I NEED HELP BADLY PLEASE HELP

the simplified expression is 129.

To simplify the expression 8² + 9(12 ÷ 3 × 2) - 7, we follow the order of operations, which is:

Parentheses (do the calculations inside parentheses first)

Exponents (do the calculations involving exponents)

Multiplication and Division (do these operations from left to right)

Addition and Subtraction (do these operations from left to right)

Using these rules, we can simplify the expression step by step as follows:

8² + 9(12 ÷ 3 × 2) - 7 (Apply parentheses first)

= 8² + 9(4 × 2) - 7 (Evaluate 12 ÷ 3 as 4 and then 4 × 2 as 8 inside the parentheses)

= 8² + 9(8) - 7 (Evaluate 9 times 8 as 72)

= 64 + 72 - 7 (Evaluate 8² as 64)

= 129 (Perform the final subtraction and add the remaining terms)

Therefore, the simplified expression is 129.

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Answer:

Let's use algebra to solve this problem.

Let's start by defining some variables to represent the number of cows and chickens the farmer sold. Let c be the number of cows and h be the number of chickens. We know that the farmer sold a total of 11 animals, so:

c + h = 11 (Equation 1)

We also know that the total amount of money the farmer received was $2475. The amount he received from selling cows was $350 times the number of cows, or 350c. The amount he received from selling chickens was $75 times the number of chickens, or 75h. So:

350c + 75h = 2475 (Equation 2)

Now we have two equations with two unknowns. We can solve for one variable in terms of the other in the first equation:

c + h = 11

c = 11 - h

We can substitute this expression for c into the second equation:

350c + 75h = 2475

350(11 - h) + 75h = 2475

Simplifying and solving for h:

3850 - 350h + 75h = 2475

-275h = -1375

h = 5

So the farmer sold 5 chickens. We can substitute this value of h back into the first equation to find c:

c + h = 11

c + 5 = 11

c = 6

Therefore, the farmer sold 6 cows and 5 chickens.

In the following question, among the various parts to solve on houses - 1. price = β0 + β1sqrft + β2bdrms, 2.  β2, 3. β2 + 140β1, 4. R-squared value is provided in the regression output, 5. 276.878 thousand dollars, 6. 23.122.

1. The regression equation of house price in thousands of dollars to the house size measured in square feet (sqft) and the number of bedrooms in the house (bdrms) can be written as follows: price = β0 + β1sqrft + β2bdrms Here, price refers to the house price in thousands of dollars, sqft refers to the house size measured in square feet and bdrms refers to the number of bedrooms in the house.

2. The estimated increase in price for a house with one more bedroom, holding square footage constant is equal to the coefficient of bdrms in the regression equation, which is β2.

3. The estimated increase in price for a house with an additional bedroom that is 140 square feet in size can be calculated as follows: β2 + 140β1. Comparing this to the answer in question two above, we can see that the price increase is greater when an additional 140 square feet are added to the house rather than an additional bedroom.

4. The percentage of the variation in price explained by square footage and the number of bedrooms can be found using the R-squared value. The R-squared value is a measure of how much of the variation in the dependent variable (house price) is explained by the independent variables (sqft and bdrms). In this case, the R-squared value is provided in the regression output.

5. To find the predicted price for the first house in the sample using the model estimated above, we need to plug in the values of sqft and bdrms for the first house into the regression equation. Here, sqrft = 2,438 and bdrms = 4. Thus, the predicted price for the first house is given by: price = β0 + β1sqrft + β2bdrms = -14.973 + 0.128sqrft + 15.204bdrms = -14.973 + 0.128(2,438) + 15.204(4) = 276.878 thousand dollars.

6. The residual for the first house in the sample can be calculated as follows: Residual = Actual price - Predicted price = 300 - 276.878 = 23.122. The fact that the residual is positive suggests that the buyer overpaid for the house.

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Answer:

R466.66...

