Which graph is sequenced by definition by the function F(x)=3(2)x-1

Answer:

5/7

Step-by-step explanation:

ratio of boys to girls---> 2:5

ratio sum: 2+5=7

girls ratio =5/7

OR

total of people in 7th grade choir=28

hence, no of girls in 7th grade choir=5/7 × 28

=20

proportion of girls in choir= 20/28 = 5/7

x represents a length, we can discard the negative sοlutiοn and cοnclude that x ≈ 15.7.

A triangle is a three-sided pοlygοn with three angles. It is a fundamental geοmetric shape and is οften used in geοmetry and trigοnοmetry.

Using the Law οf Cοsines:

c² = a² + b² - 2ab cοs(C)

where c is the side οppοsite angle C, a and b are the οther twο sides, and C is the angle between sides a and b.

Substituting the knοwn values:

26² = x² + (x+7)² - 2x(x+7) cοs(52°)

Simplifying and sοlving fοr x:

676 = x² + x² + 14x + 49 - 2x² cοs(52°) - 14x cοs(52°)

676 = 2x² - 14x cοs(52°) + 49

2x² - 14x cοs(52°) - 627 = 0

Using the quadratic fοrmula:

[tex]x = [14 cos(52^\circ) \± \sqrt{((14 cos(52^\circ))}^2 - 4(2)(-627))] / (2(2))[/tex]

x ≈ 15.7 οr x ≈ -21.6

Since x represents a length, we can discard the negative sοlutiοn and cοnclude that x ≈ 15.7.

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Answer:

  21

Step-by-step explanation:

You want the length of segment AB, given similar triangles AEB and ADC with AE=14, ED=12, and BC=18.

Corresponding sides (or segments) of similar triangles are proportional:

  AB/AE = BC/ED

  AB = AE·BC/ED = 14·18/12 = 14·3/2

  AB = 21

The proportion of company breadth between the widget's weight of [tex]0.13[/tex]and its weight of [tex]71[/tex]ounce is [tex]81.53[/tex].

Meaning of the word "company" A company is a collective of individuals. The word "company" is frequent and has several distinct meanings, but they are all related to gatherings or interactions of people. The most frequent usage of the term "company" is to describe a firm.

a) Using the Empirical Rule, we know that for a bell-shaped distribution, approximately [tex]68[/tex]% of the data falls within one standard deviation of the mean, [tex]95[/tex]% falls within two standard deviations of the mean, and [tex]99.7[/tex]% falls within three standard deviations of the mean. Therefore, for this problem, we can say:

[tex]95[/tex]% of the widget weights lie between [tex](59-2(6)) = 47[/tex] and [tex](59+2(6)) = 71[/tex] ounces.

b) To find the percentage of widget weights that lie between 53 and 71 ounces, we need to find the z-scores for each value and use a standard normal distribution table to find the areas under the curve.

The z-score for [tex]53[/tex] ounces is: [tex](53-59)/6 = -1.00[/tex]

The z-score for [tex]71[/tex] ounces is: [tex](71-59)/6 = 2.00[/tex]

Using a standard normal distribution table, we can find the area to the left of each z-score:

The area to the left of [tex]z = -1.00[/tex] is 0.1587

The area to the left of [tex]z = 2.00[/tex] is 0.9772

To find the percentage of widget weights between [tex]53[/tex] and [tex]71[/tex] ounces, we can subtract the area to the left of [tex]z = -1.00[/tex] from the area to the left of [tex]z = 2.00[/tex] and multiply by [tex]100[/tex]:

Percentage [tex]= (0.9772 - 0.1587) * 100 = 81.85[/tex]%

Therefore, approximately [tex]81.85[/tex]% of the widget weights lie between 53 and [tex]71[/tex] ounces.

c) To find the percentage of widget weights that lie above [tex]41[/tex] ounces, we need to find the z-score for [tex]41[/tex] and use a standard normal distribution table to find the area to the right of the z-score.

The z-score for 41 ounces is: [tex](41-59)/6 = -3.00[/tex]

Using a standard normal distribution table, we can find the area to the right of [tex]z = -3.00[/tex]:

The area to the right of [tex]z = -3.00[/tex] is [tex]0.0013[/tex]

To find the percentage of widget weights above [tex]41[/tex]ounces, we can multiply the area to the right of [tex]z = -3.00[/tex] by [tex]100[/tex]:

Percentage [tex]= 0.0013 * 100 = 0.13[/tex]%

Therefore, approximately [tex]0.13[/tex] % of the widget weights lie above [tex]41[/tex]ounces.

