The number of children and adults that swam in the pool that day is given as follows:
Children: 66.Adults: 62.How to obtain the number of children and adults?The variables to the system of equations are given as follows:
Variable x: number of children.Variable y: number of adults.128 people used the public swimming pool, hence:
x + y = 128.
x = 128 - y.
The receipts for admission totaled $223.00, hence:
1.5x + 2y = 223.
We have that x = 128 - y, hence the number of adults is obtained as follows:
1.5(128 - y) + 2y = 223
0.5y = 31
y = 62.
Then the number of children is obtained as follows:
x = 128 - 62
x = 66.
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What is the length of MN if ABC~LMN
The length of MN if ABC~LMN would be 11 unit.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
The relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.
Given that the side lengths of two similar triangles are always proportional.
Given that ∆ABC ~ ∆LMN, therefore:
AB = 10
LM = 5
MN = x + 4
BC= 3x + 1
Plug in the values
AB / LM = BC/ MN
Cross multiply
10/ 5 = (3x + 1)/(x + 4)
2 (x + 4) = (3x + 1)
2x + 8 = 3x + 1
x = 7
MN = x + 4 = 11
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PLEASE HELP let f(x) = 2x^2+x-3 and g(x) =x-1
find the domain
a x^2-4: domain; positive real numbers
b 2x^2-4 domain all real numbers
c x^2-4 domain all real numbers
d 2x^2-2 domain all real numbers
The correct options regarding the domain of the functions are given as follows:
b 2x^2-4 domain all real numbers
c x^2-4 domain all real numbers
d 2x^2-2 domain all real numbers
How to obtain the domain of the function?The domain of a function is the set composed by all the input values that the function accepts.
Quadratic functions have no restrictions, hence the domain for all of them is of all real values, and thus options b, c and d are the correct options.
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For events A, B, and C we have that
P(A)=0.24,
P(B)=0.26,
P(A n B)=0.05,
P(A n C)=0.05,
P(B n C)=0.05,
P(A n B n C)=0.02 and P((A u B uC)^/)=0.33,
find P(C)
How should i go about this??
=================================================
Work Shown:
P( (A u B u C)' ) = 0.33
P(A u B u C) = 1 - P( (A u B u C)' )
P(A u B u C) = 1 - 0.33
P(A u B u C) = 0.67
--------
P(AuBuC) = P(A)+P(B)+P(C)-P(AnB)-P(AnC)-P(BnC)+P(AnBnC)
0.67 = 0.24 + 0.26 + P(C) - 0.05 - 0.05 - 0.05 + 0.02
0.67 = P(C) + 0.37
P(C) = 0.67 - 0.37
P(C) = 0.30
For more information, search out "inclusion-exclusion principle". Also feel free to ask me if you have any questions.
A bee with velocity vector r′(t)r′(t) starts out at (3,1,−5)(3,1,−5) at t=0t=0 and flies around for 99 seconds. Where is the bee located at time 99 if ∫90r′(u)du=0∫09r′(u)du=0 ?
The bee would be located at (-4, 3, 1) at time 7. The integral of a velocity vector over a certain time period represents the net displacement of an object.
In this case, the velocity vector is represented by r'(t) and the time period is from t=0 to t=7.
If the integral of the velocity vector over this time period is equal to 0, it means that the object's net displacement over the 7 seconds is 0. This implies that the object's final position is the same as its initial position.
As the bee started at (-4,3,1) at time t=0, it means the bee would have not moved from its initial position and would be located at (-4,3,1) at time t=7.
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Amanda wants to showcase her favorite function: f(x)=1+sqrt(x+5). She has built a function machine that performs these operations on the input values. Her brother Eric is always trying to mess up Amanda's stuff, so he created the inverse of f(x), called it e(x), and programmed it into a machine.
b. What happens if the two machines are pushed together? What is e(f(-4))? Explain why this happens
Answer:
First find what f(-4) is, it is 2. Then substitute the 2 into x for e(x). Then is becomes e(2) which equals -4. The output is the same as the input because they are inverses. When putting the output of the original equation as the input of the inversed equation, the output of the inversed equation becomes the input of the original equation.
The function e(f(-4)) = -4. This happens because when two function machines are pushed together, the output of the first machine is used as the input for the second machine.
