One travel bag is 15 inches long, another is 18 inches long, and a third is 21 inches long. They are all 10 inches deep and 15 inches wide. Which travel bag can hold exactly 3,150 cubic inches? Explain your reasoning. 50 points to whoever answers

Step-by-step explanation:

a. 23.5 × 27 = 634.5

b. 23.5 × 2.7 = 63.45

c. 2.35 × 0.27 = 0.6345

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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Answer:

The answer is C unlikely and likely

Step-by-step explain

the is 50% chance of getting likely or unlikely

Answer:

Step-by-step explanation:

A 50% probabilty is interpreted as being equally ikely and unlikely.

There are many tools available to calculate the line of best fit, including Excel, R, and Python. These tools provide the best equation that approximates the data

To determine which equation could be used as a line of best fit to approximate the data in a scatter plot, we need to look for the equation that closely follows the general trend of the data. The line of best fit is the straight line that best represents the data on the graph.

One common method for finding the line of best fit is through linear regression analysis. This involves calculating the slope and intercept of the line that best fits the data. The equation for a line is y = mx + b, where m is the slope and b is the y-intercept.

In general, the equation that could be used as a line of best fit depends on the data on the scatter plot. However, we can determine the best equation by looking at the trend of the data. If the data shows a positive trend, we need to choose an equation with a positive slope. If the data shows a negative trend, we need to choose an equation with a negative slope.

To determine the equation of the line of best fit for a specific scatter plot, we need to use regression analysis. and allows us to make predictions based on the data.

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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]

To solve the problem, we need to use the given function f(x) = 4 + z(0.25) where x is the amount of 25% orange juice drink added and z is the amount of 100% orange juice drink added.

We know that initially 4 gallons of pure juice are present. So, the total amount of juice in the mixture is 4 + x gallons.

When we add 2 gallons of 25% orange juice drink, x = 2 and z = 0. So, using the function f(x), we get:

f(2) = 4 + z(0.25)

    = 4 + 0(0.25)

    = 4

Therefore, the concentration of orange juice in the drink after adding 2 gallons of 25% orange juice drink is 4%, expressed as a percent but without the percent sign.

Answer: 4

Part A: Fraction of all students who does not like Lakers = 3/8.

Part B: Fraction of all student who likes the Clippers = 9/32

Part C: Fraction of all student who does not likes the Clippers =  23/32.

Part D: Fraction of all student who does not likes either the Clippers or Lakers : 35/32

The components of a whole or group of items are represented by fractions. A fraction consists of two components. The numerator is the figure at the top of the line. It details the number of equal portions that were taken from the total or collection.

Given data:

Students who likes only Los Angeles Lakers = 5/8

Students who does not likes only Los Angeles Lakers: 1 - 5/8 = 3/8

From 3/8, only 3/4 like the Los Angeles Clippers.

= 3/8 * 3/4

= 9/32

Now,

Students who does not like the Los Angeles Clippers: 1 - 9/32 = 23/32.

Part A: Fraction of all students who does not like Lakers = 3/8

Part B: Fraction of all student who likes the Clippers = 9/32

Part C: Fraction of all student who does not likes the Clippers =  23/32.

Part D: Fraction of all student who does not likes either the Clippers or Lakers :

= 3/8 + 23/32

= 35/32

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Answer:

He took 28min eat.

Step-by-step explanation:

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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Therefore, the length of the side x is approximately 10.76.

A right-angled triangle, also known as a right triangle, is a type of triangle that has one of its angles measuring exactly 90 degrees. The other two angles in a right triangle are acute angles, which means they are less than 90 degrees. The side opposite to the right angle is called the hypotenuse, and it is the longest side of the triangle. The other two sides of the right triangle are called legs, and they are adjacent to the right angle. The Pythagorean theorem, which states that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse, is a fundamental concept in the study of right triangles. Right triangles have many practical applications in geometry, trigonometry, and physics.

In a right triangle, the sum of the measures of the two acute angles is always 90 degrees. Therefore, the third angle in this triangle is:

90 - 34 = 56 degrees

Now, we can use the trigonometric ratios to find the length of the side x.

We know that the tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. Therefore:

[tex]tan(34) = x/18[/tex]

To solve for x, we can multiply both sides by 18:

[tex]18 tan(34) = x[/tex]

Using a calculator, we can find that:

18 tan (34) ≈ 10.76.

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Answer:

1

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.

