Small baskets of tomatoes are sold at a vegetable stand for $3 per basket. Large


baskets of tomatoes are sold at the stand for $5 per basket. Only whole numbers of


baskets may be purchased.


A customer purchases a total of 8 baskets of tomatoes and pays $36.


A. Write and solve a system of equations that models the number of small


baskets (x) and the number of large baskets () that the customer purchases.


Show or explain all your work.

Trevor spent a mean amount of $0.85 per day on snacks during

Week 1.

We have,

To determine during which week Trevor spent a mean amount of $0.85 per day on snacks, we need to calculate the mean (average) amount spent per day for each week and find the week that matches $0.85.

Let's calculate the mean amount spent per day for each week:

Week 1:

= (0.50 + 0.75 + 0.50 + 1.00 + 1.25 + 1.25 + 0.75 + 0.25 + 1.00) / 9

= $0.86 (approximately)

Week 2:

= (1.00 + 0.50 + 0.75 + 0.25 + 0.25 + 1.25 + 0.25) / 7

= $0.61 (approximately)

Week 3:

= (0.75 + 1.00) / 2

= $0.88

Week 4:

= (1.25 + 1.00 + 1.00 + 1.25 + 0.50) / 5

= $1.00

From the calculations, we can see that the mean amount spent per day for Week 1 is closest to $0.85.

Therefore,

Trevor spent a mean amount of $0.85 per day on snacks during

Week 1.

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dx/dy = -sin(t)/cos(t) when x = sin^3(t) and y = cos^3(t).

To find dx/dy, we first need to find dx/dt and dy/dt, and then we can use the chain rule.

Given x = sin^3(t) and y = cos^3(t),

dx/dt = d(sin^3(t))/dt = 3sin^2(t) * cos(t) (using the chain rule)
dy/dt = d(cos^3(t))/dt = -3cos^2(t) * sin(t) (using the chain rule)

Now, we can find dx/dy by dividing dx/dt by dy/dt:

dx/dy = (dx/dt) / (dy/dt) = (3sin^2(t) * cos(t)) / (-3cos^2(t) * sin(t))

Simplify the expression:

dx/dy = -sin(t)/cos(t)

So, dx/dy = -sin(t)/cos(t) when x = sin^3(t) and y = cos^3(t).

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The drink mix that is needed to make one cup of drink is 21/20

Ali uses 21/2 scoops of drink mix to make 10 cups off drinks

The amount of drink mix needed to make one cup can be calculated as follows

21/2= 10

x= 1

cross multiply both sides

10x= 21/2

Divide by the coefficient of x which is 10

x= 21/2 ÷ 10

x= 21/2 × 1/10

x= 21/20

Hence the drink mix needed to make one cup is 21/20

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The perimeter of the bedroom is 116ft

A rectangle is a type of quadrilateral, whose opposite sides are equal and parallel.

The perimeter of a rectangle is expressed as;

P = 2(l+w)

where l is the length and w is the width of the rectangle.

length = √ 21-21)²+ 18-(-7)²

= √25²

l = 25 ft

width = √ 21-(-12)²+18-18)²

= √ 33²

= 33

Perimeter = 2(33+25)

= 2 × 58

= 116ft

therefore the perimeter of the bedroom is 116ft.

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The system of equations is:

Revenue from balcony seats = $10 × B

Revenue from main floor seats = $25 × M

Total revenue = $6250

B = 3M + 20

Let's define the following variables:

B = number of balcony seats sold

M = number of main floor seats sold

We know that the price of a balcony seat is $10 and the price of a main floor seat is $25.

From the given information, we can create the following equations:

The total revenue from balcony seats sold (B) is given by: Revenue from balcony seats = $10 × B

The total revenue from main floor seats sold (M) is given by: Revenue from main floor seats = $25 × M

The total revenue from the afternoon performance is $6250: Total revenue = Revenue from balcony seats + Revenue from main floor seats

The number of balcony seats sold (B) is 20 more than 3 times the number of main floor seats (M): B = 3M + 20

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Answer:

D. lines A and D are perpendicular

In a case whereby When finding the Quotient of 8,397 divided 12, Calida first divided 83 by 12, then she will be wrong, because the answer is 699.75.

In maths, a division can be described as the process of splitting a specific amount  which can be spread to equal parts instance of thisd is when we divide a group of 20 members into 4 groups and this can be done using the mathematical sign.