The inverse of a function f(x) is a function e(x) such that e(f(x)) = x for all x in the domain of f(x). To find the inverse of Amanda's favorite function, f(x) = 1 + sqrt(x + 5), we need to isolate x in the expression 1 + sqrt(x + 5) and call it y:
y = 1 + sqrt(x + 5)
Squaring both sides of the equation, we get:
y^2 = 1 + 2sqrt(x + 5) + (x + 5)
y^2 - 1 = 2sqrt(x + 5) + x + 5
Expanding the square on the left side, we get:
y^2 - y - 6 = x
So the inverse function, e(x), can be defined as:
e(x) = y^2 - y - 6
Finally, to evaluate e(f(-4)), we first need to find f(-4), then use the result as an input to the inverse function e(x):
f(-4) = 1 + sqrt(-4 + 5) = 1 + sqrt(1) = 1 + 1 = 2
e(f(-4)) = e(2) = 2^2 - 2 - 6 = 4 - 8 = -4
Therefore, e(f(-4)) = -4.
This happens because when two function machines are pushed together, the output of the first machine is used as the input for the second machine.
The inverse of Amanda's favourite function takes the output of the first machine and finds the original input that would have produced the same output. In this case, the original input was -4.
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Learning Task 1: Show the following decimals using grids. Write your
answer in your notebook.
1) 0. 9 2) 0. 1 3) 0. 24 4) 0. 21 50. 11 6) 0. 15
The following values are: 1) 9/10, 2) 1/10, 3) 24/100, 4) 21/100, 5) 1/10, 6) 15/100
There are a few steps for writing decimals in grids:
Step 1: Count how many grids are completely shaded. This will be the number for your one digit.
Step 2: In the grid that is partially shaded, count how many parts are shaded and how many total parts there are.
If the grid has 10 total parts, the number of shaded parts will be your tenth digit.
If the grid has 100 total parts, the number of shaded parts will end in the hundredth place.
Step 3: Combine your numbers from steps 1 and 2 to create the decimal that the grids represent.
As per the steps now we can calculate the decimal numbers:
0.9/100=0.9/10
0.1/100=1/10
0.24/1000=24/100
0.21/1000=21/100
0.15/1000=15/100.
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JK is a midsegment of AFGH. Find the values of x and y.
28
F
X =
y =
J
10
y
H
K
G
The values of x and y are 18,14 respectively.
What is Triangle ?
Triangle can be defined in which it consists of three sides , three angles and sum of three angles is always 180 degrees.
Given ,
From the given figure ,
JK is a midsegment of triangle FGH.
And
FG = 28
FJ = x
JH = 10
JK = y
As, JK is midsegment of FG
So,
JK = FG / 2
y = 28 / 2
y = 14
So, the value of y could be 14.
And FG = 28
So,
FJ + JH = FG
10+x = 28
x = 28 - 10
x = 18
So, the value of x could be 18.
Hence, The values of x and y are 18,14 respectively.
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evaluate the fraction
36^3/12^5
[tex]\huge\text{Hey there!}[/tex]
[tex]\mathsf{\dfrac{36^3}{12^5}}[/tex]
[tex]\mathsf{= \dfrac{36\times36\times36}{12\times12\times12\times12\times12}}[/tex]
[tex]\mathsf{= \dfrac{1,296\times36}{144\times144\times12}}[/tex]
[tex]\mathsf{= \dfrac{46,656}{20,736\times12}}[/tex]
[tex]\mathsf{= \dfrac{46,656}{248,832}}[/tex]
[tex]\mathsf{\mathsf{= \dfrac{46,656\div 15,552}{248,832 \div 15,552}}}[/tex]
[tex]\mathsf{= \dfrac{3}{16}}[/tex]
[tex]\huge\text{Therefore, your answer should be:}[/tex]
[tex]\ \huge\boxed{\mathsf{\dfrac{46,656}{248,832}\ or\ \dfrac{3}{16}}}\huge\checkmark\\\\\ \huge\boxed{\uparrow}}\huge\text{ Either of those should work because they are}\\\huge\text{equivalent to each other :) }} \huge\boxed{\uparrow}}[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
Answer: 46656/248832
Step-by-step explanation:
Potentiation in fractions.
Power is made up of a base and an exponent. The exponent indicates how many times the base will be multiplied by itself.
For example:The power of 2² = 2 × 2 × 2 = 8We solve:[tex]\boldsymbol{\sf{\dfrac{36^3}{12^5}=\dfrac{36\times36\times36}{12\times12\times12\times12\times12 }=\dfrac{46656}{248832 } }}[/tex]
36³/12⁵ = 46656/248832
Milas tried to perform the following division and reduce to lowest terms:
x² + 12x +36
+8
This is his work:
3x +18
+2
Rewriting the dividend:
Rewriting the divisor:
3x +18 3(x+6)
a+2
a+2
Finding excluded values: a-8 and a -2
(+6)2
+8
Dividing and reducing to lowest terms:
x² + 12x +36
x+8
30
²+12a+36
248
2.
