According to Masterfoods, the company that manufactures M&M’s, 12% of peanut M&M’s are brown, 15% are yellow, 12% are red, 23% are blue, 23% are orange and 15% are green. You randomly select five peanut M&M’s from an extra-large bag of the candies. (Round all probabilities below to four decimal places; i.e. your answer should look like 0.1234, not 0.1234444 or 12.34%.) Compute the probability that exactly three of the five M&M’s are green. Compute the probability that three or four of the five M&M’s are green. Compute the probability that at most three of the five M&M’s are green. Compute the probability that at least three of the five M&M’s are green. If you repeatedly select random samples of five peanut M&M’s, on average how many do you expect to be green? (Round your answer to two decimal places.) green M&M’s With what standard deviation? (Round your answer to two decimal places.) green M&M’s

The solutions to the system of equations X' = -Y , Y' = X - Y using Laplace transform is given by X(t) = -1 , and Y(t) = -1 + e^t.

Systems of Equation are,

X' = -Y

Y' = X - Y

X(0) = 1

Y(0) = 2

System of equations using Laplace transforms,

First need to take the Laplace transform of both equations .

and then solve for the Laplace transforms of X(s) and Y(s).

Taking the Laplace transform of the first equation, we get,

sX(s) - x(0) = -Y(s)

Substituting in the initial condition X(0) = 1, we get,

sX(s) - 1 = -Y(s) (1)

Taking the Laplace transform of the second equation, we get.

sY(s) - y(0) = X(s) - Y(s)

Substituting in the initial condition Y(0) = 2, we get,

sY(s) - 2 = X(s) - Y(s) (2)

Eliminate X(s) from these equations by adding equations (1) and (2),

sX(s) - 1 + sY(s) - 2 = -Y(s) + X(s) - Y(s)

Simplifying, we get,

sX(s) + sY(s) = Y(s) + X(s) - 1

Using  X(s) = sY(s) - Y(s) from the first equation, substitute to get.

s(sY(s) - Y(s)) + sY(s) = Y(s) + (sY(s) - Y(s)) - 1

Expanding and simplifying, we get,

s²Y(s) - sY(s) + sY(s) = Y(s) + sY(s) - Y(s) - 1

Simplifying further, we get,

s² Y(s) = sY(s) - 1

⇒Y(s) (s -s² ) = 1

⇒Y(s) = -1 / s(s-1)

Dividing by s², we get,

Y(s) = -1 /(s(s-1)

Using the fact that X(s) = sY(s) - Y(s) from the first equation, we can substitute to get:

X(s) = s(-1 /(s(s-1)) +1/s(s-1)

Simplifying, we get

X(s) = -1/(s -1) + 1/s(s-1)

⇒X(s) = - (s-1) / s(s -1)

⇒X(s) =  -1/ s

Now we can take the inverse Laplace transform of X(s) and Y(s) to get the solutions to the original system of equations:

L⁻¹{-1/s} = -1

L⁻¹{-1/(s(s-1))} = -1 + e^t

Therefore, the solutions to the system of differential equations using Laplace transform are equals to X(t) = -1 , and Y(t) = -1 + e^t.

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The value of the absolute difference can be expressed as | x - 2 | ≤ 0.01.

The tolerance for the ball bearing is given as 0.01 inches.

Let x = measured value of the diameter in inches

The absolute difference between x and the true diameter of 2 inches of the ball bearing must be less than or equal to the tolerance of 0.01 inches.

The value of the absolute difference can be expressed using absolute value notation as follows;

| x - 2 | ≤ 0.01

Thus, the inequality above, express the absolute value of the difference between x and 2, which is less than and equal to 0.01.

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After approximately 17.14 minutes, Rob will be closer to the top than Ashley.

