If the smith children are served randomly how likely is it that the two oldest smith children are served before the others ?

The probabilities are given as follows:

a) Square: 1/6.

b) Not the triangle: 43/48.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The total area of the figure is given as follows:

12 x 8 = 96 units². (rectangle).

The area of the square is given as follows:

4² = 16 units² (square of the side lengths).

Hence the probability of the square is given as follows:

p = 16/96

p = 1/6.

The area of the triangle is given as follows:

A = 0.5 x 4 x 5 = 10 units². (half the multiplication of the side lengths).

Hence the complement of the area of the triangle is of:

96 - 10 = 86 units².

And the probability of the complement is of:

86/96 = 43/48.

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The original price of the product was approximately $98.00.

To determine the original price before the 15.5% decrease, we can use the following equation:

Original price - (15.5% of original price) = $82.81

Let's solve for the original price:

Original price - (0.155 * Original price) = $82.81

Simplifying the equation:

0.845 * Original price = $82.81

Dividing both sides by 0.845:

Original price = $82.81 / 0.845

Calculating the original price:

Original price ≈ $98.00

Therefore, the original price of the product was approximately $98.00.

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The volume of water remaining in Abigail’s cup can be found to be 25. 14 cm³ .

First, find the volume of water in the cup when it is full. This would be the volume of the cup which is the formula of the volume of a cone :

Volume = ( 1 / 3 ) × π × r² × h

Volume = ( 1 / 3 ) × π × ( 3 cm )² × ( 8 cm )

Volume = 24π cm³

If Abigail too 2 / 3 to slosh on her face, the amount of water left would be :

= 24π cm³ - ( 1 - 2 / 3 )

= 24π cm³ - 1 / 3

= 8π cm³

= 25. 14 cm³

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Answer:

380.1 square kilometers

Step-by-step explanation:

The value of   logarithm log5 5/3 is approximately equal to 0.3174.

Using the approximation of ≈ 0.4307 for log5 2 and ≈ 0.6826 for log5 3, we can approximate the value of log5 5/3 by subtracting the two approximations.

log5 5/3 = log5 5 - log5 3 ≈ 1 - 0.6826 ≈ 0.3174


To explain further, logarithms are a way to express the relationship between exponential growth or decay and the input values. In this case, we are using the base of 5 to represent the exponent and trying to find the logarithm of 5/3.

By using the approximation values of log5 2 and log5 3, we can estimate the value of log5 5/3 by subtracting the two approximations. This approximation is useful in situations where we need a quick estimate of a logarithmic function without having to do complex calculations.

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The sale price of the product would be Birr 105.80.

If the cost of the product is Birr 92 and the company has a policy of 15% mark-up on the cost, then the sale price can be found by adding 15% of the cost to the cost itself.

To calculate this, we can use the formula:

Sale price = Cost + Mark-up

where the mark-up is 15% of the cost.

Mark-up = 15% of Cost = 0.15 * 92 = Birr 13.80

So, the sale price = Cost + Mark-up = 92 + 13.80 = Birr 105.80.

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The standard deviation in the first sample was $20,00 and the standard deviation in the second sample is $18,000 so the agent's claim cannot be rejected at the 0.05 level of significance.

To test the agent's claim, we can perform a two-sample t-test with a significance level of 0.05. The null hypothesis is that there is no difference in the mean salaries of safeties and linebackers, while the alternative hypothesis is that there is a difference.

We can calculate the t-statistic using the formula:

t = (x1 - x2) / sqrt(s1²/n1 + s2²/n2)

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Plugging in the given values, we get:

t = (501580 - 513360) / sqrt((20000²/15) + (18000²/15))

t = -1.2605

Using a t-distribution table with 28 degrees of freedom (15 + 15 - 2), we find that the critical value for a two-tailed test at a significance level of 0.05 is approximately ±2.048.

Since the absolute value of the calculated t-statistic (1.2605) is less than the critical value (2.048), we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that there is a difference in the mean salaries of safeties and linebackers in the NFL.

