Can someone give me the answers in order please

A car was valued at $44,000 in the year 1992. The value depreciated to $15,000 by the year 2006.
A) What was the annual rate of change between 1992 and 2006?
r=---------------Round the rate of decrease to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r=---------------%
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2009 ?
value = $ -----------------Round to the nearest 50 dollars.

Kareem borrowed $7000 from the credit union based on the given interest rate.

We may use the simple interest formula to figure out how much Kareem borrowed:

I = Prt

Where r: yearly interest rate represented as a decimal, P: borrowed principal, I: interest paid, and t: amount of time in years.

We are aware that Kareem borrowed money for 6 years at an interest rate of 6% each year. We also know that he spent $2520 on interest in total. We may determine P using the following formula:

2520 = P0.066

2520 = 0.36P

P = 7000

Kareem thus took out a $7000 loan from the credit union.

In conclusion, we utilized the simple interest formula and the provided data on the interest rate, time, and total interest paid to calculate how much money Kareem borrowed from the credit union. We discovered that he took out a loan for $7000, the principal of which accrued $2520 in interest over the course of six years at an annual rate of 6%.

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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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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

The volume of the shape is 144π cubic inches and the total surface area of the shape is 114π square inches.

To find the volume and total surface area of the shape, we need to first determine the radius of the semicircle.

Since the length of the shape is 12 inches and the base is a semicircle, the diameter of the semicircle is also 12 inches. Therefore, the radius of the semicircle is half the diameter, which is 6 inches.

Now we can use the formula for the volume of a cylinder, which is:

V = (πr^2h)/2

where V is the volume, r is the radius, and h is the height.

Substituting in the values we have:

V = (π(6)^2(8))/2

V = 144π cubic inches

So the volume of the shape is 144π cubic inches.

Next, we can find the total surface area of the shape by adding the area of the semicircle base to the lateral surface area of the cylinder. The formula for the lateral surface area of a cylinder is:

L = 2πrh

where L is the lateral surface area.

Substituting in the values we have:

L = 2π(6)(8)

L = 96π square inches

The formula for the area of a semicircle is:

A = (πr^2)/2

where A is the area of the semicircle.

Substituting in the values we have:

A = (π(6)^2)/2

A = 18π square inches

Adding the lateral surface area and the area of the semicircle base together, we get:

Total surface area = L + A

Total surface area = 96π + 18π

Total surface area = 114π square inches

So the total surface area of the shape is 114π square inches.

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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%.

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

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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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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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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Step-by-step explanation:

When the ball hits the ground , the height = 0

  0 = 96 + 80 t - 16t^2

    Use Quadratic Formula with   a = -16      b = 80      c = 96

             to find  t=  6 seconds

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:

B

Step-by-step explanation:

[tex]\frac{1}{2}[/tex] fathom = 1 yard = 3 feet

then 1 fathom = 2 × 3 feet = 6 feet

1 furlong = 660 feet , then

660 feet ÷ 6 feet = 110

there are 110 fathoms in 1 furlong

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

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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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:

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:

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

24. The price of the motorcycle before tax was $11,500. 25. the instructor earns $28.75 per class after the raise. 26.  the sum that Kay and Shay paid, including tax, was $28.62.

An equation is a mathematical statement that asserts that two expressions are equal. Equations are used to represent relationships and constraints between variables, and are commonly used in many fields of science, engineering, and mathematics.

An equation consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an equals sign (=). Each side of the equation can contain numbers, variables, mathematical operations (such as addition, subtraction, multiplication, and division), and parentheses.

The goal of solving an equation is to determine the value(s) of the variable(s) that make the equation true. This is typically done by applying mathematical operations to both sides of the equation in order to isolate the variable on one side and simplify the expression on the other side.

24. Let the price of the motorcycle before tax be represented by x. Then, we can write an equation based on the given information:

0.125x = 1437.50

Solving for x, we get:

x = 1437.50 / 0.125 = $11,500

Therefore, the price of the motorcycle before tax was $11,500.

25. The instructor now earns 115% of $50, or:

$50 x 1.15 = $57.50

Since this is the amount earned for teaching 2 classes, the amount earned per class after the raise is:

$57.50 / 2 = $28.75

Therefore, the instructor earns $28.75 per class after the raise.

26. To find how much Kay paid, we first need to calculate the tax:

0.08 x $14.50 = $1.16

So the total amount Kay paid was:

$14.50 + $1.16 = $15.66

Similarly, the tax Shay paid was:

0.08 x $12 = $0.96

So the total amount Shay paid was:

$12 + $0.96 = $12.96

Therefore, the sum that Kay and Shay paid, including tax, was:

$15.66 + $12.96 = $28.62

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Answer:

Assuming exponential growth, the number of bacteria in the Petri dish can be modeled by the equation:

N(t) = N0 * e^(kt)

where N(t) is the number of bacteria at time t, N0 is the initial number of bacteria, e is the base of the natural logarithm, k is a constant representing the growth rate, and t is the time elapsed.

We are given that there are 47 bacteria initially, so N0 = 47. After 8 hours, there are 273 bacteria, so we can use this information to solve for k:

273 = 47 * e^(k*8)

Dividing both sides by 47 and taking the natural logarithm of both sides, we get:

ln(273/47) = k*8

Simplifying this expression, we get:

k ≈ 0.53

Now we can use this value of k to find the number of bacteria after 48 hours:

N(48) = 47 * e^(0.53*48)

N(48) ≈ 1.3 * 10^12

Therefore, assuming exponential growth, there would be approximately 1.3 * 10^12 bacteria in the Petri dish after 48 hours.

[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:

  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

Answer:

The least possible degree is 3.

We can mark the points (2, 95) and (4, 145) on the grid and then draw the x-axis and y-axis to plot the points on a coordinate grid.

A linear function is a first-degree mathematical problem with one or more variables set to one and no terms with a degree higher than one. The recipe for a direct condition is y = mx + b, where m indicates the incline and b the y-catch..

given

We may use the slope-intercept form of a linear equation to graph the relationship between the number of hours worked and the fee, which is:

y = mx + b

where m is the hourly rate of $35 and b is the fixed charge of $25, y is the total amount charged, x is the number of hours worked, and m is the hourly rate.

Hence, the equation is:

y = 35x + 25

y = 35(2) + 25 = 95

y = 35(4) + 25 = 145

Hence, the line's two points are (2, 95) and (4, 145).

We can mark the points (2, 95) and (4, 145) on the grid and then draw the x-axis and y-axis to plot the points on a coordinate grid.

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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

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.