Question 16 (6 marks) If b and c are real numbers and b^2 <3c, show that the equation x^3 + bx^2 + cx = 2022 has exactly one real solution.

Answer:

There is a cylinder with a height of 6 centimeters and a radius of r1, which has a volume of 302 cubic centimeters. It's fascinating to see how the volume of different shapes can vary based on their dimensions.

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Using  net worth statement, Andre's net worth is $159,573.

Andre's net worth can be calculated by subtracting his total liabilities from his total assets.

Total assets = $155,000 (condominium) + $8,100 (stock portfolio) + $9,100 (car value) + $2,600 (bank account) = $174,800

Total liabilities = $3,300 (car loan) + $7,600 (student loans) + $4,327 (credit card balance) = $15,227

Net worth = Total assets - Total liabilities = $174,800 - $15,227 = $159,573

Therefore, Andre's net worth is $159,573.

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Let's label the 3 points as A;B;C, where BC is the hypotenuse and AB is the shortest side, and x as AH.

We can see that triangle AHC is similar to triangle CAB, and triangle ACH is similar to triangle BCA, so therefore triangle AHC is similar to triangle CHA (i might've messed up the order but you get it)

Because they are similar, we get the ratio : x/12 = 40/x, or x^2 = 480.

So, x = [tex]\sqrt{480}[/tex], or 21.9 (1 d.p)

Answer:

-----------------------

According to the right triangle altitude theorem:

We see x is the altitude and 12 & 40 are the segments formed on the hypotenuse.

Applied to the given triangle, we get following equation:

The intermediate step in completing the square for x^2= -16x-37 is (x+8)^2=27.

To complete the square for the given equation, we need to add a constant value to both sides of the equation such that we can factor the left-hand side as a perfect square.

x^2 + 16x = -37

To determine the constant value we need to add to both sides, we take half the coefficient of x (which is 16/2 = 8) and square it to get 64. Then we add 64 to both sides of the equation:

x^2 + 16x + 64 = 27

Now we can factor the left-hand side as a perfect square:

(x + 8)^2 = 27

So the intermediate step in completing the square for x^2= -16x-37 is (x+8)^2=27.

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Answer:

36!

Step-by-step explanation:

There is 3 brands of clothing each brand has short and long sleeve, and both come in 6 different colors.

multiply 6*2 (because 6 different colors per long and short sleeve) get 12, then you multiply 12 times 3 because each one is from a different brand.

IT MAY BE A BI CONFUSING I SUCK AT EXPLAINING BUT

YEAHHH!

Answer:

18shirts

Step-by-step explanation:

3×6=18 shirts

The length of the trail is determined as 1.3 miles.

The length of the trail is calculated as follows;

The circumference of the circle is calculated as;

S = πd

where;

S = π x 2 ft

S = 2π ft

I revolution = 1 circumference = 2π ft

1 rev = 2π ft

1,100.5 rev = ?

= 1,100.5 rev/rev x 2π ft

= 6,914.65 ft

5280 ft -------> 1 mile

6,914.65 ft ------> ?

= 1.3 miles

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The volume of the obtained solid is 36 units³.

Given that a rectangle of dimension 9 units x 2 units, has been rotated to form a solid we need to find its volume,

So we know that a rectangle rotated to form a rectangular prism.

Volume of a rectangular prism = product of the dimensions.

The dimensions of the obtained solid will be 9 units x 2 units x 2 units,

So the volume = 9 x 2 x 2 = 36 units³

Hence the volume of the obtained solid is 36 units³.

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The expression (x + 1)(x - 3) represents the Area of base of the prism.

a crystal is a polyhedron containing a n-sided polygon base, a respectable halfway point which is a deciphered duplicate of the first, and n different countenances, fundamentally all parallelograms, joining relating sides of the two bases. Translations of the bases exist in every cross-section that runs parallel to the bases.

The volume of a rectangular prism is given by the formula V = Bh, where B is the area of the base and h is the height of the prism. In this case, the base is a square with edge length (x + 1), so its area is (x + 1)^2. The volume of the prism is given as (x + 1)^2(x - 3).

We can find the height of the prism by dividing the volume by the area of the base:

B = V/h = (x + 1)^2(x - 3)/(x + 1) = (x + 1)(x - 3)

Therefore, the expression (x + 1)(x - 3) represents the Area of base of the prism.