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Between the widget's weight of 0.13 and its weight in ounces, there is an 68 % company breadth.

Percentage is a way of expressing a quantity or value as a fraction of 100. It is denoted by the symbol %, which means "per cent" or "out of 100"

a) Using the Empirical Rule, we know that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% of the data falls within two standard deviations of mean, and approximately 99.7% of the data falls within three standard deviations of mean.

Since the mean weight of the widgets is 59 ounces and the standard deviation is 6 ounces, we can use the Empirical Rule to determine that:

Approximately 68% of the widget weights lie between 53 and 65 ounces (one standard deviation below and above the mean).

Approximately 95% of widget weights lie between 47 and 71 ounces.

Approximately 99.7% of widget weights lie between 41 and 77 ounces.

b) To find the percentage of widget weights that lie between 53 and 71 ounces, we can use the Empirical Rule and subtract the percentage of widget weights that lie outside of this range from 100%.

we can estimate that approximately 68% of the widget weights lie between 53 and 71 ounces.

Therefore, the percentage of widget weights that lie between 53 and 71 ounces is approximately 68%.

c) To find the percentage of widget weights that lie above 41 ounces, we can use the Empirical Rule and subtract the percentage of widget weights that lie within three standard deviations below the mean from 100%.

Therefore, the percentage of widget weights that lie above 41 ounces is approximately:

100% - 99.7% = 0.3%

So, approximately 0.3% of the widget weights lie above 41 ounces.

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The equation is written as

Aisha has 15 points

Let's use the given information to set up an equation to represent the relationship between the points each friend has.

Let x be the number of points Aisha has.

Then, according to the problem statement, we know that:

Carter has 7 1/2 more points than Aisha, so he has

x + 7 1/2 points.

Jamila has 4 - 1/2 more points than Carter, so she has

x + 7 1/2 + 4 - 1/2 points,

which simplifies to x + 11 points.

Jamila has 26 points, so we can set x + 11 equal to 26 and solve for x:

x + 11 = 26

x = 26 - 11

x = 15

Therefore, Aisha has 15 points.

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The probability that no more than one of them need correction of her eyesight is 5.960.

What is probability?

Probability is a way of calculating how likely something is to happen. It is difficult to provide a complete prediction for many events. Using it, we can only forecast the probability, or likelihood, of an event occurring. The probability might be between 0 and 1, where 0 denotes an impossibility and 1 denotes a certainty.

Here using formula , Binomial distribution: P(X) = [tex]nC_x \times p^x\times q^{n-x}[/tex]

=> P(an adult need correction), p = 0.75

=>q = 1 - p = 0.25

Sample size, n = 12

P(no more than 1 of them need correction for their eyesight) = P(none of them need correction) + P(only 1 need correction)

=> [tex]0.25^{12}[/tex] + [tex]12\times0.75\times0.25^{11}[/tex]

=> 5.960

If the probability of an event is less than 0.05, it can be considered significantly low

Therefore, 1 is a significantly low number of adults requiring eyesight​ correction.

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H_0: mu=10, H_a: mu not equal 10. n=25, sigma-3, sample mean xbar =11. The Z-value is 1.67. Option B

To calculate the Z-value, we can use the formula:

Z = (xbar - mu) / (sigma / sqrt(n))

Where xbar is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size.

Given:

H_0: mu = 10

H_a: mu ≠ 10

n = 25

sigma = 3

xbar = 11

We can plug in the values and get:

Z = (11 - 10) / (3 / sqrt(25))

Z = 5/3

Therefore, the Z-value is 1.67.

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Answer:

In chemistry, an element is typically represented by its atomic symbol, which consists of one or two letters. The regions in an element refer to different parts of its atom.

The three main regions of an atom are the nucleus, the electron cloud, and the space in between. The nucleus is located at the center of the atom and contains positively charged protons and neutrally charged neutrons. The electron cloud surrounds the nucleus and contains negatively charged electrons, which are distributed in different energy levels or orbitals. The space in between the nucleus and the electron cloud is mostly empty.

To identify the different regions in an element, one can look at the atomic structure of the element, which is typically represented by an electron configuration or an orbital diagram. These representations show the number and distribution of electrons in the different energy levels or orbitals. By analyzing the electron configuration or orbital diagram, one can identify the number of electrons in the electron cloud and the number of protons and neutrons in the nucleus. The space in between the nucleus and the electron cloud is relatively small and is not usually a focus of study in chemistry.