Step-by-step explanation:

To decrease A in the ratio x : y (where x < y), we have to perform the following

[tex]A*\frac{x}{y}[/tex]   (as a proper fraction)

R700 * 2/3 = R466.66...

Hello,

3m + 5 > 2(m - 7) =

3m + 5 > 2m - 14

3m - 2m > - 14 - 15

x > - 29

Step-by-step explanation:

3m±2m>-14-5

5m>-19

m>-19/5

m>3.8

Answer:

Step-by-step explanation:

To find the normal plane and osculating plane, we first need to find the required derivatives.

x = 5t, y = t^2, z = t^3

dx/dt = 5, dy/dt = 2t, dz/dt = 3t^2

So, the velocity vector v and acceleration vector a are:

v = <5, 2t, 3t^2>

a = <0, 2, 6t>

Now, let's evaluate them at t = 1 since the point (5, 1, 1) is given.

v(1) = <5, 2, 3>

a(1) = <0, 2, 6>

The normal vector N is the unit vector in the direction of a:

N = a/|a| = <0, 1/√10, 3/√10>

Using the point-normal form of the equation for a plane:

normal plane equation = 0(x-5) + 1/√10(y-1) + 3/√10(z-1) = 0

Simplifying this equation we get:

5x + 2y + 3z = 30

The osculating plane can be found using the formula:

osculating plane equation = r(t) · [(r(t) x r''(t))] = 0

where r(t) is the position vector, and x is the cross product.

At t = 1, the position vector r(1) is <5, 1, 1>, v(1) is <5, 2, 3>, and a(1) is <0, 2, 6>.

r(1) x v(1) = <-1, 22, -5>

r(1) x a(1) = <12, -6, -10>

v(1) x a(1) = <-12, 0, 10>

Substituting these values into the formula, we get:

osculating plane equation = (x-5, y-1, z-1) · <12, -6, -10> = 0

Simplifying this equation we get:

3x - 15y + 5z = 5

Therefore, the equations for the normal plane and osculating plane at (5, 1, 1) are:

(a) 5x + 2y + 3z = 30

(b) 3x - 15y + 5z = 5

Therefore, the compound inequality for the diameter of the washers is: 3.150 ≤ d ≤ 3.240.

In mathematics, an inequality is a statement that compares two values or expressions, indicating that one is greater than, less than, or equal to the other. The symbols used to represent inequalities are:

">" which means "greater than"

"<" which means "less than"

"≥" which means "greater than or equal to"

"≤" which means "less than or equal to"

Inequalities can be solved by applying algebraic techniques, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. The solution to an inequality is a range of values that satisfy the inequality.

Here,

The formula for the circumference of a circle in terms of its diameter is:

C = πd

where π (pi) is approximately 3.14.

We are given that the acceptable range for the circumference of the washer is 9.9 ≤ C ≤ 10.2 centimeters. Substituting C = 3.14d into this inequality, we get:

9.9 ≤ 3.14d ≤ 10.2

Dividing all sides of the inequality by 3.14, we obtain:

3.15 ≤ d ≤ 3.24

Rounding to three decimal places, the corresponding interval for the diameters of the washers is:

3.150 ≤ d ≤ 3.240

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Jerry will have $1825 in his account after 3 years and Jerry will have $6125 in his account after 10 year when compounded.

Simple interest is computed just using the principle, which is the initial sum borrowed or put into an investment. The interest rate is constant throughout time and solely applies to the principal sum. Short-term loans or investments frequently employ simple interest.

The given situation can be modeled as a linear equation given by:

y = mx + c

For Jerry we have:

y = 50x + 125

For 3 years = 36 months we can substitute x = 36:

y = 50(36) + 125

y = 1825

For x = 10:

y = 50(120) + 125

y = 6125

Hence, Jerry will have $1825 in his account after 3 years and Jerry will have $6125 in his account after 10 year.