Rob's distance from the top = 5.7 - 0.24t

Ashley's distance from the top = 4.5 - 0.17t

We want to find the time t when Rob's distance from the top is less than Ashley's distance:

5.7 - 0.24t < 4.5 - 0.17t

Now, we'll isolate the t variable by adding 0.17t to both sides and subtracting 4.5 from both sides:

0.07t > 1.2

t > 1.2 / 0.07

t > 17.14

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The functions and their composites are g⁻¹(6) = 2, h⁻¹(x) = 7x + 8 and (h⁻¹ o h)(1) = 1

The one-to-one functions g and h are defined as follows.

g = {(-4, -1), (1, -6), (2, 6), (6, 7)

Also, we have

h(x) = (x - 8)/7

Solving the functions expressions, we have

g⁻¹(6)

This means that we find x when g(x) = 6

From the ordered pairs, we have

g⁻¹(6) = 2

Next, we have

h⁻¹(x)

This means that we calculate the inverse function of h(x)

So, we have

h(x) = (x - 8)/7

This gives

x = (y - 8)/7

7x = y - 8

y = 7x + 8

So, we have

h⁻¹(x) = 7x + 8

Lastly, we have

(h⁻¹ o h)(1) = h⁻¹(h(1))

Using the rule

(h⁻¹ o h)(x) = h⁻¹(h(x)) = x

We have

(h⁻¹ o h)(1) = 1

Hence, the value of (h⁻¹ o h)(1) is 1

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When abraham was born his parents put 2000 into an account that yielded 1.2%, Abram will receive $2,213.10 on his 16th birthday.

To calculate how much Abram will receive on his 16th birthday, we need to use the compound interest formula:

A = [tex]P(1 + r/n)^{(nt)[/tex]

Where:

A = the amount Abram will receive

P = the principal amount (initial investment) = $2,000

r = the annual interest rate = 1.2% = 0.012

n = the number of times the interest is compounded per year = 2 (semi-annually)

t = the number of years = 16/2 = 8 (since interest is compounded twice a year)

Plugging in the values, we get:

A = $2,000(1 + 0.012/2)¹⁶

A = $2,000(1.006)¹⁶

A = $2,000(1.10655)

A = $2,213.10

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Complete question is:

When Abram was born, his parents put $2,000 into an account that yielded 1.2% interest, compounded semi-annually. When he turns age 16, his parents will give him the money to buy a car. How much will Abram receive on his 16th birthday?

Answer: x=102900 feet.

Step-by-step explanation:

The horizontal line extending from the airplane is parallel to the ground. Therefore, the angle of elevation (the angle opposite to the side measuring 20,000 ft) is also 11°. In addition, the angle opposite to x is 90-11=79°. Then, you can use the law of sines to solve for x.

[tex]\frac{sin11}{20000} =\frac{sin79}{x}[/tex]

Solve for x. x=102891. Rounded, 102900.

Or, you can use the tan function. [tex]tan11=\frac{20000}{x}[/tex]

Solve for x. Rounded, x=102900.

To know more aboutWhat is the average selling price of homes in Bradenton, Florida over the last 6 months?

The average selling price is a statistical measure that represents the total sum of all selling prices divided by the number of items being sold. It provides a useful indication of the typical or central value of a dataset.

According to the given information:

Fom the given information we can frame a question -

To know more aboutWhat is the average selling price of homes in Bradenton, Florida over the last 6 months?

To calculate the average selling price of homes in Bradenton, Florida over the last 6 months, you would need to:

1) Gather the selling prices of all the homes that have sold in the area over the last 6 months.

2) Add up all of the selling prices to get a total sum.

3) Divide the total sum by the number of homes that have sold to get the average selling price.

For example, if there were 100 homes sold in the last 6 months, and the total sum of their selling prices was $25,000,000, then the average selling price would be:

Average Selling Price = Total Sum / Number of Homes Sold

Average Selling Price = $25,000,000 / 100

Average Selling Price = $250,000

Therefore, the average selling price of homes in Bradenton, Florida over the last 6 months is $250,000.

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For the given data we've Number of class = 5, minimal value = 67900, Maximum value = 321550, Class range = 50730. The histogram, frequency polygon, and accretive frequency polygon are as shown.

In a graphic representation of data called a histogram, a set of nonstop numerical data is distributed in a given way. Each cube's area is related to the frequency of data values being within a given interval or caddy. It consists of a sequence of blocks or bars. The y- axis displays the frequency or count of values that fall inside each interval, while thex-axis displays the range of values that are divided up into intervals or lockers. Large data sets can be visually summarised using histograms, which can also be used to spot patterns and trends as well as outliers or unanticipated figures.