Answer:

To solve this problem, we need to use the Poisson distribution, which describes the probability of a certain number of events occurring in a fixed interval of time or space, given the average rate of occurrence.

Let λ be the average number of calls per minute. From the problem statement, we have λ = 2.35.

Now we need to find the probability of having at most 2 phone calls in one minute. Let X be the random variable representing the number of phone calls in one minute. Then we have:

P(X ≤ 2) = e^(-λ) * (λ^0/0!) + e^(-λ) * (λ^1/1!) + e^(-λ) * (λ^2/2!)

Substituting λ = 2.35, we get:

P(X ≤ 2) = e^(-2.35) * (2.35^0/0!) + e^(-2.35) * (2.35^1/1!) + e^(-2.35) * (2.35^2/2!)

≈ 0.422

Therefore, the probability that during one particular minute there will be at most 2 phone calls is about 0.422, or 42.2%.

The two expressions are not equivalent.

To simplify the expression (p^2n^1/2)^0 √ p^5n^4, we first need to understand the properties of exponents and radicals.

Recall that any number raised to the power of zero is equal to 1, so (p^2n^1/2)^0 = 1.

Next, we can simplify the radical term by multiplying the exponents inside and outside the radical:

√ p^5n^4 =

(p^5n^4)^(1/2)

= p^(5/2)n^(4/2)

= p^(5/2)n^2

Combining this result with the previous step, we have:

(p^2n^1/2)^0 √ p^5n^4

= 1 * p^(5/2)n^2

= p^(5/2)n^2

This is not equivalent to p^18n^6 √p. In fact, p^(5/2)n^2 cannot be simplified any further without additional information about the expression.  ,

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Step-by-step explanation:

a. 23.5 × 27 = 634.5

b. 23.5 × 2.7 = 63.45

c. 2.35 × 0.27 = 0.6345

Answer:

a) the percentage change in the firm's variable costs is 66.67%.

b) Crystal Clear would need to sell 200 of its smallest greenhouses per month to break even.

Step-by-step explanation:

a) The percentage change in variable costs can be calculated using the formula:

((New Value - Old Value) / Old Value) * 100%

Substituting the values, we get:

((£100 - £60) / £60) * 100% = 66.67%

Therefore, the percentage change in the firm's variable costs is 66.67%.

b) The break-even point is the point at which the total revenue equals total costs. The total cost is the sum of fixed costs and variable costs.

Let's assume that the fixed costs for Crystal Clear are £10,000 per month. Then, the total cost can be calculated as:

Total Cost = Fixed Cost + (Variable Cost per unit * Number of units)

We can rearrange this formula to find the number of units:

Number of units = (Fixed Cost + Total Cost) / Variable Cost per unit

At the break-even point, the total revenue equals the total cost. Let's assume that the selling price per unit is £150. Then, the break-even point can be calculated as:

Total Revenue = Total Cost

Number of units * Selling Price per unit = Fixed Cost + (Variable Cost per unit * Number of units)

Number of units * £150 = £10,000 + (£100 * Number of units)

Number of units * (£150 - £100) = £10,000

Number of units = £10,000 / £50

Number of units = 200

Therefore, Crystal Clear would need to sell 200 of its smallest greenhouses per month to break even.

Answer:5% = 0.05 . Then multiply the original amount by the interest rate. $1,000 * 0.05 = $50 . That's it.

Step-by-step explanation:

please give me brainliest

Answer:

y = x + 7

Step-by-step explanation:

The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept. We are given that the slope of the line is 1, so we can substitute m = 1 into the equation:

We also know that the line passes through the point (-2, 5), so we can substitute x = -2 and y = 5 into the equation and solve for b:

Therefore, the equation of the line in slope-intercept form is:

y = x + 7

Answer:

The Equation of the Line is : y = x + 7

Step-by-step explanation:

Equation of the Line in Slope-Intercept Form is y = mx + b

where m is the Slope and b is the y-intercept

since the slope (m) = 1 , Substitute by m = 1 in the above equation

Then the Equation will be y = x + b

Since the line passes through point ( -2 , 5 ) , Then y = 5 when x = - 2

Substitute , Then 5 = -2 + b

Then b = 5 + 2 =7

So the Equation will be y = x + 7

Hope this is Helpful

[tex]4 x 2 + 4 y 2 - 12 x + 16 y = 0[/tex]

a) The amount of charge stored on either plate of the capacitor is 0.0528 C. ,b) The area of the plates is 3.47 x 10⁷ m². c)  The capacitance of the capacitor is 6.62 x 10⁻⁸ F.

what is capacitance ?

