Solve the following absolute value equations. Show the solution set.

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

Answer:

Sure, let me know if you have a new question or need any assistance!

A telephone pole has a wire attached to its top that is anchored to the ground then conclude the height of the pole is approximately 51.53 feet.

Let h be the height of the pole. The equation h = (h - 69) + 6 represents the given information. Solving it gives h = 75.

Let's denote the height of the pole as "h". Then, according to the problem, the distance from the bottom of the pole to the anchor point is 69 feet less than the height of the pole, which means it is h - 69. Additionally, the wire is to be 6 feet longer than the height of the pole, so its length is h + 6.

Now we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (in this case, the wire) is equal to the sum of the squares of the lengths of the other two sides (in this case, the height of the pole and the distance from the bottom of the pole to the anchor point). So we have:

(h - 69)^2 + h^2 = (h + 6)^2

h^2 - 138h + 4761 + h^2 = h^2 + 12h + 36

h^2 - 75h - 1602 = 0

h = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = -75, and c = -1602.

h = (75 ± sqrt(75^2 - 4(1)(-1602))) / 2(1)

h ≈ 51.53 or h ≈ -31.53

Since the height of the pole cannot be negative, we can ignore the negative solution and conclude that the height of the pole is approximately 51.53 feet.

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The volume of the regular pentagonal prism with an edge length of 9 m and a height of 13 m is approximately 1811.6 m³ to the nearest tenth.

To find the volume of a regular pentagonal prism with an edge length of 9 m and a height of 13 m, follow these steps:

Step 1: Find the apothem (a) of the base pentagon. Use the formula a = s / (2 * tan(180/n)), where s is the edge length and n is the number of sides (5 for a pentagon).

a = 9 / (2 * tan(180/5))
a ≈ 6.1803 m

Step 2: Calculate the area (A) of the base pentagon. Use the formula A = (1/2) * n * s * a.

A = (1/2) * 5 * 9 * 6.1803
A ≈ 139.3541 m²

Step 3: Determine the volume (V) of the pentagonal prism. Use the formula V = A * h, where h is the height.

V = 139.3541 * 13
V ≈ 1811.6033 m³

So, the volume of the regular pentagonal prism with an edge length of 9 m and a height of 13 m is approximately 1811.6 m³ to the nearest tenth.

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Answer:

(3x+4) ÷ (x+6)

Step-by-step explanation:

3x²-14x-24 = (3x+4) (x-6)

x²-36 = (x+6) (x-6)

= (3x+4) (x-6) ÷ (x+6) (x-6)

Eliminate the (x-6)

= (3x+4) ÷ (x+6)

The line of regression equation for the mileage rating of a ( point) four wheel drive vehicle is [tex]\hat y = 16.111 + 0.365x,[/tex] and the best predicted new mileage rating of a ( point) four-wheel drive vehicle when x = 19 mi/gal, is equals to the 23.046 mi/gal. So, option(b) is right one.

A linear regression line has an equation of the form [tex]\hat y = a + bx,[/tex]

where x is the independent variable and y is the dependent variable. The slope of the line is b, and a is the estimated intercept (the value of y when x = 0). We have a table form data of old and new rating of four-wheel-drive vehicles. We have to determine the line of regression. Now, we have to calculate the value of 'a' and 'b'. Let the old and new mileage rating of four-wheel-drive vehicles be represented by vaiables 'x' and 'y'. Using the following formulas, [tex]b =\frac{ S_{xy}}{S_{xx}}[/tex] where, [tex]S_{xx} = \sum x² - \frac{ (\sum x)² }{n} [/tex]

where , [tex]\bar x = \frac{\sum x }{n}[/tex]

Here, n = 11, [tex]\sum x[/tex] = 235

[tex]\sum xy[/tex] = 263, [tex]\sum x²[/tex] = 5733, [tex]\sum xy[/tex] = 5879, so

[tex]S_{xx}[/tex] = 5733 - (235)²/11

= 5733 - 5020.454 = 712.546

[tex]S_{xy}[/tex] = 5879 - (235×263)/11

= 260.364

Now, b = 260.364/712.546 = 0.365

a = (263/11) - 0.365 ( 235/11)