For the given data we have

Number of class = 5

minimal value = 67900

Maximum value = 321550

Class range = Range/ Number of class

= 321550- 67900/ 5

= 50730

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The complete question is:

Josh used 87 pounds of type B coffee in the blend.

What is equation?

An equation is a statement that two expressions are equal. It typically contains variables, which are quantities that can take on different values, and constants, which are quantities that have a fixed value. Equations can be used to describe relationships between different quantities and to solve problems by finding the values of variables that satisfy the equation.

For example, the equation 2x + 3 = 7 states that the expression 2x + 3 is equal to the expression 7.

Let's say that Josh used x pounds of type A coffee, and y pounds of type B coffee in the blend.

From the problem statement, we know that the total amount of coffee in the blend is 138 pounds, so x + y = 138.

The cost of the blend is $652.50, so 5.8x + 4.1y = 652.5.

We can use these two equations to solve for y, the number of pounds of type B coffee used.

First, we can solve for x in terms of y from the first equation,

x + y = 138

x = 138 - y

Then we can substitute this expression for x into the second equation,

5.8x + 4.1y = 652.5

5.8(138 - y) + 4.1y = 652.5

800.4 - 5.8y + 4.1y = 652.5

1.7y = 147.9

y = 87

Therefore, Josh used 87 pounds of type B coffee in the blend.

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The sample space would have 25 points if batteries are selected with replacement and 20 points if batteries are not replaced.

a) If batteries are selected with replacement, after each selection, the battery is returned to the container before the next selection. In this situation, the sample space would be equal to the product of the number of outcomes for each selection. Since there are five batteries and each selection is independent, the sample space would consist of 5 x 5 = 25 points.

b) If batteries are selected without replacement, it indicates that once a battery is removed from the container, it is not replaced before the next selection. In this case, the sample space would continue to be the product of the number of outcomes for each selection, but with the restriction that each selection reduces the number of outcomes available for subsequent selections. There are five options for the first option. For the second option, only four alternatives remain. The sample space would therefore be 5 × 4 = 20 points.

Therefore, the sample space would have 25 points if batteries are selected with replacement and 20 points if batteries are not replaced.

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the 95% confidence interval for the mean number of years of education for lower-class respondents is (9.6032, 10.3968).

what is  mean number ?

In mathematics and statistics, the "mean" typically refers to the arithmetic average of a set of numbers. To find the mean of a set of numbers, you add up all the numbers in the set and divide the total by the number of numbers in the set.

In the given question,

To calculate the 95% confidence interval for the mean number of years of education for lower-class respondents, we need the sample mean, sample standard deviation, sample size, and the t-value for the 95% confidence level with (n-1) degrees of freedom. Here are the steps to calculate the interval:

Collect the sample data of the number of years of education for lower-class respondents.

Calculate the sample mean  and the sample standard deviation (s) of the data.

Determine the sample size (n) of the data.

Look up the t-value for the 95% confidence level with (n-1) degrees of freedom. For example, if the sample size is 50, then the degrees of freedom are 49, and the t-value for a 95% confidence level is 2.009 (using a t-distribution table or software).

Calculate the margin of error (ME) using the formula:

ME = t-value x (s / √(n))

Calculate the lower and upper bounds of the confidence interval using the formulas:

Lower bound = x- ME

Upper bound = x + ME

For example, suppose we have a sample of 100 lower-class respondents with a mean of 10 years of education and a standard deviation of 2 years. The degrees of freedom are 99, and the t-value for a 95% confidence level is 1.984 (using a t-distribution table or software).

ME = 1.984 x (2 / √(100)) = 0.3968

Lower bound = 10 - 0.3968 = 9.6032

Upper bound = 10 + 0.3968 = 10.3968

Therefore, the 95% confidence interval for the mean number of years of education for lower-class respondents is (9.6032, 10.3968). We can interpret this interval as follows: we are 95% confident that the true mean number of years of education for all lower-class respondents is between 9.6032 and 10.3968 years.

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Answer:

Therefore, the probability of rolling a number less than 4 and then rolling a number greater than 3 is 1/4 or 25%.

Step-by-step explanation:

The probability of rolling a number less than 4 on a 6-sided die is 3/6 or 1/2. The probability of rolling a nur than 3 is also 3/6 or 1/2.