The value of the functions are;

1. (f+g(x)) = -x² -x + 1,  (f-g(x)) = -5x²+ 5x - 7

2. (f+g(x)) = 6x² - 4x + 6,  (f-g(x)) = 2x² - 6x + 10

3. (f+g(x)) = 3√m . √4, (f-g(x)) = √m· √4

4. (f+g(x)) = 7x² - 3x - 5, (f-g(x)) = x² + 3x + 5

It is important to note that functions are described as an equation or expression showing the relationship between two variables.

From the information given, we have the functions

1.

f(x) = -3x² + 2x - 3

g(x) = 2x² - 3x + 4

To add the functions, we have;

(f+g(x)) =  -3x² + 2x - 3 + 2x² - 3x + 4

collect the like terms and add

(f+g(x)) = -x² -x + 1

(f-g(x)) =  -3x² + 2x - 3 - 2x² +3x - 4

collect the like terms and subtract

(f-g(x)) = -5x²+ 5x - 7

2. f(x) = 4x² - 5x + 8

g(x) = 2x² + x - 2

(f+g(x)) = 4x² - 5x + 8 + 2x² + x - 2

(f+g(x)) = 6x² - 4x + 6

(f-g(x)) =4x² - 5x + 8 -  2x² - x +2

subtract the values

(f-g(x)) = 2x² - 6x + 10

3. f(x) = 2√m

g(x) = 3√4m

(f+g(x)) = 2√m + √m. √4

(f+g(x)) = 3√m . √4

(f-g(x)) = 2√m -√m. √4

(f-g(x)) = √m· √4

4. f(x) = √16x^4

g(x) = 3x² - 3x - 5

(f+g(x)) = √16x^4 +  3x² - 3x - 5

find the square root

(f+g(x)) = 7x² - 3x - 5

(f-g(x)) =√16x^4 -  3x² + 3x +5

(f-g(x)) = x² + 3x + 5

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Answer:

10cm

Step-by-step explanation:

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

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

V = 10

Answer:

3p^3

pq

There is probably more

Step-by-step explanation:

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:

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!

The value of length BD in the given parallelogram is 10 units.

A quadrilateral having two sets of parallel sides is referred to as a parallelogram. In other words, opposing sides and angles are congruent and parallel. There are numerous crucial characteristics of parallelograms, including:

A parallelogram's opposing sides are congruent.

A parallelogram's opposing angles are congruent.

The parallelogram's subsequent angles are additional (add up to 180 degrees).

A parallelogram's diagonals cut each other in half (i.e. they intersect at their midpoints).

According to the properties of parallelogram the diagonals of a parallelogram bisect each other.

Thus, BE = ED

Given BE = 5

Thus, ED = 5.

Now, the value of BD = BE + ED = 5 + 5 = 10

Hence, the value of BD in the given parallelogram is 10 units.

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Using division, we can prove that there are models that we can use to prove 7 divided by 2 is = 7/2.

A division is one of the fundamental mathematical operations that divides a larger number into smaller groups with the same number of components. How many groups will be created, for instance, if a sports event requires that 30 students be divided into groups of 5 students. The division operation makes it simple to tackle such issues. Divide 30 by 5 in this case. The outcome is 30 x 5 = 6. There will thus be 6 groups, each with 5 pupils. By multiplying 6 by 5, which yields the initial figure of 30, you can confirm this value.

Here in the question,

We have to model to find 7 divided by 2 = 7/2.

For the two math classes, 7 pizzas have been delivered.

Half of the first pizza is distributed to each class.

A half of the second pizza is distributed to each class.

Third pizza, cut in half, with half going to each class.

Continue until each class receives 7 halves, or 7/2, of the pizzas that were delivered.

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Step-by-step explanation:

(f○g)(x) = f(g(x))

so, we use the expression of g(x) as the "x" in f(x).

(f○g)(x) = f(g(x)) = 3(x² + 3) - 1 = 3x² + 9 - 1 = 3x² + 8

so,

(f○g)(3) = 3×3² + 8 = 27 + 8 = 35

check :

g(3) = 3² + 3 = 9 + 3 = 12

f(g(3)) = f(12) = 3×12 - 1 = 36 - 1 = 35

correct.

The linear inequality for the shaded region with slope [tex]-4[/tex] is:

[tex]y < -4x+4[/tex]

In mathematics, a linear inequality is an inequality involving a linear function in one or more variables. It describes a region in the coordinate plane that satisfies the inequality.

In mathematics, the slope is a measure of the steepness of a line. It describes how much a line rises or falls as we move from left to right along it.