In the case of Calida above, the division can be made as

8,397 divided 12

=8,397 / 12

=699.75

Therefore we can say that the right answer to the querstion is 699.75

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The approximate length of the crust when x = 1.2 is 17.28 centimeters. The correct option is D.

To find the total perimeter of Sydney's sandwich, we need to add up the lengths of all four sides. From the given dimensions, we can see that the top and bottom sides each have a length of 2x²8, and the right and left sides each have a length of 2x²9. Therefore, the total perimeter can be expressed as:

2(2x²8) + 2(2x²9)

Simplifying this expression gives:

4x²8 + 4x²9

And further simplifying by factoring out 4x² gives:

4x²(8 + 9)

Which equals:

4x²17

Now, to find the approximate length of the crust when x = 1.2, we simply plug in this value for x into the expression we just found:

4(1.2)²17

Simplifying this expression gives:

4(1.44)17

Which equals:

5.76 + 11.52 = 17.28

Therefore, the approximate length of the crust when x = 1.2 is 17.28 centimeters. The answer is option D, which is 4x²17; 22.76 centimeters.

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The measurement of the two shorter sides in Monica's wrapping paper is 17 inches.

We can use proportions to solve this problem. Since Francisco's piece of paper is similar to Monica's piece of paper, the ratios of the corresponding sides will be equal. Specifically, we have:

17 / x = 34 / y

where x is the length of one of Francisco's shorter sides, and y is the length of one of Monica's shorter sides.

To solve for y, we can cross-multiply and simplify:

17y = 34x

y = 2x

So the length of one of Monica's shorter sides is half the length of one of her longer sides, or:

y = 1/2 * 34 = 17

Therefore, the measurement of the two shorter sides in Monica's wrapping paper is 17 inches. Answer: a. inches.

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To understand how to approach problems involving similar triangles, parallel lines, and proportional parts.

When you have two similar triangles, their corresponding sides are proportional. This means that the ratio between the sides of one triangle is the same as the ratio between the sides of the other triangle.

If you have parallel lines cut by a transversal, corresponding angles will be congruent, which can help establish similarity between triangles.

For example, if you have two similar triangles ABC and DEF, where AB/DE = BC/EF = AC/DF, then you can use these ratios to solve for unknown side lengths or other values.

To answer questions 1-16, identify the given information and determine whether the triangles are similar. If they are, use the proportional relationships to solve for the unknowns.

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How do you solve these problems on Unit 6: similar triangles homework 5: parallel lines and proportional parts?

7. X-5/6 = 14/2x+3

16. X-7/35 = 4/x-3

The first derivative of the implicit function given by x^(2/3) + y^(2/3) = 14 can be found using implicit differentiation. We take the derivative of both sides with respect to x and use the chain rule to differentiate the terms involving y:(2/3)x^(-1/3) + (2/3)y^(-1/3) * dy/dx = 0Then, we solve for dy/dx:dy/dx = -(x/y)^(1/3)This is the first derivative of the implicit function. To evaluate it at a specific point, we need to substitute the coordinates of that point into the equation above.

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[tex]dy/dx = -x^{-1/3} / y^{-1/3}[/tex]

To find the first derivative of the given equation x^{2/3} + y^{2/3} = 14, we will differentiate both sides of the equation with respect to x and then solve for dy/dx (the first derivative of y with respect to x).

Step 1: Differentiate both sides of the equation with respect to x.
[tex]d/dx (x^{2/3} + y^{2/3}) = d/dx (14)[/tex]

Step 2: Apply the chain rule to differentiate y^{2/3}.
[tex]d/dx (x^{2/3}) + d/dx (y^{2/3}) = 0(2/3)x^{-1/3} + (2/3)y^{-1/3}(dy/dx) = 0[/tex]

Step 3: Solve for dy/dx.
[tex](2/3)y^{-1/3}(dy/dx) = -(2/3)x^{-1/3}dy/dx = -(2/3)x^{-1/3} / (2/3)y^{-1/3}[/tex]

Step 4: Simplify the expression.
[tex]dy/dx = -x^{-1/3} / y^{-1/3}[/tex]

Your answer: [tex]dy/dx = -x^{-1/3} / y^{-1/3}[/tex]

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Answer:

-12+3(-11)-40-10

Step-by-step explanation:

Answer:

Step-by-step explanation:

-12+12-45-40+1

0-85+1

-84

The equation of the perpendicular line to y = -3x + 4 and passing through (-6, 2) is y = (1/3)x + 4.

There are a lot of perpendicular lines that we may see in reality. The corners of a chalkboard, a window, the sides of a set square, and the Red Cross symbol are a few examples.