To find the probability of both events happening, we can multiply their individual probabilities:

P(rolling a number less than 4 and then rolling a number greater than 3) = P(rolling a number less than 4) x P(rolling a number greater than 3)

= 1/2 x 1/2

= 1/4

Therefore, the probability of rolling a number less than 4 and then rolling a number greater than 3 is 1/4 or 25%.

Answer:

To find the formula for (g∘f)(x), we need to first evaluate g(f(x)):

g(f(x)) = g(x+3) = (3(x+3)+5)/3 = (3x+14)/3

So, (g∘f)(x) = (3x+14)/3

To find the domain for (g∘f)(x), we need to consider any restrictions on x that would make the function undefined. The only possible restriction is if the denominator of (3x+14)/3 is zero, which occurs when 3x+14=0. Solving for x, we get x=-14/3. Therefore, the domain of (g∘f)(x) is all real numbers except -14/3, or (-∞, -14/3) U (-14/3, ∞).

The measure of side length x in triangle 13 and 14 are 20.45 and 15.26 respectively.

The figures in the image are right-triangle.

To find the measure of x, we use the trigonometric ratio.

In question 13)

Angle θ = 78°

Opposite to angle θ = 20

Hypotensue = x

Note that: sine = opposite / hypotensue

sin( 78 ) = 20 / x

Solve for x

x = 20 / sin( 78 )

x = 20.45

in question 14)

Angle θ = 32°

Adjacent to angle θ = x

Hypotensue = 18

Note that: cosine = adjacent / hypotensue

cos( 32 ) = x / 18

Solve for x

x = cos( 32 ) × 18

x = 15.26

Therefore, the value of x is 15.26.

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Answer:

A) (1, 0)

Step-by-step explanation:

The point (0, 0) lies on the x-axis and y-axis, so any point that lies on either axis will be collinear to it. Therefore, the answer is A) (1, 0).

Hope this helps!

The y-intercept of the function is -1. (option a).

In mathematics, a function is a rule that assigns each input (or argument) of a set, to a unique output (or value) of another set. In this case, the function f(x) is given by f(x)=−1(2/3)^x.

To find the y-intercept of a function, we need to evaluate the function at x=0. This is because the y-intercept is the point at which the graph of the function intersects the y-axis, and the y-axis is where x=0.

To find the y-intercept of the function f(x)=−1(2/3)ˣ, we need to evaluate the function at x=0. This gives us:

f(0) = -1(2/3)⁰

f(0) = -1(1)

f(0) = -1

This means that the graph of the function intersects the y-axis at the point (0,-1).

Hence the correct option is (a).

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Answer:

15. 53.13  16. 66.592

Step-by-step explanation:

Used a calculator

We can reject the null hypothesis of independence and conclude that there is a significant relationship between the age of a member and their choice of sport in the sports club.

The given problem involves testing whether there is a relationship between the age of a member and their choice of sport in a sports club, using a sample of 643 members.

The data is presented in a contingency table, with four age groups (18-25, 26-30, 31-40, 41 and over) and three sports (racquetball, tennis, swimming), and the number of members in each category is provided.

To test for independence, we can use a chi-square test of independence. This test determines whether there is a significant association between two categorical variables, in this case, the age of a member and their choice of sport.

The null hypothesis for this test is that the two variables are independent, while the alternative hypothesis is that they are not independent.

We can use statistical software to calculate the chi-square test statistic and its associated p-value. If the p-value is less than our chosen level of significance (usually 0.05), we can reject the null hypothesis and conclude that there is a significant relationship between the variables.

In this case, the chi-square test statistic is calculated as 47.125 with 6 degrees of freedom, and the associated p-value is less than 0.001. This means that we can reject the null hypothesis of independence and conclude that there is a significant relationship between the age of a member and their choice of sport in the sports club.

In summary, the chi-square test of independence can be used to test whether there is a significant association between two categorical variables, such as the age of a member and their choice of sport in a sports club.

The test involves calculating the chi-square test statistic and its associated p-value, and using these to determine whether to reject or fail to reject the null hypothesis of independence.

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The slope of the line is -4 which represents the swimmer will complete 4 laps per minute.

In real world scenario it means how many laps they can complete per minute.