3x + 18
x+2
3(x+6)
x+2
(x+6) (x + 2)
3(x + 8)
(x+6)²
x + 2
x+8 3(x+6)
(x+6) (x+6)(x + 2)
3(x+8)(x+6)
What mistake did Milas make?
(x+6)²
+8
By lowest terms -4x + 20 / x² - x - 12 * x + 3 / x² - 25 this we get -4 / (x - 4) (x + 5).
How do you explain lowest terms?A numerator and denominator's lowest value after being divided by their biggest common factor.
By multiplying the reciprocal of a fraction, divide it:
-4x + 20 / x² - x - 12 * x + 3 / x² - 25
The expression is multiplied:
4(-x + 5 ) / x² - x - 12 * x + 3 / x² - 25
The expression is multiplied:
4(-x+5) / (x-4)(x+3) * x+3 / x² - 25
Applying factoring to the phrase
a² - b² = (a+b) (4-b):
4(-x + 5 )/(x-4)(x+3) * x+3 / (x+5)(x-5)
Eliminate all but the lowest term from the expression:
-4 / (x - 4) * 1 / ( x + 5)
Put the following in one fraction:
-4 / (x - 4 ) (x + 5 )
Answer : -4 / (x - 4) (x + 5).
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3: What is the intensity in watts per meter squared of 85.0-dB sound?
The intensity in watts per meter squared of 85.0-dB sound is 0.316 watts per meter squared.
What is the watts per square meter of an 85.0 dB sound?To calculate the intensity in watts per meter squared of 85.0-dB sound, we first need to convert the dB sound level to a ratio.
This is done by calculating 10 raised to the power of the dB sound level divided by 10.
Therefore, 10^(85.0/10) = 316.
Next, we need to convert the ratio to intensity in watts per meter squared. This is done by dividing the ratio by 1,000,000.
Therefore, 316 / 1,000,000 = 0.000316.
Finally, we need to convert the intensity in watts per meter squared to a more readable number. This is done by multiplying the intensity by 1,000,000.
Therefore, 0.000316 * 1,000,000 = 0.316 watts per meter squared.
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The intensity in watts per meter squared of 85.0-dB sound is 0.316 watts per meter squared.
What is an 85.0 dB sound's watts per square meter?We must first translate the dB sound level into a ratio in order to determine the intensity of 85.0-dB sound in watts per square meter.
To do this, divide the decibel sound level by 10 and multiply the result by 10 raised to that power.
Consequently, 10 (10/85.0) = 316.
The ratio must then be changed to intensity in watts per square meter. To do this, double the ratio by 1,000,000.
316 divided by 1,000,000 equals 0.000316.
Finally, we must translate the intensity measured in watts per square meter into a more understandable value. The intensity is multiplied by 1,000,000 to achieve this.
Therefore, 0.316 watts per square meter are equal to 0.000316 * 1,000,000.
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Which inequality is NOT part of the system of inequalities that defines a polygonal convex set that includes a
the interior of a square whose vertices are: 44. 4). B(4. -4). C(4. 4) and D(4. 4)?
y2-4
y<-4
b.
a
C.
X2-4
d. X4
The inequality y ≤ -4 is NOT part of the system of inequalities that defines a polygonal convex set that includes a the interior of a square whose vertices are A(4, 4), B(4, -4), C(-4, -4) and D(4, -4).
As per the given data the interior of a square whose vertices are:
A(4, 4), B(4, -4), C(-4, -4) and D(4, -4)
Diagram attached at the end of solution.
The shaded portion is the polygonal convex set, which represents the interior of the square
Clearly for points inside the set,
x ≤ 4, x ≥ - 4, y ≤ 4 and y ≥ -4
Hence options
(a) The inequality y ≥ -4 represent points inside the set.
(b) The inequality x ≥ - 4 represent points inside the set.
(c) The inequality y ≤ -4 does not represent points inside the set.
(d) The inequality x ≤ 4 represent points inside the set.
Therefore option (c), the inequality y ≤ -4, does not represent points inside the set.
Thus (c) is the correct one.
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A barn is 40 feet wide by 100 feet long. Make a scale drawing of the barn that has a scale of 1/2 inch = 10 feet
As per the given measurement, the scale drawing of the barn is attached below.