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Let n be a positive integer. We can use the properties of modular arithmetic to calculate this remainder. Let's start with a = (32n + 4)-1 mod 9. We can rewrite this as a = 9 - (32n + 4)-1 because 9 = 0 mod 9.

We can use Fermat's Little Theorem to calculate (32n + 4)-1. This theorem states that (32n + 4)-1 mod 9 = (32n + 4)8 mod 9.

Using the identity (a + b)n mod m = ((a mod m) + (b mod m))n mod m, we can simplify the equation to (32n mod 9 + 4 mod 9)8 mod 9.

32n mod 9 = 0, so (32n mod 9 + 4 mod 9)8 mod 9 = 48 mod 9 = 1.

Finally, a = 9 - 1 = 8 mod 9, so the remainder when a is divided by 9 is 8.

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In order for each of the 25 students in the class to get a full glass of milk during the break, six bottles of milk are required.

Each student in a class of 25 drinks a full 210 millilitre glass of milk, hence the amount of milk consumed overall during the break is:

25 students times 210 millilitres each equals 5250 millilitres.

Milk comes in 1000 millilitre bottles, thus to determine how many bottles are needed, divide the entire amount eaten by the volume of milk in each bottle.

5.25 bottles are equal to 5250 millilitres divided by 1000 millilitres.

We must round up to the nearest whole number because we are unable to have a fraction of a bottle. This results in:

6 bottles in 5.25 bottles

In order for each of the 25 students in the class to get a full glass of milk during the break, six bottles of milk are required.

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In the following question, among the conditions given, It can be given by Ymax = H(N*)So the maximum sustainable yield for a population governed by a Gompertz model and subjected to harvesting (either constant yield or constant effort) can be calculated using the above formula.

To find the maximum sustainable yield for a population governed by a Gompertz model and subjected to harvesting (either constant yield or constant effort) we will start with the formula of the Gompertz model that is, IN = r Nlog(K)Solution: Given that the population is governed by a Gompertz model = r Nlog(K)We know that the harvesting term that is responsible for the overexploitation of the resources, hence we can write the Gompertz model as dN/dt = rN[1 - N/K] - H(x)WhereH(x) can be a constant yield or constant effort in the harvesting term.

To find the maximum sustainable yield, we will use the concept of steady-state or equilibrium population level. Steady-state condition:dN/dt = 0Solving this equation gives us the steady-state population level, say N*.Now we need to calculate the maximum amount of resources that can be harvested sustainably. This maximum amount of resources harvested sustainably is called the maximum sustainable yield (MSY). It can be given by Ymax = H(N*)So the maximum sustainable yield for a population governed by a Gompertz model and subjected to harvesting (either constant yield or constant effort) can be calculated using the above formula.

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The given third-order linear equation is (t - 2t^2)y' - 4y'' = -2t with y(3) = -2, y'(3) = 2, y''(3) = -3.

We can write this equation as a system of first-order linear equations with initial values by introducing three new variables x_1, x_2, and x_3 such that:

x_1 = y

x_2 = y'

x_3 = y''

with initial values x_1(3) = -2, x_2(3) = 2, x_3(3) = -3.

The resulting system of equations is:

x_1' = x_2

x_2' = x_3

x_3' = (2t^2 - t)x_2 - 4x_3 + 2t

This system can be solved numerically for the unknown functions x_1, x_2, and x_3 with the initial conditions given.

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The approximate slope and y-intercept of the line of best fit is y = -0.17x + 33.3.

In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]

Where:

At data point (20, 20), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:

[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - 20 = \frac{(10- 20)}{(80-20)}(x -20)[/tex]

y - 20 = -1/6(x - 20)

y = -x/6 + 20/6 + 20.

y = -x/6 + 200/6

y = -0.17x + 33.3

In this context, we can reasonably infer and logically deduce that a linear function that represent the line of best fit in slope-intercept form is y = -0.17x + 33.3.