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The change to move the graph down 3 units is given as follows:

the value of b to 2.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

The function in this problem is given as follows:

y = -x + 5.

Moving the graph down 3 units, we subtract by three, hence:

y = -x + 5 - 3

y = -x + 2.

Meaning that the value of b is of b = 2.

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Answer:

[tex]\frac{-5}{34}[/tex]

Step-by-step explanation:

[tex]\frac{5}{12} * \frac{-6}{17}[/tex]  = [tex]\frac{-30}{204}[/tex]

We can simplify.

[tex]\frac{-15}{102}[/tex]  ⇒ Divided both by 2

[tex]\frac{-5}{34}[/tex]  ⇒  Divided both by 3

[tex]\frac{-5}{34}[/tex] is the final answer

Kearney's retirement savings when he retired at age 68, assuming monthly compounding of interest, was $429,336.69.

To calculate Kearney's retirement savings at age 68, we need to use the formula for the future value of an annuity due, which is:

FV = PMT x [((1 + r/n[tex])^(n*t)[/tex] - 1) / (r/n)] x (1 + r/n)

Where:

Plugging in the numbers, we get:

FV = 200 x [((1 + 0.055/12[tex])^(12*38)[/tex] - 1) / (0.055/12)] x (1 + 0.055/12)

FV = $429,336.69

Therefore, Kearney's retirement savings at age 68 would be approximately $429,336.69, assuming he invested $200 per month in securities with an average annual return of 5.5% and monthly compounding of interest. It's important to note that this calculation assumes that Kearney did not withdraw any money from his retirement savings during the 38-year period. Additionally, the actual value of his retirement savings could be different based on fluctuations in the market and any fees or taxes associated with his investments.

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Answer:

x = -4, y = 3.

Step-by-step explanation:

Substitute y = 3 into the first equation:

-5x - 6(3) = 2

-5x = 2 + 18

-5x = 20

x = -4

The rectangles with dimensions 1 x 12 and 2 x 6 do not have a perimeter of 14 inches. These rectangles have perimeters of 26 inches and 16 inches, respectively.

To determine which rectangles with an area of 12 square inches do not have a perimeter of 14 inches, we will first find the possible dimensions of the rectangles and then calculate their perimeters.

1. Since the area of a rectangle is given by length x width, let's find the factors of 12:
  - 1 x 12
  - 2 x 6
  - 3 x 4
2. Now, let's calculate the perimeters for each of these rectangles using the formula 2(length + width):
  - For the 1 x 12 rectangle, the perimeter is 2(1+12) = 2(13) = 26 inches
  - For the 2 x 6 rectangle, the perimeter is 2(2+6) = 2(8) = 16 inches
  - For the 3 x 4 rectangle, the perimeter is 2(3+4) = 2(7) = 14 inches
The rectangles with dimensions 1 x 12 and 2 x 6 do not have a perimeter of 14 inches. These rectangles have perimeters of 26 inches and 16 inches, respectively.

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The number that is equal to the place values, 7 hundred thousands 4 thousands 3 tens and 6 ones, is 704,036

From the question, we are to determine the number that is equal to the given place values

From the given information, the given place value is

7 hundred thousands 4 thousands 3 tens and 6 ones

Now, we will write each of the values in figures

7 hundred thousands = 700,000

4 thousands = 4,000

3 tens = 30

6 ones = 6

To determine the number that is equal to the place values, we will sum all the digits

700,000 + 4,000 + 30 + 6

704,036

Hence,

The number that is equal to the place value is 704,036

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The equation of the curve is y = x2 + 4.

To solve this problem, we need to use the fact that the curve y = ax2 + bx + c passes through the point (2, 12) and is tangent to the line yat at the origin.

First, we know that the tangent to the curve at the origin is the line y = 0x + c = c. Since the curve is tangent to this line at the origin, we know that the derivative of the curve at x = 0 is equal to 0.