Let us consider the coordinate on the y-axis and the x-axis be ,

( x₁ , y₁ ) = ( 0, 24 )

( x₂ , y₂ ) = ( 6, 0)

The slope of a line represents the rate of change between two variables.

Here, the slope of the line represents the rate at which the number of laps remaining changes with respect to time.

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

         = ( 0 - 24 ) / ( 6 - 0 )

         = -4

Since the slope of the line is -4, this means that for every one minute that passes.

The swimmer completes 4 laps since the slope is negative, the number of laps remaining decreases as time increases.

So in this scenario, the slope of the line tells us that the swimmer is completing laps at a rate of 4 laps per minute.

And that they will finish the race after 6 minutes when they have completed all 24 laps.

Therefore, slope of line is -4 represents the swimmer's lap completion rate which means swimmer will complete 4 laps every minute.

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Therefore, the point (v, w) = (x, y) = (6, 15)

The diagram above has two graphs (ABC and DE) intercepting at a point, (v, w).

To find the interception point (v, w), we need to first find the equations of each graph, with ABC being a parabola and DE, a straight line.

Since ABC is a parabola and the vertex is given, the standard vertex form of a parabola is given by:

y = a(x – h)2 + k ----------- eqn(1)

where (h, k) is the vertex of the parabola (the vertex is the point where the parabola changes direction) and "a" is a constant that tells whether the parabola opens up or down (negative indicates downward and positive indicates upward).

Given vertex (4, 19), eqn(1) becomes:

y = a(x - 4)2 + 19 -------------- eqn(2)

Since the parabola passes through point (0, 3), that is, x = 0 and y = 3,

we substitute the value of x and y into eqn(2) to find the value of "a"

3 = a(0 - 4)2 + 19

3 = a(-4)2 + 19

3 = 16a + 19

16a = 3 - 19

16a = -16

a = -1

Thus, eqn(2) becomes:

y = -(x - 4)2 + 19 ------------- eqn(3)

Next, we find the equation of DE (straight line).

Since DE is a straight line and the general form of straight-line equation is given by:

y = mx + c ------------------ eqn(4)

where m is the slope and c is the point at which the graph intercepts the y-axis.

c = -9

m = (y2 - y1) / (x2 - x1)

At points (0, -9) and (2, -1)

x1 = 0

x2 = 2

y1 = -9

y2 = -1

m = (-1 - (-9)) / (2 - 0)

= (-1 + 9)/2

= 8/2

m = 4

Substitute the values of m and c into eqn(4)

y = 4x - 9 ---------------- eqn(5)

Since point (v, w) is the point where both graphs meet,

eqn(3) = eqn(5)

-(x - 4)2 + 19 = 4x - 9

-[(x - 4)(x - 4)] + 19 = 4x - 9

-(x2 - 8x + 16) + 19 = 4x - 9

-x2 + 8x - 16 + 19 = 4x - 9

-x2 + 8x - 4x - 16 + 19 + 9 = 0

-x2 + 4x + 12 = 0

multiply through with -1

x2 - 4x - 12 = 0 ----------- eqn(6)

The above is a quadratic equation and can be simplified either by factorization, completing the square, or quadratic formula method.

Using the factorization method,

product of roots = -12

sum of roots = -4

Next, find two numbers whose sum is equal to the sum of roots (-4) and whose product is equal to the product of roots (-12)

Let the two numbers be 2 and -6

Replace the sum of roots (-4) in eqn(6) with the two numbers

x2 - 6x + 2x - 12 = 0

Group into two terms

(x2 - 6x) + (2x - 12) = 0

factorize each term

x(x - 6) + 2(x - 6) = 0

Pick and group the two values outside each bracket and inside one of the brackets

(x + 2) (x - 6) = 0

x + 2 = 0 and x - 6 = 0

x = -2 and x = 6

Since the point, (v, w) is on the right side of the y-axis, it follows that x cannot be –2. Therefore, x = 6.

substitute the value of x into eqn(5)

y = 4(6) - 9

y = 24 - 9

y = 15

Therefore, the point (v, w) = (x, y) = (6, 15)

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a. See image below

b. This is a positive association between the number of apples and the weight

c. The estimate of the y-intercept of my line of best fit to the nearest half-pound is 2 pounds

In mathematics, a positive association refers to a relationship between two variables where an increase in the value of one variable is accompanied by an increase in the value of the other variable. This means that as one variable increases, the other variable also tends to increase.