Knowing that the slopes of perpendicular lines are the negative reciprocals of one another is necessary to determine the equation of a line perpendicular to a given line.

The given line has a slope of -3, so the slope of a line perpendicular to it would be the negative reciprocal of -3, which is 1/3.

We also know that the line passes through the point (-6, 2).

The equation of the line passing through (-6, 2) and perpendicular to y = -3x + 4 can be found using the point-slope form of a line:

y - 2 = (1/3)(x + 6)

Simplifying this equation, we get:

y = (1/3)x + 4

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Place value is the value of a digit based on its position within a number system. To solve the equation 19 + 50, we use place value to add the digits in the ones and tens place.

Place value is a fundamental concept in mathematics that helps in understanding the value of digits in a number system. In the decimal system, each digit's value depends on its position relative to the decimal point, with the position to the left representing higher values than those to the right.

To solve 19 + 50, we add the digits in the ones place (9 + 0 = 9) and the digits in the tens place (1 + 5 = 6) separately, then combine the results to get the final answer of 69. This is possible due to the place value concept. The digit 9 in the ones place of 19 represents a value of 9 units, while the digit 1 in the tens place represents 10 units.

Similarly, the digit 5 in the tens place of 50 represents 50 units, and the digit 0 in the ones place represents 0 units. By adding the digits based on their place value, we get the answer 69, where 9 is in the ones place and 6 is in the tens place.

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If Ali took 6 pencils from a new box of pencils and there are 18 pencils left in the box, then there were 24 pencils in the brand new box.

To see why, you can add the number of pencils Ali took to the number of pencils left in the box:

6 + 18 = 24

Therefore, there were 24 pencils in the brand new box before Ali took 6 of them.

Probability or p-vale that the sample mean would differ from the population mean by less than 221 miles in a sample of 56 tires is equals to zero if the manager is correct.

We have data of an operation manager at a tire manufacturing company.

Mean mileage of a tire, [tex] \mu[/tex]

= 37,014 miles

standard deviation, [tex] \sigma[/tex]

= 4617 miles.

Sample size, n = 56

We have to determine the probability that the sample mean would differ from the population mean by less than 221 miles. Using Z-score formula in normal distribution, [tex]\small z= \frac{ \bar x-\mu }{\frac{\sigma }{\sqrt{n}}},[/tex]

Plugging all known values in above formula, [tex]z = \frac{ 221 - 37,014} {\frac{4617}{ \sqrt{56}}}[/tex]

= 59.634

[tex]P( \bar x < 221) = P ( \frac{ \bar x-\mu }{\frac{\sigma }{\sqrt{n}}} < \frac{ 221 - 37,014} {\frac{4617}{ \sqrt{56}}}) \\ [/tex]

=> P ( z < 59.63) = P( \bar x < 221)

Using the Z-distribution table, probability value is equals to 0. Hence, required probability is zero.

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Answer:they cannot conclude the mean price

Step-by-step explanation:

khan

At the α=0.01 significance level, there is not enough evidence to conclude that the mean price of gasoline in your state is more than $3 per gallon on that day.

Here you collected a random sample of 40 gas stations and calculated the sample mean (bar x) and the sample standard deviation (sₓ).

In this case, you found that the test statistic t was approximately 0.65, and the P-value was approximately 0.260. The P-value is the probability of observing a test statistic as extreme as the one you calculated, assuming that the null hypothesis is true.

A P-value of 0.260 means that if the null hypothesis were true, there is a 26% chance of observing a sample mean as extreme or more extreme than the one you calculated.

To make a decision about the hypothesis, you need to compare the P-value to the significance level (α), which represents the maximum probability of rejecting the null hypothesis when it is actually true. In this case, the significance level is set to α=0.01, which means that you want to be 99% confident in your decision.

If the P-value is less than the significance level, you reject the null hypothesis in favor of the alternative hypothesis.

If the P-value is greater than the significance level, you fail to reject the null hypothesis.

In this case, the P-value is greater than the significance level, which means that you fail to reject the null hypothesis.

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Complete Question:

An economist studying fuel costs suspected that the mean price of gasoline in her state was more than $3 per gallon on a certain day. On that day, she sampled 40 gas stations to test H0:μ=$3

Ha​:μ>$3

where μ is the mean price of gasoline per gallon that day in her state.

The sample data had a mean of bar x=$3.04 and a standard deviation of sₓ=$0.39

These results produced a test statistic of t≈0.65 and a P-value of approximately 0.260

Assuming the conditions for inference were met, what is an appropriate conclusion at the α=0.01  significance level?