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The total volume of juice in the 5 glasses is 925 milliliters. To find the total volume of juice in the 5 glasses, we simply need to multiply the volume of juice in one glass by the number of glasses.

In this case, each glass contains 185 milliliters of juice and Ian has stacked 5 glasses, so we can use the formula:
Total volume of juice = Volume of juice per glass x Number of glasses
Plugging in the numbers, we get:
Total volume of juice = 185 ml/glass x 5 glasses
Total volume of juice = 925 ml

Therefore, the total volume of juice in the 5 glasses is 925 milliliters. This is a simple example of using multiplication to find the total quantity of something when we know the amount in one unit and the number of units. In this case, we were able to find the total volume of juice by multiplying the volume in one glass by the number of glasses.

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A number cube has sides numbered 1 through 6. the probability of rolling a 2 is 1/6. The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 is 5/6 (in simplest form). The correct answer is 5/6.

The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 can be calculated as follows:
Determine the total number of outcomes, which is 6 (1, 2, 3, 4, 5, and 6).
Determine the number of favorable outcomes, which is 5 (1, 3, 4, 5, and 6), since you're looking for the probability of not rolling a 2.
Calculate the probability by dividing the number of favorable outcomes by the total number of outcomes.
The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 is 5/6 (in simplest form).

So, the correct answer is 5/6.

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Answer:

Step-by-step explanation:

There are 8 equally likely sandwich options: rye and ham, rye and turkey, white and ham, white and turkey, rye and ham with cheese, rye and turkey with cheese, white and ham with cheese, white and turkey with cheese.

Since each type of sandwich was made in equal number, there are 4 sandwiches with cheese.

Therefore, the chances that Mary will get a sandwich with cheese is 4/8 or 1/2.

Answer: one half

If and only if the alternating sum of the two-digit chunks of a number n is divisible by m, then n is divisible by m.

Let's denote the two-digit chunks of n as a₁a₂, a₃a₄, ..., where a₁, a₂, a₃, a₄, ... are the digits from the ones place onward.

The alternating sum of these two-digit numbers is given by a₁a₂ - a₃a₄ + a₅a₆ - a₇a₈ + ...

We can rewrite n as (a₁a₂ × 100) + (a₃a₄ × 10) + (a₅a₆ × 1) + ...

The alternating sum expression can be written as (a₁a₂ × 100) - (a₃a₄ × 10) + (a₅a₆ × 1) - ...

If n is divisible by m, we have n ≡ 0 (mod m).

Rewriting n in terms of the alternating sum, we get (a₁a₂ × 100) - (a₃a₄ × 10) + (a₅a₆ × 1) - ... ≡ 0 (mod m).

Factoring out each two-digit chunk, we have (a₁a₂ - a₃a₄ + a₅a₆ - ...) ≡ 0 (mod m).

This implies that n is divisible by m if and only if the result of the alternating sum process is divisible by m.

m is the value that makes the alternating sum of the two-digit chunks of n a divisibility test for n.

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The probability of the first toy being defective is 1/3 and the probability of both toys being defective is 1/15.

Hence, option 'a'.

Use the concept of probability defined as:

Probability is equal to the percentage of positive outcomes compared to all outcomes for the occurrence of an event.

Given that,

Mrs. Jones buys her son two toys.

The probability that the initial plaything is broken =  1/3.

The probability that the second toy has a problem.,

Also given the original toy is defective, is 1/5.

The goal is to calculate the probability that both toys are defective.

To calculate the probability that both toys are defective:

Multiply the individual probabilities.

Given that the probability of the first toy being defective = 1/3,

The probability of the second toy being defective given that the first toy is defective= 1/5,

Calculate the probability that,

Both toys have flaws, multiplying these probabilities together:

(1/3) x (1/5) = 1/15

Hence,

The probability that both toys are defective is [tex]1/15[/tex] which is option 'a'.

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The complete question is:

Mrs. Jones buys two toys for her son. the probability that the first toy is defective is 1/3, and the probability that the second toy is defective given /that the first toy is defective is 1/5. what is the probability that both toys are defective?

a.  1/15

b. 1/4

c. 1/8

d. 3/5

Answer:

Step-by-step explanation:

To add fractions with unlike denominators, we need to find the least common multiple of the denominators. In this case, the least common multiple of 3 and 4 is 12. So, we must find equivalents of each fraction with a denominator of 12.