The term scale drawing is defined as a real object with accurate sizes reduced or enlarged by a certain amount.
Here we have given that a barn is 40 feet wide by 100 feet long.
Here we have given in the question that if you draw a line that is 1/2 an inch on paper its proportional to 10 feet long on the house.
Which means that 1 inch would be 20, then 1.5 would be 30 feet, and then 2 inches would be 40.
Here we have given the proportion would be written as,
=> 2 in: 40 ft
As we know that if we make the ratios equal, then we have,
=> 1/20 = x/40
When we simplify this then we get the value of x as 2
Similarly, for the another one,
=> 1/20 = x/100
Now, we have to multiply both sides by 100:
=> 100/20 = x
Therefore, the value of x is 5.
Hence the scale drawing of the house would be 2 inches wide and 5 inches long.
And the drawing is attached below.
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Find x and y from -7x+3y=12 2x-8y=32
Answer:
x = -3.84
y = -4.96
Step-by-step explanation:
-7x+3y=12
2x-8y=32
We multiply the first equation by 2 and the second equation by 7
-14x + 6y = 24
14x -56y = 224
-50y = 248
y = -4.96
Now put -4.96 in for y and solve for x
-7x+3(-4.96) = 12
-7x - 14.88 = 12
-7x = 26.88
x = -3.84
Let's check
-7(-3.84) + 3(-4.96) = 12
26.88 - 14.88 = 12
12 = 12
So, x = -3.84, and y = -4.96 is the correct answer.
Today only, a table is being sold at a 30% discount. The sale price is $252 .
What was the price yesterday?
The set of all points in a plane that are the same distance from a given point called the center. Radius. A straight line from the center to the circumference of a circle or sphere. Diameter
1. The set of all points in a plane that are the same distance from a given point called Radius. So the option B is correct.
2. A straight line from the center to the circumference of a circle or sphere is Diameter. So the option A is correct.
The set of all points in the plane that are a fixed distance (the radius) away from a fixed point is referred to as a circle (the center).
Radius is the name for any distance that connects a point on a circle to its center. Any two radii have the same length when compared to the definition of a circle. Take note of the fact that the term "radius" is being used to describe both these intervals and their typical length.
The term "chord" refers to the space that connects two points on a circle.
The term "diameter" refers to a chord that runs through the center. Since a diameter is made up of two radii that are connected at their ends, each diameter has a length that is twice as long as its radius.
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The complete question is:
1. The set of all points in a plane that are the same distance from a given point called
1. Center
2. Radius
C. Surface area
D. Circumference
2. A straight line from the center to the circumference of a circle or sphere is
A. Diameter
B. Radius
C. Surface area
D. Circumference
If the florist uses 10 white roses in a bouquet,
how many
red roses does she use?
When the florist uses 10 white roses in a bouquet, the red roses that she uses will be 25.
How to calculate the number of red roses?Ratio in Mathematics simply demonstrates how many times one number can be able to fit into another number. It should be noted that ratios contrast two numbers by dividing them. In this case, A/B will be the formula if one is comparing one variable (A) to another variable (B).
In this case, the ratio is given as 5:2. Therefore, when the florist uses 10 white roses in a bouquet, the red roses that she uses will be:
= 5/2 × 10
= 25
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A florist uses 5 red roses for every 2 white roses in her bouquets. If the florist uses 10 white roses in an arrangement, how many red roses does she use?
Buddy walks 5000 feet due North, then turns and walks 4000 feet due East. What is the direction and magnitude of the resultant vector?
The direction and magnitude of the resultant vector are:
i. Magnitude = 6403
ii. Direction = 51.3^o
What is a resultant vector?A resultant vector is just a single vector which represent two or more vectors acting at a point in both magnitude and direction.
To determine the magnitude of the resultant vector, we have;
let the resultant vector be represented by R, so that;
/Hyp/^2 = /Opp/^2 + /Adj/^2
R^2 = 5000^2 + 4000^2
= 25000000 + 16000000
R^2 = 41000000
R = (41000000)^1/2
= 6403.12
The resultant of the two vectors is 6403.
ii. To determine the direction (θ) of the vector,
Tan θ = y/ x
Tan θ = 5000/ 4000
θ = Tan^-1 1.25
= 51.341
θ = 51.3^o
The direction of the resultant vector is 51.3^o.