Stella drove 150 miles this week to and from school in the van.

To determine how many miles Stella drove this week, we can use the given information about the van's gas mileage.

First, we know that the van can drive 60 miles using 10 gallons of gasoline. We can calculate the miles per gallon (mpg) by dividing the miles driven by the gallons of gasoline used:
[tex]Miles per gallon (mpg) = \frac{60 miles}{10 gallons}  = 6 mpg[/tex]

Now, we know that Stella used 25 gallons of gas driving to and from school this week in the van. To find out how many miles she drove, we can multiply the gallons of gas she used by the van's mpg:
Miles driven = 25 gallons x 6 mpg = 150 miles

So, Stella drove 150 miles this week to and from school in the van. We know this by calculating the van's gas mileage (6 mpg) and multiplying it by the gallons of gas Stella used (25 gallons).

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The researcher can be 95% confident that the mean number of hours all the residents of the city are watching news on television is between 1.7538 and 1.8462 hours, with a margin of error of 0.0462 hours.

Based on the information provided, the researcher surveyed 220 residents of the city and found that the mean number of hours they spend watching news on television each day is 1.8, with a standard deviation of 0.35.

The researcher wants to know the margin of error at a 95% confidence level for the mean number of hours all residents of the city are watching news on television.

To calculate the margin of error, we need to use the formula:
Margin of error = Critical value x Standard error

The critical value for a 95% confidence level is 1.96, and the standard error can be calculated as:
Standard error = Standard deviation /square root of sample space

Substituting the values given:
Standard error = 0.35 / sqrt(220) = 0.0236

Therefore, the margin of error can be calculated as:
Margin of error = 1.96 x 0.0236 = 0.0462

So, the researcher can be 95% confident that the mean number of hours all the residents of the city are watching news on television is between 1.7538 and 1.8462 hours, with a margin of error of 0.0462 hours.

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The number of triangles formed or the interior angle sum should be matched to each regular polygon as follows;

In Mathematics and Geometry, the sum of the interior angles of both a regular and irregular polygon can be calculated by using this formula:

Sum of interior angles = 180 × (n - 2)

1,440 = 180 × (n - 2)

1,440 = 180n - 360

180n = 1,440 + 360

n = 1,800/180

n = 10 (decagon).

Sum of interior angles = 180 × (n - 2)

1,800 = 180 × (n - 2)

1,800 = 180n - 360

180n = 1,800 + 360

n = 2,160/180

n = 12 (dodecagon).

Generally speaking, the number of triangles in a regular polygon (n-gon) can be calculated by using this formula;

Number of triangles = n - 2

4 = n - 2

n = 4 + 2 = 6 (regular hexagon).

Number of triangles = n - 2

6 = n - 2

n = 6 + 2 = 8 (regular octagon).

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Answer:

The cost of Ticket = £63Booking Fee = 3% = 63*(3/100) = 1.89 = £ 1.89The total price of the ticket = £63 + £1.89 = £64.89. Therefore, the total price of the ticket is £64.89.

A circumscribed angle is an angle whose sides are tangent to a circle. In other words, the angle is formed by two intersecting chords of the circle. The vertex of the angle is located outside of the circle, while the two endpoints of the angle lie on the circle.

Circumscribed angles have some important properties in geometry. For example, the measure of a circumscribed angle is half the measure of its intercepted arc (the arc of the circle that lies inside the angle). Additionally, if two angles intercept the same arc of a circle, they are congruent.

Circumscribed angles also appear frequently in trigonometry, where they are used to define the sine, cosine, and tangent functions. The sine of a circumscribed angle is defined as the ratio of the length of the opposite side of the angle to the length of the circle's radius. The cosine of a circumscribed angle is defined as the ratio of the length of the adjacent side of the angle to the length of the radius.

Finally, the tangent of a circumscribed angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

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The equation of the tangent line to f(x) is y = -4x + 4

The equation of the tangent line to f(x) = 1/x at a point x = a is given by:

y - f(a) = f'(a) * (x - a)

where f'(x) is the derivative of f(x) with respect to x.

We can find a "nice" point near 0.504 by choosing a = 0.5, which is close to 0.504 and makes the calculation easy.