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Answer:

y = 90°

x = 63°

Step-by-step explanation:

The unknown y angle is a right angle, meaning it is a 90° angle.

We know a triangle is 180°. We know 2 angles one is 90°, one is 27°, so to find the other missing angle

We Take

180 - (27 + 90) = 63°

So,  x = 63°      y = 90°

The total amount repaid by Melvin was $1,440. Which is option C. $1,440.

Melvin borrowed $1,200 for furniture

His monthly payments were $60 for 24 months

To find the total amount repaid, we can multiply the monthly payment with the number of months. We can write this in mathematical terms as:

The total amount repaid = Monthly payment × Number of months

Using the above formula, let's calculate the total amount repaid by Melvin.

Total amount repaid = $60 × 24= $1,440

Therefore, the total amount repaid by Melvin was $1,440. Therefore, the correct option is C. $1,440.

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The probability of 0.0475 is less than 0.05, so it is unlikely to happen. Hence, it would be unusual if less than 30% of the sampled teenagers owned smartphones.

a) Find the probability that the proportion of the sampled teenagers who own a smartphone is between 0.35 and 0.45b) Find the probability that less than 45% of sampled teenagers own smartphonesc) Would it be unusual if less than 30% of the sampled teenagers owned smartphones?a)Find the probability that the proportion of the sampled teenagers who own a smartphone is between 0.35 and 0.45We know that the proportion of teenagers aged 12-17 who own smartphones is 0.38. Now, we need to find the probability that the proportion of sampled teenagers who own smartphones is between 0.35 and 0.45.The sample size is 120.Let P be the probability that a teenager has a smartphone. The population mean is P = 0.38, while the standard deviation is σ =√((P (1 - P)) /n).P (0.38) and n (120) are given, so σ is calculated as follows:σ=√((P (1 - P)) /n) =√((0.38 (1 - 0.38)) /120) =√((0.38 × 0.62) /120) =0.048Now, let us define the Z scores for both sides, i.e. for 0.35 and 0.45.z1 = (0.35 - 0.38) /0.048 = -0.625z2 = (0.45 - 0.38) /0.048 = 1.458Hence, P(0.35 < p < 0.45) is equal to P(-0.625 < z < 1.458).The probability can be calculated using standard normal tables, which gives 0.5763. Therefore, the probability that the proportion of the sampled teenagers who own a smartphone is between 0.35 and 0.45 is 0.5763.b) Find the probability that less than 45% of sampled teenagers own smartphonesLet P be the probability that a teenager has a smartphone. The population mean is P = 0.38, while the standard deviation is σ =√((P (1 - P)) /n).P (0.38) and n (120) are given, so σ is calculated as follows:σ=√((P (1 - P)) /n) =√((0.38 (1 - 0.38)) /120) =√((0.38 × 0.62) /120) =0.048Now, let us define the Z score as follows:z = (0.45 - 0.38) /0.048 = 1.458We must find the probability that a randomly selected teenager from the sample has less than 0.45 probability of owning a smartphone. This probability can be calculated using a standard normal table. P(z < 1.458) = 0.9265, thus, the probability that less than 45% of sampled teenagers own smartphones is 0.9265.c) Would it be unusual if less than 30% of the sampled teenagers owned smartphones?For this, we need to find the Z-score of 0.30.z = (0.30 - 0.38) /0.048 = -1.667The probability is obtained using a standard normal table. The probability is 0.0475. The probability of 0.0475 is less than 0.05, so it is unlikely to happen. Hence, it would be unusual if less than 30% of the sampled teenagers owned smartphones.

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The distance that the particle has traveled after t=8 seconds is 640 meters.

Distance covered by the particle in the first 8 seconds is given by

S(8) = ∫v(t)dt = ∫[90/t^2+12t+35]dt = [ -90/t + 6t^2 + 35t ]  from 0 to 8S(8) = [-90/8 + 6(8^2) + 35(8)] - [-90/0 + 6(0^2) + 35(0)]S(8) = [360 + 280] - [-Infinity]S(8) = 640 meters.