Taking the derivative of y = ax2 + bx + c, we get y' = 2ax + b. Setting x = 0, we get y' = b. Since y' = 0 at x = 0, we know that b = 0.

So now we have y = ax2 + c. We can use the fact that the curve passes through the point (2, 12) to solve for a and c.

Substituting x = 2 and y = 12 into the equation y = ax2 + c, we get 12 = 4a + c.

Since we know that a = 4, 6, or 1, we can substitute each of these values into the equation and solve for c.

When a = 4, we get 12 = 4(4)(2) + c, which simplifies to 12 = 32 + c. Solving for c, we get c = -20.

When a = 6, we get 12 = 4(6)(2) + c, which simplifies to 12 = 48 + c. Solving for c, we get c = -36.

When a = 1, we get 12 = 4(1)(2) + c, which simplifies to 12 = 8 + c. Solving for c, we get c = 4.

So the values of a, b, and c are:
a = 1
b = 0
c = 4

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The probability of a defect in the manufacturing process, assuming that the weight of the products follows a normal distribution, is 0.1587 to four decimal places.

To calculate the probability of a defect, we first need to calculate the z-score of the weight that would classify the product as a defect. The z-score is a measure of how many standard deviations a value is from the mean. In this case, the z-score is -1 or 1, depending on whether the weight is less than one standard deviation below the mean or greater than one standard deviation above the mean.

Once we have calculated the z-score, we can use a standard normal distribution table or a calculator to find the probability of a product being classified as a defect. If the z-score is -1, the probability of a product being classified as a defect is 0.1587. If the z-score is 1, the probability of a product being classified as a defect is also 0.1587.

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The volume of the cylinder with a height of 6 inches and a diameter of 8 inches is 904.78 cubic inches.

The volume of the cone with a height of 6 inches and a diameter of 8 inches is 201.06 cubic inches.

The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. Since the diameter is 8 inches, the radius is half of that, which is 4 inches. So, the volume of the cylinder is:

V = π(4)²(6)

V = π(16)(6)

V = 96π

V ≈ 301.59 cubic inches (rounded to two decimal places)

The formula for the volume of a cone is V = (1/3)πr²h. Again, since the diameter is 8 inches, the radius is 4 inches. So, the volume of the cone is:

V = (1/3)π(4)²(6)

V = (1/3)π(16)(6)

V = (1/3)(96π)

V ≈ 100.53 cubic inches (rounded to two decimal places)

However, since the problem only asked for the diameter and not the radius, we can simplify the calculations by using the formula for the volume of a cylinder with diameter D directly, which is:

V = π(D/2)²h

V = π(8/2)²(6)

V = π(4)²(6)

V = 16π(6)

V ≈ 301.59 cubic inches (rounded to two decimal places)

Similarly, we can use the formula for the volume of a cone with diameter D directly, which is:

V = (1/3)π(D/2)²h

V = (1/3)π(8/2)²(6)

V = (1/3)π(4)²(6)

V = (1/3)(16π)(6)

V ≈ 100.53 cubic inches (rounded to two decimal places)

Thus, the main answer is the volume of the cylinder is 904.78 cubic inches and the volume of the cone is 201.06 cubic inches, both rounded to two decimal places.

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The derivative of the given function is f'(x) = 2(sec x * tan x) + sec^2 x. b) The critical numbers for the function are x = 0 and x = π.of the given function is f'(x) = 2(sec x * tan x) + sec^2 x.

The critical numbers for the function are x = 0 and x = π.

Derivative and critical numbers,

a) Find the derivative: We're given the function f(x) = 2 sec x + tan x.

To find its derivative, we need to find the derivatives of the individual terms (sec x and tan x) and then add them together.

The derivative of sec x is sec x * tan x. So, for the term 2 sec x, the derivative is 2 * (sec x * tan x).

The derivative of tan x is sec^2 x.