Capacitance is a physical property of a capacitor that represents its ability to store an electric charge. It is defined as the ratio of the amount of electric charge that can be stored on the capacitor's plates to the potential difference (voltage) between them. Capacitance is measured in farads (F)

In the given question,

:Capacitance (C) = 4.4 mF = 4.4 x 10⁻³ F

Voltage (V) = 12 V

The amount of charge (Q) stored on either plate of the capacitor.

 Q = C x V

Q = 4.4 x 10⁻³ F x 12 V

Q = 0.0528 C

The amount of charge stored on either plate of the capacitor is 0.0528 C.

Capacitance (C) = 1.4 F

Plate separation (d) = 2.2 x 10⁻³m

Find: The area (A) of the plates.

Equation:

C = ε0 x (A/d)

where ε0 is the permittivity of free space and has a value of 8.85 x 10⁻¹² F/m

A = C x d / ε0

A = 1.4 F x 2.2 x 10⁻³ m / (8.85 x 10⁻¹² F/m)

A = 3.47 x 10⁷ m²

The area of the plates is 3.47 x 10⁷ m².

Dielectric constant (k) = 2.5

Plate separation (d) = 0.0003 m

Area (A) = 30 m²

The capacitance (C) of the capacitor.

C = k x ε0 x (A/d)

where ε0 is the permittivity of free space and has a value of 8.85 x 10⁻¹² F/m

C = 2.5 x 8.85 x 10⁻¹² F/m x 30 m² / 0.0003 m

C = 6.62 x 10⁻⁸ F

The capacitance of the capacitor is 6.62 x 10⁻⁸ F.

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Answer:

He took 28min eat.

Step-by-step explanation:

Answer:

The total amount of money collected from selling tickets is: 1900 tickets × $4/ticket = $7,600 Since the cash prize is $3,800, the expected value can be calculated as follows: Expected value = (Probability of winning) × (Cash prize) + (Probability of losing) × (Cost of ticket) The probability of winning is 1 out of 1900 tickets sold, or 1/1900. The probability of losing is 1899/1900. So, the expected value is: (1/1900) × $3,800 + (1899/1900) × $4 = $1.999 Rounding to the nearest cent, the expected value is $2.00.

95% of adult males have diastolic blood pressure readings that are at least 62 mmHg

The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule is further illustrated below

68% of data falls within the first standard deviation from the mean.

95% fall within two standard deviations.

99.7% fall within three standard deviations.

From the information given, the mean is 80 mmHg and the standard deviation is 9 mmHg.

95% fall within two standard deviations.

2*9=`18

80 - 18 = 62 mmHg

Therefore, 95% of adult males have diastolic blood pressure readings that are at least 62 mmHg.

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The correct equation is:  5/6 - 1/6 = 4/6

A number line is a visual representation of numbers, ordered from left to right, where each point on the line corresponds to a number. The number line can be used to represent a wide range of numbers, including integers, fractions, decimals, and even negative numbers.

On a basic number line, 0 is located in the center, with positive numbers to the right and negative numbers to the left. The numbers are usually evenly spaced, with tick marks or dots indicating each value. The distance between any two points on the number line represents the difference between the corresponding numbers.

According to question Option D is correct

This is because starting at 5/6 and ending at 1/6 involves moving in the negative direction on the number line. To find the distance between these two points, we need to subtract the smaller number (1/6) from the larger number (5/6).

So, 5/6 - 1/6 = 4/6, which simplifies to 2/3. This means that the distance between 5/6 and 1/6 on the number line is 2/3.

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The solutions of given functions are : 1. (f + g)(x) = 2x² + 3x - 2, 2. (fog)(x) = 6x² + 4, 3. (gof)(x) = 18x² - 60x + 53, 4. (f.g)(x) = 6x³ - 10x² + 9x - 15, 5. (f - g)(x) = -2x² + 3x - 8, 6. (fog)(3) = 58.

What are the functions?

We have the given functions:

f(x) = 3x - 5

g(x) = 2x² + 3

Here are the solution steps of the given functions f(x) and g(x).

1. (f + g)(x) = f(x) + g(x) = (3x - 5) + (2x² + 3) = 2x² + 3x - 2

So, (f + g)(x) = 2x² + 3x - 2.