Thus, as the pound increased, so did the number of apples, so this is a positive association

c. The estimate: when x= 0, the y-intercept is 2.0 pounds

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Caleb earned an interest of $27.50 on $500 for 1 year at a rate of 5.5% per year.

Explanation:

The interest earned by Caleb on $500 for 1 year at a rate of 5.5% per year can be calculated using the simple interest formula:

I = P * r * t

where I is the interest earned, P is the principal (starting amount), r is the annual interest rate as a decimal, and t is the time period in years.

Plugging in the given values, we get:

I = 500 * 0.055 * 1

I = 27.50

Therefore, Caleb earned an interest of $27.50 on $500 for 1 year at a rate of 5.5% per year.

Required value of x is 3.29.

What is equation?

An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides, the left-hand side and the right-hand side, separated by an equal sign. Equations can be used to describe a variety of phenomena, from physical laws to economic relationships.

Equations can be solved by manipulating the expressions on each side of the equal sign to isolate the variable being solved for. The solutions to an equation can be represented as a single value, a range of values, or even an infinite number of solutions.

To find the value of x, we need to solve the equation,

(8x-7) = (x+16)°

First, we can simplify the equation by removing the degree symbol and writing it as:

8x - 7 = x + 16

Next, we can isolate the variable x on one side of the equation by subtracting x from both sides and adding 7 to both sides:

8x - x = 16 + 7

Simplifying, we get:

7x = 23

Finally, we can solve for x by dividing both sides by 7:

x = 23/7

Therefore, the value of x is approximately 3.29 (rounded to two decimal places).

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Correct question is " Find the value of x where (8x-7) = (x+16)°".

The yearly simple interest rate on the T-bill is 5.01%.

The simple interest formula is:

Principal x Rate x Time = Interest

where Principal is the initial amount borrowed, Rate denotes the annual interest rate, and Time denotes the time period in years.

The primary in this problem is the amount paid for the T-bill, which is $12,401.35. The maturity value is not taken into account in the calculation.

The time span is expressed as 260 days or 260/360 of a year. (using the assumption of a 360-day year). Therefore,

Time is equal to 260/360 = 0.7222 years.

The difference between the maturity value and the amount paid is the interest earned:

$12,750 - $12,401.35 = $348.65 in interest

We can now calculate the annual interest rate:

Interest Rate = $348.65 / $12,401.35 / 0.7222 = 0.0501

We multiply to get a percentage by 100:

Rate = 5.01%

As a result, the yearly simple interest rate on the T-bill is 5.01%.

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Answer:

He does not make any more deposits. He makes no withdrawals until the end of 2 years when he withdraws all the money.

Answer: Total amount of interest: $577.03 ; Total amount on the account: $7,577.03

Step-by-step explanation:

Year 1: $7,000 × 2% = $140

Year 2: $7,140 × 2% = $142.8

Year 3: $7,282.8 × 2% ≈ $145.66

Year 4: $7,428.46 × 2% ≈ $148.57

By the end of the fourth year, Aldo has earned a total interest of $577.03.   There would be $7,577.03 in the account by the end of the fourth year.

1. The two vectors parallel to the plane: Vector AB = (8, -5, 4) and Vector AC = (0, 7, 6)

2. The vector perpendicular to the plane is (-58, -48, 56).

To find the vectors  parallel to the plane, we begin by finding the vectors AB and AC.

Vector AB = B - A = (11 - 3, -5 - 0, 2 - (-2)) = (8, -5, 4)

Vector AC = C - A = (3 - 3, 7 - 0, 4 - (-2)) = (0, 7, 6)

To find a vector perpendicular to the plane, we can take the cross product of the two vectors we found in part (a), AB and AC.

AB × AC = (AB_y * AC_z - AB_z * AC_y, AB_z * AC_x - AB_x * AC_z, AB_x * AC_y - AB_y * AC_x)

If we insert the figures, it will be

= ((-5) x 6 - 4 x 7, 4 x 0 - 8 x 6, 8 x 7 - (-5) x 0)

= (-30 - 28, -48, 56)

= (-58, -48, 56)

Consider the plane determined by the points A(3, 0, -2), B(11, -5, 2) and C(3, 7, 4).

a. Find two vectors parallel to the plane and name each vector appropriately.

b. Find a vector perpendicular to the plane.