The derivative of f(x) = 1/x is given by:

[tex]f'(x) = -1/x^2[/tex]

At x = 0.5, we have:

f(0.5) = 1/0.5 = 2

[tex]f'(0.5) = -1/(0.5)^2 = -4[/tex]

Plugging these values into the equation of the tangent line, we get:

y - 2 = -4 * (x - 0.5)

Simplifying, we get:

y = -4x + 4

Now we can use this tangent line to approximate (1/0.504) as follows:

(1/0.504) ≈ y(0.504)

Plugging x = 0.504 into the equation of the tangent line, we get:

y(0.504) = -4(0.504) + 4 = 1.784

Therefore, (1/0.504) ≈ 1.784

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Following shows (sin 7x)/sin x = 64(cos x)^6 - 80(cos x)^4 +24(cos x)^2 - 1:

12sin^3 x - 8sin^

To show that (sin 7x)/sin x is equal to 64(cos x)^6 - 80(cos x)^4 + 24(cos x)^2 - 1, we can use trigonometric identities and algebraic manipulation.

Let's start with the left-hand side of the equation:

(sin 7x)/sin x

Using the trigonometric identity for sin(A + B):

sin(A + B) = sin A cos B + cos A sin B

We can rewrite sin 7x as sin (6x + x):

sin (6x + x) = sin 6x cos x + cos 6x sin x

Now we can substitute sin 7x with sin 6x cos x + cos 6x sin x:

(sin 6x cos x + cos 6x sin x)/sin x

Next, we can simplify this expression by dividing both terms by sin x:

(sin 6x cos x)/sin x + (cos 6x sin x)/sin x

The sin x term cancels out, leaving us with:

sin 6x cos x + cos 6x

Now, we can use the double-angle identity for sin 2A:

sin 2A = 2sin A cos A

To rewrite sin 6x cos x, we can treat it as sin 2A with A = 3x:

sin 6x cos x = 2sin 3x cos 3x

Next, we can use the triple-angle identity for sin 3A:

sin 3A = 3sin A - 4sin^3 A

To rewrite sin 3x, we can treat it as sin A with A = x:

sin 3x = 3sin x - 4sin^3 x

Substituting this into our expression:

2sin 3x cos 3x = 2(3sin x - 4sin^3 x) cos 3x

Expanding further:

= 6sin x cos 3x - 8sin^3 x cos 3x

Now, we can use the double-angle identity for cos 2A:

cos 2A = cos^2 A - sin^2 A

To rewrite cos 3x, we can treat it as cos A with A = x:

cos 3x = cos^2 x - sin^2 x

Substituting this into our expression:

6sin x cos 3x - 8sin^3 x cos 3x = 6sin x (cos^2 x - sin^2 x) - 8sin^3 x (cos^2 x - sin^2 x)

Expanding further:

= 6sin x cos^2 x - 6sin x sin^2 x - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x

Now, we can use the Pythagorean identity for sin^2 x + cos^2 x:

sin^2 x + cos^2 x = 1

To rewrite sin^2 x, we can subtract cos^2 x from both sides:

sin^2 x = 1 - cos^2 x

Substituting this back into our expression:

= 6sin x (cos^2 x - (1 - sin^2 x)) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x

= 6sin x (2sin^2 x) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x

= 12sin^3 x - 8sin^

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Carmen needs (6/7) cups of strawberry frosting and (1 1/7) cups of lemon frosting to make 2 cups of the frosting mixture.

To determine the amount of strawberry frosting and lemon frosting that Carmen needs to make 2 cups of the frosting mixture, we need to use a proportion.

Let x be the amount of strawberry frosting needed in cups, and y be the amount of lemon frosting needed in cups.

From the given information, we know that Carmen mixed 1/4 cup of strawberry frosting with 1/3 cup of lemon frosting. Thus, the ratio of the amounts of strawberry frosting to lemon frosting is:

x/y = (1/4)/(1/3)

Simplifying this ratio, we get:

x/y = 3/4

We also know that the total amount of frosting needed is 2 cups, so:

x + y = 2

Using substitution, we can solve for x:

x + (4/3)x = 2

(7/3)x = 2

x = (6/7) cups

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The length of the shadow cast on level ground by a radio mast 90m high when the elevation of the sun is 40 degrees is approximately 85.3 meters.

To calculate the length of the shadow, we need to use trigonometry. We can imagine a right-angled triangle, where the height of the mast is the opposite side, the length of the shadow is the adjacent side, and the angle of elevation is 40 degrees.

Using the trigonometric function tangent (tan), we can find the length of the shadow, which is equal to the opposite side (90m) divided by the tangent of the angle of elevation (40 degrees). Therefore, the length of the shadow is approximately 85.3 meters.