Now, we add both derivatives to find the derivative of f(x): f'(x) = 2(sec x * tan x) + sec^2 x

b) Find the critical numbers: Critical numbers are the points where the derivative of the function is either 0 or undefined.

To find the critical numbers, we'll set f'(x) equal to 0 and solve for x, as well as identify where the derivative is undefined.

First, let's set f'(x) to 0: 0 = 2(sec x * tan x) + sec^2 x

We need to solve this equation for x. It's a bit tricky, so let's rewrite the equation in terms of sin and cos: 0 = 2((1/cos x) * (sin x/cos x)) + (1/cos x)^2

Now let's simplify the equation: 0 = 2(sin x/cos^2 x) + 1/cos^2 x

To eliminate the denominators, we'll multiply through by cos^2 x: 0 = 2(sin x) + cos x

Now, we can use the unit circle to find the values of x in the interval 0 ≤ x ≤ 2π that satisfy this equation: For sin x = 0, x = 0, π For cos x = -2, there's no solution in the given interval because the range of cosine is -1 ≤ cos x ≤ 1.

Therefore, the critical numbers are x = 0 and x = π. Your answer:

a) The derivative of the given function is f'(x) = 2(sec x * tan x) + sec^2 x.

b) The critical numbers for the function are x = 0 and x = π.

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The number of bracelets that can be made using all the colors one time only is 720.

Given that Diana is making bracelet with 6 different colors we need to find the number of bracelets that can be made using all the colors one time only,

Since there are 6 beads so, the number of bracelets can be made = 6!

= 720

Hence the number of bracelets that can be made using all the colors one time only is 720.

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The measure of angle B, rounded to the nearest degree, is 37 degrees.

In trigonometry, the tangent function (tan) relates the ratio of the length of the side opposite an angle to the length of the side adjacent to that angle in a right triangle.

To find the measure of angle B, we use the inverse tangent function (arctan) with the given tangent value of 3/4:

B = arctan(3/4)

Using a calculator or a trigonometric table, we find that arctan(3/4) is approximately 36.87 degrees. Round the result to the nearest degree to obtain the final measure of angle B.

Therefore, the measure of angle B, rounded to the nearest degree, is 37 degrees.

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a) To determine if the limit exists, we need to check if the limit from all directions approaching (0,0) are equal. Let's approach (0,0) along the x-axis first, so y = 0:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim x -> 0 [(x) + 2(0) + 3(x)(0)] = lim x -> 0 x = 0

Next, let's approach (0,0) along the y-axis, so x = 0:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim y -> 0 [(0) + 2(y) + 3(0)(y)] = lim y -> 0 2y = 0

Now, let's approach (0,0) along the line y = mx, where m is some constant:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim x -> 0 [(x) + 2(mx) + 3(x)(mx)]
= lim x -> 0 [(1+3m)x + 2mx^2]
= 0 if m=0, and DNE (does not exist) for all other values of m.

Since the limit is not equal from all directions, the limit DNE at (0,0).

b) To determine if the limit exists, we need to check if the limit from all directions approaching (0,0) are equal. Let's approach (0,0) along the x-axis first, so y = 0:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim x -> 0 [(2.4) + (0)] = 2.4

Next, let's approach (0,0) along the y-axis, so x = 0:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim y -> 0 [(2.4) + (y)] = 2.4

Now, let's approach (0,0) along the line y = mx, where m is some constant:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim x -> 0 [(2.4) + (mx)]
= 2.4 if m=0, and DNE (does not exist) for all other values of m.

Since the limit is equal from all directions, the limit exists and is equal to 2.4 at (0,0).

c) To determine if the function is continuous at (0,0), we need to check if the limit as (x,y) approaches (0,0) of f(x,y) exists and is equal to f(0,0).