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what is the value of 7 in 2784
Answer:
Step1
First look at the value for 4 and then rest numbers 1234
Step2
4 is Unit 3 is Tens 2 is Hundred and 1 is Thousand
Step3
So in the question 4 is Unit 8 is Tens 7 is Hundred and 4 is thousand
The figure is made up of a cylinder and a cone. What is the exact volume of the figure? Enter your answer in the box. Ft3 $\text{Basic}$ $x$$y$$x^2$$\sqrt{ }$$\frac{x}{ }$ $x\frac{ }{ }$ $x^{ }$$x_{ }$$\degree$$\left(\right)$$\abs{ }$$\pi$$\infty$ A shape made up of a cone on top of a cylinder. The cylinder has a height of four feet and a radius of three feet. The cone has a radius of three feet and a height of five feet
The total volume is 160.2cubic feet.
for the cone,
we can take π times the radius squared times the height.
volume = π x 9 x 4
volume = 113.0973
for the cone,
you would take π times the radius squared times the height and divide it all by 3
volume= (π x 9 x 5)/3
volume= 47.123
so, The total volume is 160.2cubic feet.
Cone
A cone is defined as a distinctive three-dimensional geometric figure with a flat and curved surface pointed towards the top.
Volume
Volume is a three-dimensional quantity that is used to measure the capacity of a solid shape. It means the amount of three-dimensional space a closed figure can occupy is measured by its volume.
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The complete question is :--
The figure is made up of a cylinder and a cone.
What is the exact volume of the figure?
Enter your answer in the box.
ft3
A shape made up of a cone on top of a cylinder. The cylinder has a height of four feet and a radius of three feet. The cone has a radius of three feet and a height of five feet.
John ran 2.75 km on Sunday. If he ran the same distance every day of the week, then find the total distance he ran in the whole week
Answer:
19.25 km
Step-by-step explanation:
7 days = 1 week
1 day = 2.75 km
7 days = ?
Then you cross multiply.
Therefore: 7×2.75/1
19.25/1
Answer: 19.25 km
If the market value of a share of common stock is 3. 3 times Book value (9. 42) for 20X2 what is the firm's p/e ratio for 20X2?
As per the given market value, the firm's P/E ratio for 20X2 is $2.85
The term P/E ratio is defined as measures a company's market price compared to its earnings and it shows what the market is willing to pay today for a stock based on a company's past or future earnings.
Here we have given that the market value of a share of common stock is 3. 3 times Book value (9. 42) for 20X2.
in the given values we have know that
Price of common stock = 3.3
Earning per share = 9.42
Then the value of P/E ratio is calculated as,
=> P/E ratio = 9.42/3.3
=> P/E Ratio = 2.85
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Yusuf drove 156 miles in 3 hours. If he continued at the same rate, how far would he travel in 2 hours?
the answer would be 104
How you would get this answer if by dividing 156 ÷3 =52
then you would take the 52 and multiply it by 2 and get 104
I have also provided the work on how to write it out!
Hope this helps and have a good day!
Among the 1018 respondents, 97 said that cellular phones are "not at all annoying " What percentage of respondents said that cellular phones are "not at all annoying?" (Round to two decimal places as needed)
The percentage of respondents said that cellular phones are "not at all annoying" is 9.52%
Now, According to the question:
Among the 1018 respondents, 97 said that cellular phones are "not at all annoying "
To find the percentage of respondents said that cellular phones are "not at all annoying"
Based on the given condition:
If we divide 97 divide by 1018, we get :
[tex]\frac{97}{1018}[/tex] ≈ 0.09528
If we want to express this number as a percentage, we need to multiply by 100. Then we get:
0.09528 x 100 = 9.52%
Hence, The percentage of respondents said that cellular phones are "not at all annoying" is 9.52%
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56% of the students in the chorus are also in the musical. If there are 75 students in the chorus, how many are in the musical?
Answer:
There are 42 students in the musical.
Step-by-step explanation:
Let x be the number of students in both chorus and musical.
75 ⇒100%
x⇒56%
Find x,
x=56% divided by 100% times 75
Which equals 42 is your answer.
Find half of the time that Devon skied without falling, and write inequalities to compare it with the times others in the family skied without falling
The inequality to compare it with the times others in the family skied without falling is 2f < 223.
In math the term inequality is defined as a relation which makes a non-equal comparison between two numbers or other mathematical expressions
Here we have given that Kelvin wants to know whether he skied without falling more than twice as long as anyone else in his family.
Here we have also given that his dad tells him that he can check by using the inequality.
Based on the given details we have identified that the inequality that compare it with the times others in the family skied without falling is written as
=> 2f < 223.
here the term f refers the time skied in seconds for each person.