To add fractions with unlike denominators, we need to find the least common multiple of the denominators. In this case, the least common multiple of 3 and 4 is 12. So, we must find equivalents of each fraction with a denominator of 12.

Multiplying the numerator and denominator of 2/3 by 4, we get 8/12. On the other hand, multiplying the numerator and denominator of 1/4 by 3, we get 3/12.

Therefore, we can rewrite the sum as:

8/12 + 3/12

And adding the numerators, we get:

11/12

So, 2/3 + 1/4 = 11/12.

The final answer is a proper fraction, which means that the numerator is less than the denominator, and can be further simplified if necessary.

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A piggy bank is a small container typically used by children to save money. In this scenario, Todd had a piggy bank holding $384 and began taking out money each month. The table provided shows the amount remaining in the piggy bank, in dollars, after each of the first four months. This information can be used to track Todd's spending and savings habits.

In the first month, Todd took out $60, leaving him with $324 in his piggy bank. In the second month, he took out an additional $48, leaving him with $276. By the third month, Todd had taken out a total of $105, leaving him with $279 in his piggy bank. Finally, in the fourth month, he took out $62, leaving him with $217.

By tracking his spending and savings over the course of these four months, Todd can assess his financial habits and make any necessary adjustments. It is important for individuals to develop good financial habits early on in life, and using a piggy bank can be a fun and effective way to do so.

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we find that the solutions to the equation are:x ≈ -1.283, x ≈ 4.283.

To solve the equation 2x(x+0.5) = -(x+2.5)² + 5.5 by graphing, we can first rearrange it to the standard form:

x² - 3x - 2 = 0.5(x+2.5)²

Then we can plot the two functions y = x² - 3x - 2 and y = 0.5(x+2.5)² - 5.5 on the same graph and find their intersection points, which represent the solutions to the equation.

Enter the two functions in separate equations:

[tex]y_1 = x^2 - 3x - 2[/tex]

[tex]y_2 = 0.5(x+2.5)^2 - 5.5[/tex]

Adjust the zoom level and position of the graph to see the intersection points.

Click on the intersection points to see their coordinates.

Round the coordinates to the nearest thousandth.

After following these steps, we find that the solutions to the equation are:

x ≈ -1.283

x ≈ 4.283

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Prerequisites for the MVT are not met due to the discontinuity and non-differentiability of f(x) at x = 0 within the interval [-1, 1].

Let's first understand the Mean Value Theorem (MVT). The MVT states that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that:

f'(c) = (f(b) - f(a)) / (b - a)

Now, consider the function f(x) = 1/x^2. This function is continuous and differentiable for all x ≠ 0. However, in the interval [-1, 1], the function is not continuous nor differentiable at x = 0. Therefore, the Mean Value Theorem does not apply to this interval.

Since the MVT does not apply, we cannot say there exists a c in the interval (-1, 1) such that f'(c) = (f(1) - f(-1)) / (1 - (-1)). This is because the prerequisites for the MVT are not met due to the discontinuity and non-differentiability of f(x) at x = 0 within the interval [-1, 1].

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Approximately 378 people can fit in a passenger car with a length of 70 feet and a width of 9 feet.

To determine the number of people who can fit in a passenger car with a length of 70 feet and a width of 9 feet, we need to consider the available space inside the car and the amount of space required per person.

Assuming that the passenger car is a rectangular box shape, the volume of the car can be calculated as follows:

Volume of car = Length x Width x Height

Since we are only interested in the number of people who can fit in the car, we will assume a standard height of 6 feet for simplicity. Therefore, the volume of the car can be calculated as follows:

Volume of car = 70 feet x 9 feet x 6 feet

Volume of car = 3,780 cubic feet

Next, we need to determine the amount of space required per person. This can vary depending on the seating arrangement and the size of the individuals, but as a general rule, we can estimate that each person requires approximately 10 to 12 cubic feet of space in a seated position.

Assuming we use the lower end of that estimate and allocate 10 cubic feet of space per person, we can calculate the number of people who can fit in the car as follows:

Number of people = Volume of car / Space per person

Number of people = 3,780 cubic feet / 10 cubic feet per person

Number of people = 378 people

Therefore, if we assume a standard height of 6 feet and allocate 10 cubic feet of space per person, approximately 378 people can fit in a passenger car with a length of 70 feet and a width of 9 feet. However, it is important to note that this is just an estimate and the actual number of people who can fit in the car will depend on factors such as the seating arrangement and the size of the individuals.