Let g(x,y) = sqrt(x^2 + y^2), which satisfies 0 <= |(x,y)| <= g(x,y) for all (x,y). We have:

|f(x,y)| = |(x+y)/(4+x^2+y^2)| <= |(x+y)/4| <= (1/4)g(x,y)

So, we can bound f(x,y) by (1/4)g(x,y). By the sandwich (or squeeze) theorem, we have:

lim (x,y)->(0,0) (1/4)g(x,y) = 0

Thus, by the sandwich theorem, we have:

lim (x,y)->(0,0) f(x,y) = 0

Since the limit exists and is equal to f(0,0) = 0, the function is continuous at (0,0).

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Answer:

The answer is -84,484

Step-by-step explanation:

using Bodmas

multiplication first

5774+252-(2586×35)

5774+252-90510

6026-90510

-84,484

The expected amount of time that the 5:30 appointment spends at the barber shop is,  E[W] = 45 + 45/e.

Given that, the barber has scheduled two appointments, one at

5 pm and the other at 5:30 pm.

Since the amount of time that appointments last are independent exponential random variables with mean 45 minutes.

Let W be the time the 2nd person has to wait in chamber Let X be the time the barber takes checking 1st person X-exp(45)

The distribution is,

W= X-45            if X >45

otherwise.

Expected time 2nd person spends in barber chamber

= E (W)+45

[ 45 is the mean time barber takes checking 2nd person]

[tex]E(W) = \int\limits^{\infinity }_0 {WP(X=45+W)} \, dw\\ \\\\=\int {W.1/45e^{\frac{-45+w}{45} } \, dw\\\\[/tex]

[tex]=e^{-1} \int\frac{W}{45} e^{\frac{-w}{45} } dw\\=\frac{45}{e}[/tex]

The expected amount of time that the 5:30 appointment spends at the barber's office is,

[tex]E[W]=45+\frac{45}{e}[/tex].

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The 95% confidence interval for the population mean of the division is (49.08, 56.92).

We can use the formula for the confidence interval for a population mean:

CI = [tex]\bar{X}[/tex] ± z*(σ/√n)

where [tex]\bar{X}[/tex] is the sample mean, z is the z-score for the desired confidence level (95% in this case), σ is the population standard deviation (which we assume to be equal to the sample standard deviation), and n is the sample size.

In this problem, [tex]\bar{X}[/tex] = 53, σ = 10, n = 25, and the z-score for a 95% confidence level is 1.96 (from a standard normal distribution table).

Plugging in these values, we get:

CI = 53 ± 1.96*(10/√25) = 53 ± 3.92

Therefore, the 95% confidence interval for the population mean of the division is (49.08, 56.92).

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The length of the diagonal of the square is 22 meters.

A square is a four-sided two-dimensional geometric shape in which all sides are equal in length and all angles are right angles (90 degrees).It is a unique instance of a rectangle with equal sides. The opposite sides of a square are parallel to each other and the diagonals bisect each other at right angles.

A square is divided into two 45-45-90 triangles by its diagonal.

In a 45-45-90 triangle, the hypotenuse (the side opposite the right angle) is √2 times as long as each leg.

Therefore, in this square, the length of the diagonal (d) can be found by multiplying the length of one side (s) by √2:

d = s√2

In this case, the side length of the square is 11√2 meters, so:

d = 11√2 × √2 = 11 × 2 = 22 meters

Therefore, the length of the diagonal of the square is 22 meters.

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Answer:

Amal's mother was 11.4 years old when Amal was born.

Step-by-step explanation:

Let's start by using variables to represent the ages of each person:

From the problem, we know:

Substituting the first three equations into the fourth, we get:

[tex]\sf:\implies A + 0.5A + 3A + 4(3A) = 94[/tex]

Simplifying:

[tex]\sf:\implies A + 0.5A + 3A + 12A = 94[/tex]

[tex]\sf:\implies 16.5A = 94[/tex]

[tex]\sf:\implies A = 5.7[/tex]

So Amal is 5.7 years old. To find the age of Amal's mother when Amal was born, we need to subtract Amal's age from his mother's age:

[tex]\sf:\implies M - A = 3A - A = 2A[/tex]

So Amal's mother was 2A = 2(5.7) = 11.4 years old when Amal was born.