2. (fog)(x) = f(g(x)) = f(2x² + 3) = 3(2x² + 3) - 5 = 6x² + 4

So, (fog)(x) = 6x² + 4.

3. (gof)(x) = g(f(x)) = g(3x - 5) = 2(3x - 5)² + 3 = 18x² - 60x + 53

So, (gof)(x) = 18x² - 60x + 53.

4. (f.g)(x) = f(x)g(x) = (3x - 5)(2x² + 3) = 6x³ + 9x - 10x² - 15

So, (f.g)(x) = 6x³ - 10x² + 9x - 15.

5. (f - g)(x) = f(x) - g(x) = (3x - 5) - (2x² + 3) = -2x² + 3x - 8

So, (f - g)(x) = -2x² + 3x - 8.

6. (fog)(3) = f(g(3)) = f(2(3)² + 3) = f(21) = 3(21) - 5 = 58

So, (fog)(3) = 58.

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Answer: To find the weight of the sumo wrestlers at the highest 10%, we need to find the z-score that corresponds to the top 10% of the distribution.

Using a standard normal table or a calculator, we can find that the z-score that corresponds to the top 10% is approximately 1.28.

Next, we can use the formula for a z-score to find the weight that corresponds to this z-score:

z = (x - mu) / sigma

where z is the z-score, x is the weight we want to find, mu is the mean weight, and sigma is the standard deviation.

Substituting in the values we know, we get:

1.28 = (x - 330) / 15

Solving for x, we get:

x = 1.28(15) + 330 = 349.2

Rounding to the nearest whole number, the minimum weight of the sumo wrestlers at the highest weight of the league is 349 lbs.

Step-by-step explanation:

Step-by-step explanation:

We can solve this problem using the hypergeometric distribution.

The total number of apples is 24 (4 bad + 20 good). Let X be the number of bad apples in a draw of 2 apples.

The probability of drawing 0 bad apples is:

P(X=0) = (20 choose 2) / (24 choose 2) = 190 / 276

The probability of drawing 1 bad apple is:

P(X=1) = ((4 choose 1) * (20 choose 1)) / (24 choose 2) = 64 / 276

The probability of drawing 2 bad apples is:

P(X=2) = (4 choose 2) / (24 choose 2) = 1 / 276

Therefore, the probability distribution of the number of bad apples in a draw of 2 apples at random is:

X | 0 | 1 | 2

----|------|------|-----

P(X) |190/276|64/276|1/276

Answer:

10cm

Step-by-step explanation:

V = (1/3)Length x width x height

V = (1/3)(2)(3)(5)

V = 10

Step-by-step explanation:

1. 84 cm^2

2. 616 cm^2

3. 2496 cm^2

Given:

A triangle

l (length) = 12 cm

w (width) is 2 cm less than 3/4 of its length

Find: A (area) - ?

First, let's find the width of the rectangle according to the given information:

[tex]w = (\frac{3}{4} \times 12) - 2 = 9 - 2 = 7 \: cm[/tex]

Now, we can find the area:

[tex]a = w \times l[/tex]

.

2. Given:

A circular clock

C (circumference) = 88 cm

π = 22/7

Find: A (area) - ?

[tex]c = 2\pi \times r[/tex]

First, let's find the radius of the clock:

[tex]2 \times \frac{22}{7} \times r = 88[/tex]

[tex] \frac{44}{7} \times r = 88[/tex]

Multiply both sides of the equation by 7 to eliminate the fraction:

[tex]44r =616[/tex]

Divide both parts of the equation by 44 to make r the subject:

[tex]r = 14[/tex]

Now, we can find the area:

[tex]a = \pi {r}^{2} [/tex]

[tex]a = \frac{22}{7} \times {14}^{2} = 616 \: {cm}^{2} [/tex]

.

3. Given:

A rectangle

l (length) = 52 cm

P = 200 cm

Find: A (area) - ?

First, let's find the width of the rectangle (the perimeter is equal to the sum of all side lengths):

P = 2l + 2w

2w = P - 2l

2w = 200 - 2 × 52

2w = 96 / : 2

w = 48 cm

Now, we can find the area:

A = w × l

A = 48 × 52 = 2496 cm^2

Answer:

The least possible degree is 3.

Answer:

he get 100$ in day time u can understand

Answer:

You can save money by putting aside part of your income on a regular basis or by putting aside extra income, or by reducing your expenses.