Complete Question:
Kelvin wants to know whether he skied without falling more than twice as long as anyone else in his family. The total amount of time spent on skied without falling 223. Then write the inequality to compare it with the times others in the family skied without falling.
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Recall that |x| is the distance between x and 0 on the number line. How does |x-y| relate to the distance between x and y on the number line
The expression |x-y| represents the distance of the subtraction of x by y to the number zero on the number line.
What is the absolute value of a number?The absolute value of a number is the number without the signal, representing the difference between the number and zero.
For example:
|-2| = |2| = 0.
The definition of the absolute value function is given as follows:7
|x| = x, x >= 0.|x| = -x, x < 0.Hence, for the difference between x and y, the absolute value is given as follows:
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Is this relation a function? Justify your answer.
OA. Yes, because all the points are in the first quadrant.
B. No, because the points do not form a pattern.
C. Yes, because there are three x-values and three y-values.
D. No, because there is one x-value that corresponds to two y-values
No, because there is one x-value that corresponds to two y-values. Hence option D is the correct answer.
What is a function?A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a connection between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input. A function is often represented as y = f. (x).
We know that each input is connected to precisely one output in a function. In the graph given one 2 input value correspond to the same output value and hence, it does not form a function.
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1. Which of the following scenarios represent exponential relationships? Select all that apply.
A bank offers an interest rate of 3% compounded yearly.
A population of rabbits doubles every two months.
A contract worker earns $12.50 for every hour that she works.
A radioactive substance decays at a rate of 2.3% per hour.
PLEASE ITS DUE
A surveyor measures the lengths of the sides of a triangular plot of land. what is the measure of the angle of the triangular plot at which the surveyor stands? approximate to the nearest degree. cos–1(0.75) = 41° cos–1(0.125) = 83° cos–1(0.563) = 56° cos–1(0.15) = 89°
The measure of the angle of the triangular plot at which the surveyor stands is cos(0.125) = 83°
Since we have given that the lengths of the sides of triangle from figure :
a = 300 m
b = 250 m
c = 200 m
We need to find the measure of angle i.e. A
As we know the "Law of Cosine rules ":
Cos A = b^2 + c^2 - a^2 / 2bc
= 250^2 + 200^2 - 300^2 / 2.250.200
= 125 / 1000
= 0.125
therefore , A = 82.81° approximate and close to 83°
Hence second option is correct.
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A man has three children. They are not triplets. By a remarkable coincidence,
however, they were all born on the same day in different years. By another
remarkable coincidence, their ages right now form a Pythagorean triple. The
oldest child is nine years older than the youngest, and the middle child is two
years younger than the oldest.
a) How old are the three children?
b) The man knows that a multiple of a Pythagorean triple is also a Pythagorean triple. He reasons that in the year when the ages of his three children have doubled, their ages will form a pythagorean triple? Is he correct? Why or why not?
a) The current ages of his children are: 8 years, 15 years and 17 years
b) In the year when the ages of his three children have doubled, their ages also form a pythagorean triple.
Let x be the age of the youngest child, y represents the age of the middle child and z be the age of oldest child.
The oldest child is nine years older than the youngest.
so, we get an equation,
z = 9 + x
⇒ x = z - 9
and the middle child is two years younger than the oldest.
so, we get an equation,
y = z - 2
The age of his children now form a Pythagorean triple.
⇒ x² + y² = z²
substitute the values for x and y from above two equations.
⇒ (z - 9)² + (z - 2)² = z²
⇒ z² -18z + 81 + z² - 4z +4 = z²
⇒ 2z² -22z + 85 - z² = z² - z²
⇒ z² - 22z + 85 = 0
⇒ (z - 17)(z - 5) = 0
For z = 17:
x = 17 - 9
x = 8
and y = 17 - 2
y = 15
For z = 5, the value of x would be negative which is contradiction.
As is always greater than 0
For x = 8 , y = 15 and z = 17
Consider x² + y²
= 8² + 15²
= 64 + 225
= 289
= 17²
= z²
This means, (8, 15, 17) is a Pythagorean triple.
Thus, their current ages are: 8 years, 15 years and 17 years
If the ages of his three children have doubled then:
x = 16 , y = 30 and z = 34
Consider x² + y²
= 16² + 30²
= 256 + 900
= 1156
= 34²
= z²
This means, (16, 30, 34) also form a Pythagorean triple.
In the year when the ages of his three children have doubled, their ages will form a pythagorean triple.
Thus, the man is correct.
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