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The correct sequence of steps for copying Line Segment DE using dynamic geometry software is in the order of 2,1,4,3. So, correct option is C.

The correct sequence of steps for copying Line Segment DE using dynamic geometry software is:

1.) Draw Line Segment DE with endpoint H on the circle.

2.) Draw a point and label it G.

3.) Construct a circle centered at the point G with radius Line Segment DE.

4.) Line Segment DE ≅ Line Segment GH

Therefore, option c.) 2,1,4,3 is the correct sequence of steps. The first step is to draw the original line segment DE with endpoint H on the circle.

The second step is to draw a point and label it G. The third step is to construct a circle centered at G with the same radius as Line Segment DE. Finally, in the fourth step, the line segment GH is drawn such that it is congruent to Line Segment DE, completing the copy of Line Segment DE.

So, correct option is C.

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The shopkeeper sold an average of 40.5 eggs per day during the given interval.

To calculate the average number of eggs sold per day, add up the total number of eggs sold and divide by the number of days.

Total number of eggs sold = 42 + 49 + 61 + 35 + 27 + 36 + 50 + 34 + 31 + 40 = 405

Number of days = 10

Average number of eggs sold per day = 405 / 10 = 40.5

The shopkeeper sold an average of 40.5 eggs per day during the given interval.

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The shopkeeper sold every day the number of eggs is recorded. At equal intervals group 42, 49, 61, 35, 27, 36, 50, 34, 31, 40

What is the average number of eggs sold per day by the shopkeeper over the given interval?

Using cosine rule, the largest angle In the triangle is

The angle is found first using cosine rule by the formula

cos A = (b² + c² - a²) ÷ 2bc

Solving for the largest angle

substituting the values

cos γ = (2² + 1.5² - 3²) ÷ 2 * 1.5 * 2

< γ = arc cos ( -0.4583 )

< γ = 117.28 degrees

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To use Newton's method to find the third approximation, X2, of the positive fourth root of 3, we follow these steps:

1. Define the function f(x) and its derivative, f'(x): f(x) = x^4 - 3 f'(x) = 4x^3

2. Start with an initial guess, X0 = 1.

3. Apply Newton's method formula to find the next approximations:

Xn+1 = Xn - f(Xn) / f'(Xn)

4. Calculate the first approximation, X1: X1 = X0 - f(X0) / f'(X0) X1 = 1 - (1^4 - 3) / (4 * 1^3) X1 = 1 - (-2) / 4 X1 = 1 + 0.5 X1 = 1.5

5. Calculate the second approximation, X2: X2 = X1 - f(X1) / f'(X1) X2 = 1.5 - (1.5^4 - 3) / (4 * 1.5^3) X2 = 1.5 - (5.0625 - 3) / 13.5 X2 = 1.5 - 2.0625 / 13.5 X2 = 1.5 - 0.15296 X2 ≈ 1.3470

So, the third approximation of the positive fourth root of 3, determined by calculating the third approximation of the right 0 of f(x) = x^4 - 3, starting with X0 = 1, is approximately 1.3470 (rounded to four decimal places).

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The quotient of 4.082 and 10,000 is 0.000004082.

To find the quotient of 4.082 and 10,000, we need to divide 4.082 by 10,000. However, dividing a decimal by another decimal can be tricky, so we need to move the decimal point of the dividend (4.082) and the divisor (10,000) to make the division easier.

We can move the decimal point of 4.082 four places to the left to obtain 0.04082. This is because moving the decimal point to the left makes the number smaller. For example, moving the decimal point in 4.082 one place to the left gives us 0.4082, which is ten times smaller than 4.082. Moving the decimal point four places to the left gives us a number that is 10,000 times smaller than 4.082.

Similarly, we can move the decimal point of 10,000 four places to the right to obtain 100,000,000. This is because moving the decimal point to the right makes the number larger. For example, moving the decimal point in 10,000 one place to the right gives us 1,000, which is ten times larger than 10,000. Moving the decimal point four places to the right gives us a number that is 10,000 times larger than 10,000.

Now we can divide 0.04082 by 100,000,000, which gives us the quotient of 0.000004082. Therefore, to find the quotient of 4.082 and 10,000, we need to move the decimal point in 4.082 four places to the left and move the decimal point in 10,000 four places to the right, and then divide the resulting numbers.