Step-by-step explanation:

Kelly spending per month on different working fields are shown in attachment!

[tex](-20, -4)[/tex] and are all of the ordered pairings that fulfil this equation [tex]F = 9/5C + 32. (-10, 14)[/tex].

Lines come in two varieties: identities and dependent equations. All possible values of the parameters result in an identity. Only certain combinations of the variables' values render a conditional equation true. Two expressions joined by the equals symbol ("=") form an equations.

Ax+By=C is the classic pattern for two-variable linear equations. A typical form linear equation is, for instance, 2x+3y=5. Finding all intercepts of a solution in this format is not too difficult. The approach additionally proves useful when trying to solve solutions combining two linear systems.

To find the missing y values, we need to substitute the given [tex]x[/tex] values into the equation [tex]F = 9/5C + 32[/tex] and solve for [tex]F[/tex].

For the first ordered pair (-20, F1):

[tex]F1 = 9/5(-20) + 32[/tex]

[tex]F1 = -36 + 32[/tex]

[tex]F1 = -4[/tex]

So the missing y value is [tex]-4[/tex], and the complete ordered pair is [tex](-20, -4)[/tex].

For the second ordered pair (-10, F2):

[tex]F2 = 9/5(-10) + 32[/tex]

[tex]F2 = -18 + 32[/tex]

[tex]F2 = 14[/tex]

So the missing y value is [tex]14[/tex], and the complete ordered pair is [tex](-10, 14)[/tex].

Therefore, the complete set of ordered pairs that satisfy the equation [tex]F = 9/5C + 32[/tex] is:

[tex](-20, -4)[/tex] and [tex](-10, 14)[/tex].

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The complete question is:

Given the equation

F=9,5

C+32

F=95C+32

where C is the temperature in degrees Celsius and F is the corresponding temperature in degrees Fahrenheit, and the following ordered pairs:

(−20,F1)

(−20,F1),(−10,F2)(−10,F2)

Step 1 of 2 : Compute the missing y values so that each ordered pair will satisfy the given equation.

Answer:

to find the date of birth from the age, you need to subtract the number of years and days from the current date. In this case, the current date is 15 February 2003. The sum of their ages is 23, which means that Manoj and Sandhya have a difference of 23 years between their dates of birth. Since they were both born on Saturday, their dates of birth must be on the same day of the month. Therefore, we can assume that Manoj was born on 15 February 1980 and Sandhya was born on 15 February 2003. This is one possible solution, but there may be other solutions depending on the leap years and calendar changes.

Answer:

The dates of birth of Manoj and Sandhya are February 15th, 1981 and February 15th, 1979 respectively.

Step-by-step explanation:

Let's use a system of equations to solve the problem.

Let M be Manoj's age and S be Sandhya's age. We know that their ages sum up to 23, so:

M + S = 23

We also know that they were both born on a Saturday, which means their birthdays fall on the same day of the week. In 2003, February 15th was a Saturday, so we can assume that they were born on February 15th in different years. Let's represent the year of Manoj's birth as M_year and the year of Sandhya's birth as S_year.

Since they were both born on a Saturday, we know that the year of Manoj's birth plus his age (M_year + M) must have the same remainder when divided by 7 as the year of Sandhya's birth plus her age (S_year + S). In other words:

(M_year + M) mod 7 = (S_year + S) mod 7

We can simplify this equation by subtracting M from both sides and then substituting 23 - S for M:

(M_year + (23 - S)) mod 7 = (S_year + S) mod 7

Now we have two equations with two unknowns. We can solve for M_year and S_year by guessing values for S and then checking if there are integers M and S_year that satisfy both equations. We know that M and S must be positive integers and that their sum is 23. Here are a few guesses:

If S = 1, then M = 22 and the equation becomes:

(M_year + 22) mod 7 = (S_year + 1) mod 7

This simplifies to:

M_year mod 7 = (S_year + 6) mod 7

There are no integers M_year and S_year that satisfy this equation, since the two sides always have different remainders when divided by 7.

If S = 2, then M = 21 and the equation becomes:

(M_year + 21) mod 7 = (S_year + 2) mod 7

This simplifies to:

M_year mod 7 = (S_year + 5) mod 7

The only pair of integers that satisfies this equation is M_year = 2 and S_year = 4.

Therefore, Manoj was born on February 15th, 1981 and Sandhya was born on February 15th, 1979.

So the dates of birth of Manoj and Sandhya are February 15th, 1981 and February 15th, 1979 respectively.