Answer:

a. n < 14

b.  n ≥ 14

Step-by-step explanation:

a.

We see the line to the left of 14, meaning it will be smaller than 14. So, the inequality is n < 14

b.

The line goes to the right of 14, meaning it will be bigger than 14. This has a close circle meaning there will be an equal sign. So, the inequality is n ≥ 14

The cost of a flight from Hong Kong to Tokyo with Galois Airways is 1429.99 HKD.

We start by calculating the Porson's product-moment correlation coefficient between the distance and airfare data. The value of the correlation coefficient ranges from -1 to +1. A value of -1 indicates a perfect negative correlation, 0 indicates no correlation, and +1 indicates a perfect positive correlation.

In this case, the correlation coefficient between distance and airfare for Galois Airways flights is 0.948, indicates a strong positive correlation between the distance and airfare.

The regression line is expressed as:

y = a + bx

where y is the dependent variable (airfare), x is the independent variable (distance), a is the intercept (the value of y when x is zero), and b is the slope (the change in y for a one-unit change in x).

The regression equation for Galois Airways flights is:

y = 553.51 + 0.292x

Now, we can use the regression equation to estimate the cost of a flight from Hong Kong to Tokyo, which is 2900 km away.

y = 553.51 + 0.292(2900) = 1429.99 HKD

Therefore, we estimate that the cost of a flight from Hong Kong to Tokyo with Galois Airways is 1429.99 HKD.

Finally, we need to explain why it is valid to use the regression equation to estimate the airfare between Hong Kong and Tokyo. We can do this by examining the assumptions of linear regression. The two main assumptions are that there is a linear relationship between the variables, and that the residuals (the differences between the actual and predicted values) are normally distributed with constant variance.

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Complete question is  Galois Airways has flights from Hong Kong International Airport to different destinations. The following table shows the distance, x kilometres, between Hong Kong and the different destinations and the corresponding airfare, y, in Hong Kong dollars (HKD) Destination Bali, Sydney, Bengaluru. Auckland, Bangkok, Indonesia Australia India Singapore New Thailand Zealand 3400 7400 4000 2600 9200 1700 Distance x, (km Airfare y, (HKD) 1550 3600 2800 1300 4000 1400 The Porson's product-moment correlation coefficient for this data is 0.948, correct to three significant figures. Use your prophio display calculator to find the equation of the regression line y on x.  b. The distance from Hong Kong to Tokyo is 2900 km. Use your regression equation to estimate the cost of a flight from Hong Kong to Tokyo with Calois Airways. c. Explain why it is valid to use the regression equation to estimate the airfare between Hong Kong and Tokyo.

The human heart beats approximately 4.1975 x 10⁸ times in one year and it expressed in scientific notation.

According to the question, the average human heart beats 1.15 x 10⁵ times a day. We need to find out how many times the heart beats in one year, which is 3.65 x 10² days.

To calculate the total number of heartbeats in one year, we can multiply the number of heartbeats in a day by the number of days in a year. Therefore, we have:

Total number of heartbeats in one year = 1.15 x 10⁵ beats/day x 3.65 x 10² days/year

= (1.15 x 3.65) x (10⁵ x 10²) beats/year

= 4.1975 x 10⁸ beats/year

This number may seem large, but it is necessary for the heart to pump blood throughout the body to keep us alive and healthy.

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The company will have a Loss of $194,060

The lady paid premiums for 6 years, which amounts to a total premium of$ 5,940($ 990 * 6).

Still, the insurance company will pay the face value of the policy, which is   $ 00, If she dies.

Thus, the company's total payout would be $200,000, while their total income would be $ 5,940 in premiums.

The loss for the company would be the difference between the payout and the income

200,000-$ 5,940 = $ 194,060

Thus, the insurance company's loss in this scenario would be $194,060.

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