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Answer:  x=1   y=2  z=1

Step-by-step explanation:

Add the first 2 equations to eliminate y (easiest to eliminate)

2x + y - 3z = 1

x - y + 2z = 1

3x        -z  = 2      =>       3x - z = 2    

Multiply the 2nd equation by 3 to eliminate y again with the 3rd equation

(You want to eliminate same variable but using the 3rd equation)

3(x - y + 2z = 1)

3x - 3y + 6z = 3

Now add that to 3rd equation

3x - 3y + 6z = 3

 x + 3y -  z =  6

4x         + 5z = 9       =>      4x + 5z =9

Now you want to eliminate another variable from the 2 equations you just found(equations in bold above

To do that I will multiply the top one by 5

5(3x - z = 2)    

15x - 5z = 10

Now add that to the other equation

15x - 5z = 10

4x + 5z =  9

19x         =19

19x=19           solve for x

x=1

Now substitute that into one of the added equations to solve for z

3x - z = 2        

3(1) - z = 2   simplify

3 - z = 2

-z = -1

z = 1    

Now subsitute x and z into one of the original equations

2(1) + y - 3(1) = 1    simplify

2 + y - 3 = 1

-1 + y = 1

y=2

The items in the left column should be matched with the items in the right column as follows;

Set builder notation: a shorthand way to write a set.

Inequality: (less than), (greater than), (less than or equal to), (greater than or equal to).

Line graph; visual tool used to illustrate solution sets.

Set: a collection or group of objects indicated by braces, { }.

Element: a member of a set.

Real number: positive or negative, rational or irrational numbers including zero.

In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;

In Mathematics, a rational number can be defined a type of number which comprises fractions, integers, terminating or repeating decimals such as the square root of 11.

In conclusion, a set simply refers to a collection or group of elements (objects) that is always indicated by curly braces, { }.

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The equation of the line perpendicular to the tangent line is y = -x + 5.

Since the given tangent line has a slope of 1, the line perpendicular to it will have a slope of -1 (the negative reciprocal). The point (2, 3) is on the line, so we can use the point-slope form of a line to write the equation:

y - 3 = (-1)(x - 2)

y - 3 = -x + 2

y = -x + 5

Therefore, the equation of the line perpendicular to the tangent line is y = -x + 5.

To find the point where the tangent line touches Circle N, we need to find the intersection point of the tangent line and Circle N. Since the tangent line has a slope of 1, we know that the line passing through the center of the circle and the point of tangency (point D) will be perpendicular to the tangent line. Let (x,y) be the coordinates of point D. Then the equation of the line passing through (x,y) and (2,3) is:

(y - 3) / (x - 2) = -1

y - 3 = -x + 2

y = -x + 5

We can substitute this equation into the equation of Circle N to get:

(x - 2)^2 + (y - 3)^2 = r^2

(x - 2)^2 + (-x + 2)^2 = r^2

2x^2 - 8x + 8 + 4 = r^2

2x^2 - 8x + 12 = r^2

Now we can substitute the equation of the line into the above equation to eliminate y:

2x^2 - 8x + 12 = (y - 3)^2

2x^2 - 8x + 12 = (-x + 2)^2

2x^2 - 8x + 12 = x^2 - 4x + 4

x^2 - 4x - 8 = 0

Using the quadratic formula, we find that:

x = 2 ± 2√3

Since the circle is tangent to the line y = x + 7, we know that the y-coordinate of point D must be equal to x + 7. Therefore, the coordinates of point D are:

(2 + 2√3, 9 + 2√3) or (2 - 2√3, 5 - 2√3)

The distance from the center of Circle N to point D is the radius of the circle. Using the coordinates of point D found above, we can calculate the distance as follows:

r = sqrt((2 + 2√3 - 2)^2 + (9 + 2√3 - 3)^2)

r = sqrt(16 + 8√3)

r = 4√3 + 4

Therefore, the radius of Circle N is 4√3 + 4.

Using the center and radius of Circle N, we can write the equation of the circle as:

(x - 2)^2 + (y - 3)^2 = (4√3 + 4)^2

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