Given that the velocity of a particle moving along a line is a function of time given by v(t)=[tex]90/t^2+12t+35.[/tex]

The time taken by the particle to travel 8 seconds is t = 8 seconds.

Initial velocity of the particle = v(0) = 90/(0^2) + 12*0 + 35 = 35m/sFinal velocity of the particle = v(8) = 90/(8^2) + 12*8 + 35 = 59.125 m/s

The distance covered by the particle in time t is given by the integral of the velocity of the particle between 0 to 8 seconds.

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The function f(x) = x⁴ - 2x³ + 5 is concave downwards on the interval (0,1) and concave upwards everywhere else.

To determine the concavity of the function f(x) = x⁴ - 2x³ + 5, we need to find its second derivative f''(x).

f(x) = x⁴ - 2x³ + 5

f'(x) = 4x³ - 6x²

f''(x) = 12x² - 12x

The concavity of the function is determined by the sign of its second derivative.

If f''(x) > 0 for all x, the function is concave upwards.

If f''(x) < 0 for all x, the function is concave downwards.

Let's solve for f''(x) = 12x² - 12x:

12x² - 12x = 0

12x(x - 1) = 0

x = 0 or x = 1

When x < 0, f''(x) > 0, so the function is concave upwards.

When 0 < x < 1, f''(x) < 0, so the function is concave downwards.

When x > 1, f''(x) > 0, so the function is concave upwards.

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Answer:

18+m=24, 6

Step-by-step explanation:

You will get the first part by understanding that 24 is the whole and 18 is the part. Part + the other part, m, is the whole. You will then solve this by isolating the variable m, and subtracting 18 on both sides of the equation. Since 24-18=6, that is the final answer.

The solution to the equation √(1 + 25/144) = 1 + x/12 is x = 5/6. This was achieved by simplifying the left side of the equation and isolating x on one side.

To solve the equation √(1 + 25/144) = 1 + x/12, we start by simplifying the left side of the equation. The expression inside the square root can be simplified to (144 + 25)/144 = 169/144. Taking the square root of this fraction gives us √(169/144) = (13/12).

Next, we subtract 1 from both sides of the equation to isolate x on one side: (13/12) - 1 = x/12. This simplifies to 1/12 = x/12.

Finally, we multiply both sides by 12 to solve for x: x = (1/12)*12 = 5/6.

So the solution to the equation √(1 + 25/144) = 1 + x/12 is x = 5/6.

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Solving for Cartesian product n(B), we have n(B) = 72 / 24 = 3.

The Cartesian product is a mathematical operation that takes two sets and produces a set of all possible ordered pairs of elements from both sets.

In other words, if A and B are two sets, their Cartesian product (written as A × B) is the set of all possible ordered pairs (a, b) where a is an element of A and b is an element of B.

For example, if A = {1, 2} and B = {3, 4}, then A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.

By the question.

We know that n (Ax B) represents the number of elements in the set obtained by taking the Cartesian product of sets A and B.

Using the formula for the size of the Cartesian product, we have:

n (Ax B) = n(A) x n(B)

Substituting the given values, we get: 72 = 24 x n(B)

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If triangle ABC has points A(2, -4) B(-3, 1) C(-2, -6) and you perform the following transformations, B' would be located at B' (3, 2).

In Geometry, a reflection over or across the y-axis is represented and modeled by this transformation rule (x, y) → (-x, y). This ultimately implies that, a reflection over or across the y-axis would maintain the same y-coordinate (y-axis) while the sign of the x-coordinate (x-axis) would change from positive to negative or negative to positive.

By applying a reflection over the y-axis to the coordinate of the given point B (-3, 1), we have the following coordinates:

Coordinate B = (-3, 1)   →  Coordinate B' = (-(-3), 1) = (-3, 1).

