1458x - 2916 = 0

Using a numerical method such as the Newton-Raphson method or the bisection method, we can find the approximate solution to this equation:

x ≈ 9.449

Therefore, the value of x is approximately 9.449 units.

4. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the distance we want to find as b. We can use the Pythagorean theorem to write:

h² = b² + d²

We are told that the hypotenuse is divided into segments of length 9 and 4 units, so we can write:

h = 9 + 4 = 13

Substituting this expression into the first equation, we get:

13² = b² + d²

Simplifying and solving for d, we get:

d = √(13² - b²)

Now, we need to find the value of b. We know that the hypotenuse is divided into segments of length 9 and 4 units, so we can use similar triangles to write:

b / 4 = 9 / 13

Simplifying and solving for b, we get:

b = 36 / 13

Substituting this expression for b into the equation we found earlier for d, we get:

d = √(13² - (36/13)²)

Simplifying and finding a common denominator, we get:

d =√ (169*13 - 36²) / 13²

Simplifying further, we get:

d = √(169169 - 3636) / 169

Calculating this expression, we get:

d ≈ 0.829

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Answer: 6y + 6

Step-by-step explanation:

To simplify the expression 8(3/4y -2) + 6(-1/2+4) + 1, we can follow the order of operations (PEMDAS):

First, we simplify the expression within parentheses, working from the inside out:

6(-1/2+4) = 6(7/2) = 21

Next, we distribute the coefficient of 8 to the terms within the first set of parentheses:

8(3/4y -2) = 6y - 16

Finally, we combine the simplified terms:

8(3/4y -2) + 6(-1/2+4) + 1 = 6y - 16 + 21 + 1 = 6y + 6

Therefore, the expression 8(3/4y -2) + 6(-1/2+4) + 1 is equivalent to 6y + 6.

Answer:

Step-by-step explanation:

The probability of a baby being a boy or a girl is 1/2 or 50% each. Since the couple has already had three boys, the probability of the fourth child being a boy or a girl is still 1/2 or 50%, as the gender of each child is independent of the others. Therefore, the probability of the couple's last child being a baby boy is 50%.

The required probability of a couple having a baby boy when their third child is​ born is 1/2 / 50%.

probability is the ratio of the number of favorable outcomes and the total number of possible outcomes. The chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%.

Given:

Assuming boys and girls are equally​ likely.

The first two children were both boys

According to given question we have

The probability of having a baby girl is an independent probability.

The first two children were both boys

So, it is not related to the previous child.

So required probability = 1/2 / 50%

Therefore, the required probability of a couple having a baby boy when their third child is​ born is 1/2 / 50%.

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