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The type of fee that will be charged depends on the specific service provided by the broker to the client. Non-transaction fees are typically charged for account maintenance or access to investment tools, while transaction fees are charged for specific trades or transactions.

To determine the type of fee that will be charged in each situation, we need to consider the specific services that were provided by the broker to the client.

For Joanne, her brokerage firm charged her a fee for providing investment information to help her make informed choices. This type of fee is a non-transaction fee, as it is not related to a specific transaction or trade.

Pete, who hasn't bought or sold any stock this year, was charged an inactivity fee of $75 for the year by his broker. This fee is also a non-transaction fee, as it is not related to a specific trade but rather to the inactivity of the account.

John's broker provided him with stock trading software to manage his trading. This type of service is typically included in a brokerage firm's platform fee, which is a non-transaction fee that covers the cost of maintaining the software and other tools that clients can use to manage their investments.

Lisa sold 50 shares of stock A on a given day and her broker charged her $12 for this trade. This type of fee is a transaction fee, which is charged for each specific trade or transaction that a client makes.

Finally, Reggie's broker charged him a front-end load of 2.75% when he purchased a fund. This fee is a type of sales charge or commission that is applied to the initial investment and is considered a transaction fee.

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The required probability is approximately 0.1253 or 12.53%.

We can use the central limit theorem to approximate the sampling distribution of the mean of the weights of clothes washed by 50 families of four. According to the central limit theorem, the sampling distribution of the mean will be approximately normal if the sample size is large enough (n > 30), regardless of the shape of the population distribution.

The mean of the sampling distribution of the mean will be equal to the population mean, which is 2000 lbs

[tex]SEM=\frac{\sigma}{\sqrt{n} }[/tex]

σ = population standard deviation

n = sample size.

[tex]SEM=\frac{187.5}{\sqrt{50} }[/tex]

= 26.5

Now we need to find the z-scores corresponding to the two values of the mean.

[tex]z_1=\frac{1980-2000}{26.5}[/tex]

= -0.75

[tex]z_2=\frac{1990-2000}{26.5}[/tex]

= -0.38

Using a standard normal table, we can find the probability that a z-score is between -0.75 and -0.38.

P(-0.75 < Z < -0.38) = 0.1253

Therefore, the required probability is approximately 0.1253 or 12.53%.

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Answer:

30π cm

Step-by-step explanation:

if r = 15 cm

circumference = π2r

= π × 2 × 15

= 30π cm

A value that is less than 2 1/5% from the given data is 0. 0215. Option A is the correct answer.

A fraction represents a part of a number or any number of equal parts. There is a fraction, containing numerator and denominator.

To find a value that is less than 2 1/5% we need to find the decimal number of a given fraction. to convert the given fraction into decimal form we need to divide the given fraction by 100.

= 2 1/5% / 100

= (2 + (1/5)) / 100

= 0.022

A value that is less than 0.022 from the given data is 0. 0215.

Therefore, a value less than 2 1/5% is 0. 0215.

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The final exponential function using the values of 'a' and 'b':
y = (1/2)(4^x)

To find an exponential function that passes through the points (2,8) and (4,128), follow these steps:

Step 1: Recall the general form of an exponential function: y = ab^x
Here, 'a' and 'b' are constants that need to be determined using the given points.

Step 2: Substitute the first point (2,8) into the equation:
8 = ab^2

Step 3: Substitute the second point (4,128) into the equation:
128 = ab^4

Step 4: Divide the second equation by the first equation to eliminate 'a':
(128 = ab^4) / (8 = ab^2)
16 = b^2

Step 5: Solve for 'b':
b = √16
b = 4

Step 6: Substitute the value of 'b' back into the first equation:
8 = a(4^2)

Step 7: Solve for 'a':
8 = 16a
a = 8/16
a = 1/2

Step 8: Write the final exponential function using the values of 'a' and 'b':
y = (1/2)(4^x)

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The Experimental probability of  rolling the color green is 0.15

The experimental probability of rolling the color green can be calculated by dividing the number of times the die lands on green by the total number of rolls:

Experimental probability of green = Number of green rolls / Total number of rolls

Experimental probability of green = 18 / 120

Experimental probability of green = 0.15 or 15%

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