Next, we would apply a rotation of 90° clockwise as follows;

(x, y)                               →            (y, -x)

Coordinate B' = (-3, 1) → Coordinate B' = (1, (-3)) = (1, 3)

Lastly, we would apply a translation (x + 2, y - 1) as follows:

Coordinate B' = (1, 3) → (1 + 2, 3 - 1) = B' (3, 2).

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The answer is option C: 109 shares. If the broker charges a set free of $7 per transaction. If the shares are worth $7.69/sh and you have $850 to invest 109 shares can you purchase.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

To calculate the number of shares you can purchase, you need to subtract the broker's fee from the total amount you have to invest and then divide that by the share price:

$850 - $7 = $843 (the amount available to purchase shares)

$843 ÷ $7.69/share = 109.48

Since you cannot purchase a fraction of a share, you would be able to purchase 109 shares.

Therefore, the answer is option C: 109 shares.

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Answer:

A. Scale factor is 2

B. Scale factor is 1/2

Step-by-step explanation:

Answer:

Step-by-step explanation:

42

The equation that can be used to find d, the diameter of a new soccer ball, is: |d - 8.65| = 0.05

What is diameter ?

Diameter is the distance across a circle passing through its center, and it is defined as the length of a straight line passing through the center of a circle and connecting two points on the circumference of the circle. In the case of a soccer ball, the diameter is the distance between two opposite points on the surface of the ball, passing through its center.

Radius is a term used in geometry to refer to the distance from the center of a circle to any point on its circumference. It is defined as half of the diameter of a circle. In the case of a soccer ball, the radius is the distance from the center of the ball to its surface. The radius is related to the diameter by the formula:

radius = diameter / 2

The minimum possible diameter of a soccer ball is:

d = 8.65 - 0.05 = 8.60 inches

The maximum possible diameter is:

d = 8.65 + 0.05 = 8.70 inches

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Using trigonometry, the height of the lighthouse from the top of the cliff is approximately 617.38 meters.

Trigonometry is a tool we can utilize to tackle this issue. Let's use "h" to represent the lighthouse's height and "x" to represent the distance between both the boat and the cliff's edge. Next, we have:

tan(24.7º) = h/x (1)

tan(28.4º) = (h + y)/x (2)

where "y" is the height of the cliff.

We want to find "h + y". To eliminate "x", we can use equation (1) to express "x" in terms of "h", and substitute it into equation (2):

x = h/tan(24.7º)

tan(28.4º) = (h + y)/(h/tan(24.7º))

Simplifying and solving for "h", we get:

h = y/tan(28.4º - 24.7º)

Now we can substitute this expression for "h" into equation (1) to find "y":

tan(24.7º) = y/(y/tan(28.4º - 24.7º))

Simplifying and solving for "y", we get:

y = 750 tan(24.7º) / (tan(28.4º - 24.7º))

y ≈ 237.78 meters

Therefore, the height of the lighthouse from the top of the cliff is:

h + y ≈ y/tan(28.4º - 24.7º) + y ≈ 617.38 meters.

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The question is -

A lighthouse is located at the edge of a cliff, as shown. A boat at sea level is 750 meters away from the base of the cliff. The angle of elevation between sea level and the base of the lighthouse measures 24.7º. The angle of elevation between sea level and the top of the lighthouse measures 28.4°. Find the height of the lighthouse from the top of the cliff.

The probability that more heads are tossed using coin A than coin B is 5/16.

The given data is: Coin A is tossed three times and coin B is tossed two times. We have to find the probability that more heads are tossed using coin A than coin B.

P(E) = Number of favorable outcomes/ Total number of possible outcomes

Coin toss:

There are two possible outcomes in a coin toss, Head or Tail. The probability of getting a head in a coin toss is

1/2 = 0.5.