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A family camping in a national forest used a 4-foot pole and a tarp to build a temporary shelter. One corner of the tarp was staked 5 feet from the bottom of the pole. The slope of the tarp from that corner to the top of the pole is 0.8 or 4/5.

We can draw a right triangle with the pole being the height, the distance from the pole to the stake being the base, and the slope of the tarp being the hypotenuse. The hypotenuse is the longest side of the triangle and is opposite to the right angle.

Using the Pythagorean theorem, we can find the length of the hypotenuse

hypotenuse² = height² + base²

hypotenuse² = 4² + 5²

hypotenuse² = 41

hypotenuse = √(41)

Therefore, the slope of the tarp is the ratio of the height to the base, which is

slope = height / base = 4 / 5 = 0.8

So the slope of the tarp from that corner to the top of the pole is 0.8 or 4/5.

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The actual percent loss on the original $7000 investment is 46%. This means that the investment consultant's statement of a 10% overall increase is not accurate.

To calculate the actual percent gain or loss on the original $7000 investment, we can use the following formula:

Actual percent gain or loss = (Ending value - Beginning value) / Beginning value * 100%

At the end of the first year, the investment decreased by 70% of its original value, which means its value was only 30% of $7000, or $2100.

At the end of the second year, the investment increased by 80% of the value it had at the end of the first year. So, its value at the end of the second year was:

Value at end of second year = $2100 + 80% of $2100
Value at end of second year = $2100 + $1680
Value at end of second year = $3780

Therefore, the actual percent gain or loss on the original $7000 investment is:

Actual percent gain or loss = ($3780 - $7000) / $7000 * 100%
Actual percent gain or loss = -46%

So, the actual percent loss on the original $7000 investment is 46%.

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1 x 10⁻⁹ expresses a very small number i.e. nanosecond using a negative power of 10. The correct answer is option b).

Computer calculation speeds are incredibly fast, and they are usually measured in very small units of time. One of these units is a nanosecond, which is equal to one billionth of a second, or 0.000000001 seconds. This unit is used to measure the time it takes for a computer to perform basic operations such as adding two numbers or accessing data from memory.

To express 0.000000001 in scientific notation using a negative power of 10, we need to determine the number of decimal places to the right of the decimal point until we reach the first non-zero digit. In this case, we count nine decimal places to the right of the decimal point before we reach the first non-zero digit, which is 1. This means that 0.000000001 can be written as 1 x 10⁻⁹.

In scientific notation, any number can be expressed as the product of a number between 1 and 10, and a power of 10. The power of 10 tells us how many places we need to move the decimal point to the left or right to express the number in standard form. In the case of 0.000000001, we need to move the decimal point nine places to the right to express the number in standard form.

By writing this number in scientific notation as 1 x 10⁻⁹, we can easily perform calculations with it and compare it to other values measured in nanoseconds. Hence option b) is the correct option.

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A. The joint probability distribution is:
- P(X=2, Y=0) = 6/1332
- P(X=1, Y=1) = 36/1332
- P(X=0, Y=2) = 30/1332

B. The expected value of playing this game is approximately $0.19 each time you play.

A. To find the joint probability distribution, we need to determine the probabilities of all possible outcomes for X and Y when selecting two marbles from the bag.

There are a total of 37 marbles in the bag (3 gold, 6 silver, and 28 black).

1. Probability of selecting 2 gold marbles (X=2, Y=0):
(3/37) * (2/36) = 6/1332

2. Probability of selecting 1 gold and 1 silver marble (X=1, Y=1):
(3/37) * (6/36) + (6/37) * (3/36) = 36/1332

3. Probability of selecting 2 silver marbles (X=0, Y=2):
(6/37) * (5/36) = 30/1332

So, the joint probability distribution is:

- P(X=2, Y=0) = 6/1332
- P(X=1, Y=1) = 36/1332
- P(X=0, Y=2) = 30/1332

B. To find the expected value of playing the game, we need to calculate the probability of selecting each type of marble and multiply it by its corresponding value.

1. Probability of selecting a gold marble: 3/37
Winning amount: $3

2. Probability of selecting a silver marble: 6/37
Winning amount: $2

3. Probability of selecting a black marble: 28/37
Losing amount: -$1

Expected value = (3/37 * $3) + (6/37 * $2) + (28/37 * -$1)
= 9/37 + 12/37 - 28/37
= -7/37

So, the expected value of playing this game is -$7/37, which means you can expect to lose approximately $0.19 each time you play.

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