According to the given information

To write the linear inequality for the graph passing through points [tex](0,4)[/tex] and [tex](1,0)[/tex], we need to find the equation of the line first.

The slope of the line passing through these two points is:

[tex]m =(y_{2} -y_{1} )/(x_{2} -x_{1} )\\ = (0-4)/(1-0)\\ = -4[/tex]

Using the slope-intercept form of a linear equation, [tex]y = mx +b[/tex], where m is the slope and b is the y-intercept, we can find the equation of the line passing through these two points:

[tex]y = -4x+4[/tex]

Now, to write the linear inequality for this line, we need to determine which side of the line is shaded. We can use the test point [tex](0,0)[/tex] to check which side of the line contains the solutions to the inequality.

If we plug in [tex](0,0)[/tex] into the equation [tex]y = -4x+4[/tex], we get:

[tex]0= -4(0) +4\\0 = 4[/tex]

Since [tex]0[/tex] is not less than [tex]4[/tex], the point [tex](0,0)[/tex] is not a solution to the inequality. Therefore, we need to shade the side of the line that does not contain the origin [tex](0,0)[/tex].

The linear inequality for the shaded region is:

[tex]y < -4x +4[/tex]

Therefore any point below the line [tex]y < -4x +4[/tex] satisfies this inequality.

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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.

The possible length, width, and perimeter for Small cage's floor is 4 feet, 3 feet and 14 feet respectively; for Medium cage's floor is 12 feet, 2 feet and 28 feet respectively and for Large cage's floor is 18 feet, 2 feet and 40 feet respectively.

What is perimeter?

Whether it be a two-dimensional shape or a one-dimensional length, a perimeter is a closed path that contains, surrounds, or delineates it.

We are given the floor areas of each cage.

For Small cage:

Area is given as 12 square feet which is a product of length and width.

So, here length can be 4 feet and width can be 3 feet.

Now, we get the perimeter as

⇒ Perimeter = 2 * (Length + width)

⇒ Perimeter = 2 * (4 + 3)

⇒ Perimeter = 2 * 7

⇒ Perimeter = 14 feet

For Medium cage:

Area is given as 24 square feet which is a product of length and width.

So, here length can be 12 feet and width can be 2 feet.

Now, we get the perimeter as

⇒ Perimeter = 2 * (12 + 2)

⇒ Perimeter = 2 * (14)

⇒ Perimeter = 28 feet

For Large cage:

Area is given as 36 square feet which is a product of length and width.

So, here length can be 18 feet and width can be 2 feet.

Now, we get the perimeter as

⇒ Perimeter = 2 * (18 + 2)

⇒ Perimeter = 2 * (20)

⇒ Perimeter = 40 feet

Hence, the required solution has been obtained but there are many other combinations possible.

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Answer: −1.3571428 (with a repeating bar over .3571428)

Step-by-step explanation:

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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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 labelled parts of the circle include:

A secant of a circle is a line that intersects the circle in two distinct points. In other words, a secant is a line that cuts through a circle. The length of a secant segment between the two points of intersection with the circle is called the length of the secant.

An inscribed angle is the angle formed in the interior of a circle by the intersection of two chords on the circle.

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Image transcribed:

CIRCLES ESCAPE Challenge

Directions A circle is given to the left. Parts of this circle are partially named below. Complete the names by dragging and placing the letters over the empty spaces. Though there could be more than one answer, each letter tile can only be used once

Center

Radius

Diameter

Chord

Secant

Tangent

Central Angle

Inscribed Angle

Minor Arc

Major Arc

The correct answer is: The theoretical probability, P(blue), is 25%, and the experimental probability is 30%

Theoretical probability is a concept in mathematics and statistics that refers to the likelihood or chance of an event occurring based on mathematical reasoning and analysis. Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes in a given situation or experiment. It is based on the assumption that each outcome is equally likely to occur.

The theoretical probability of pulling a blue marble from the bag is the number of blue marbles divided by the total number of marbles:

P(blue) = 10/40 = 1/4 = 25%

The experimental probability of pulling a blue marble from the bag is the number of times a blue marble was pulled divided by the total number of trials:

Experimental probability of pulling a blue marble = 12/40 = 3/10 = 30%

We can see that the experimental probability of pulling a blue marble is slightly higher than the theoretical probability, which is expected since this is based on a limited number of trials and chance variation can occur. However, as the number of trials increases, the experimental probability should converge to the theoretical probability.

The correct answer is: The theoretical probability, P(blue), is 25%, and the experimental probability is 30%.

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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.