Therefore, the probability of getting a tail in a coin toss is also 1/2 = 0.5.

Let's calculate the possible outcomes when coin A is tossed three times.

There are 2 possible outcomes when one coin is tossed.

Number of possible outcomes when three coins are tossed = 2 * 2 * 2 = 8

Likewise, the possible outcomes when coin B is tossed two times are:

The number of possible outcomes = 2 * 2 = 4

Therefore, the total number of possible outcomes = 8 * 4 = 32

Now, we will find out the cases where the number of heads is more when coin A is tossed three times.

HHH HHT HTH HTT THH THT TTH TTT HHT HTT THT TTT TTH TTT HTT TTT THT TTT TTT TTT

Therefore, the number of times when more heads are obtained when coin A is tossed three times is 10. (We have to exclude the case when there is an equal number of heads.)

Therefore, the required probability is: P = Number of favorable outcomes/ Total number of possible outcomes

P = 10/32P = 5/16

Therefore, the probability that more heads are tossed using coin A than coin B is 5/16.

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The required answers are 19) c = 8√2 units 20) a = b =a = [tex]\frac{\sqrt{10} }{14}[/tex].

A right triangle or right-angled triangle, or more formally an orthogonal triangle, formerly called a rectangled triangle, is a triangle in which one angle is a right angle, i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry.

19) Using Pythagoras theorem

[tex]c^2=a^2+a^2[/tex]

[tex]c^2=\sqrt{a^2+a^2}[/tex]

[tex]c=\sqrt{8^2+8^2}[/tex]

c = 8√2 units

20) It is isosceles triangle,So a = b

Now

[tex]$Sin(45) =\frac{a}{\frac{\sqrt{5}}{7} }[/tex]

a = [tex]\frac{\sqrt{10} }{14}[/tex]

Thus, required answers are 19) c = 8√2 units 20) a = b =a = [tex]\frac{\sqrt{10} }{14}[/tex].

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Calculating the expected value of payout using the probabilities given, we can bet on blue for all 10 races, bet on red for all 10 races and bet on purple for all 10 races

To determine the most efficient bets to increase your money during the 10 rounds, we need to consider the probability and potential payout for each racecar. We can start by calculating the expected value of each racecar's payout:

Blue: 0.3 x 2 = 0.6

Red: 0.25 x 2.5 = 0.625

Purple: 0.18 x 3 = 0.54

Yellow: 0.1 x 5 = 0.5

Black: 0.08 x 7 = 0.56

Green: 0.05 x 10 = 0.5

Orange: 0.03 x 15 = 0.45

Pink: 0.01 x 50 = 0.5

Based on the expected value of payout, the most efficient bets would be on the racecars with the highest expected value. However, we also need to consider the probability of each racecar winning or placing in the top 3.

To maximize our chances of winning, we should bet on the racecars with the highest probability of winning or placing in the top 3, even if their expected payout is not the highest.

Assuming that if one racecar is likely to win but doesn't win, they are statistically most likely to come second, and so on, we can rank the racecars in terms of their probability of winning or placing in the top 3:

1. Blue - 30% chance of winning

2. Red - 25% chance of winning

3. Purple - 18% chance of winning

4. Black - 8% chance of winning

5. Yellow - 10% chance of winning

6. Green - 5% chance of winning

7. Orange - 3% chance of winning

8. Pink - 1% chance of winning

Based on this ranking, the most efficient bets would be as follows:

1. Bet on Blue for all 10 races. The expected payout is not the highest, but the probability of winning is the highest at 30%.

2. Bet on Red for all 10 races. The probability of winning is the second-highest at 25%.

3. Bet on Purple for all 10 races. The probability of winning is the third-highest at 18%.

Betting on these three racecars will give you the highest chance of winning or placing in the top 3, and therefore, the most efficient bets to increase your money. However, keep in mind that gambling always carries risks, and there is no guarantee of winning.

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