= 23.909 - 7.798

= 16.111

So, regression line equation is

[tex]\hat y = 16.111 + 0.365x,[/tex]

The best predicted value of y, when x = 19 mi/gal, [tex]\hat y = 19× 0.365 + 16.111[/tex]

=23.046 mi/gal

Hence, the best predicted value is

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Complete question:

10. Determine the line of regression and use it to find the best predicted new mileage rating of a ( point) four-wheel-drive vehicle given that the old rating is 19 mi/gal

old 6 27 17 33 28 24 8 22 20 29 21

New 15 24 15 29 25 22 6 20 82 6 19

a) y cap = 0.863 + 0.808x; 16.2 mi/gal

b) y cap = 16.111 + 0.365x; 23.04 mi/gal

c) y cap =0.808+ 0.863x; 17.2 mi/gal

d) y cap = 0.808 0.863x; 18.1 mi/gal

(a) 1/100 (or 0.01)

(e) 0.01

This factor represents dividing the number by 100.

When Shelby multiplies 7,358.9 by a power of 10 and gets the product 73.589, we can determine the factor by comparing the two numbers.

7,358.9 → 73.589

We can see that the decimal point has moved two places to the left. Therefore, the factor is the one that will shift the decimal point two places to the left. Among the given options, the factor that does this is:

(a) 1/100 (or 0.01)

(e) 0.01

This factor represents dividing the number by 100. The other options (b, c, d, and f) do not represent the correct division by powers of 10 in this case.

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If two fair dice are rolled, the probability that the total showing is either even or less than five is 17/36. The correct answer is B.

There are 36 possible outcomes when rolling two dice, since each die has 6 possible outcomes.

To find the probability that the total showing is either even or less than five, we can first find the probability that the total showing is even, and then add to that the probability that the total showing is less than five, excluding the outcomes where the total showing is even.

To find the probability that the total showing is even, we can consider the following possibilities:

both dice show even numbers (probability 1/4)

both dice show odd numbers (probability 1/4)

So the probability of rolling an even total is 1/4 + 1/4 = 1/2.

To find the probability that the total showing is less than five, excluding the outcomes where the total showing is even, we can consider the following possibilities:

So the probability of rolling a total less than five, excluding the outcomes where the total showing is even, is 1/36 + 1/18 + 1/18 + 1/12 + 1/12 + 1/9 + 1/9 + 1/9 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 11/36.

Therefore, the probability that the total showing is either even or less than five is 1/2 + 11/36 = 17/36.

So the correct answer is (b) 17/36.

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Denise can drive at most 80 miles to stay within her budget of $55.

Let's assume that Denise drives x miles during the rental period. Then the total cost of the rental will be:

Total cost = $35 (flat rate for the day) + $0.25 per mile x (number of miles driven)

We want to find the greatest number of miles that Denise can drive and still stay within her budget of $55, so we can set up an inequality as follows:

Total cost ≤ $55

$35 + $0.25x ≤ $55

Subtracting $35 from both sides

$0.25x ≤ $20

Dividing both sides by $0.25

x ≤ 80

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To evaluate the integral of (3x+6)/(2x²-3)dx, we can use partial fraction decomposition:

(3x+6)/(2x²-3) = A/(x-√(3)/2) + B/(x+√(3)/2)

Multiplying both sides by the denominator and simplifying, we get:

3x+6 = A(x+√(3)/2) + B(x-√(3)/2)

Setting x = √(3)/2, we get:

3√(3)/2 + 6 = B(√(3)/2-√(3)/2) = 0

So B = -2√(3). Setting x = -√(3)/2, we get:

-3√(3)/2 + 6 = A(-√(3)/2+√(3)/2) = 0

So A = 2√(3). Therefore, we have:

(3x+6)/(2x^2-3) = 2√(3)/(x-√(3)/2) - 2√(3)/(x+√(3)/2)

Integrating each term, we get:

∫(3x+6)/(2x²-3)dx = 2√(3)ln|x-√(3)/2| - 2√(3)ln|x+√(3)/2| + C

where C is the arbitrary constant.
To evaluate the integral of the function 3x + 6 with respect to x, we will use the integral symbol and find the antiderivative:

∫(3x + 6) dx

To find the antiderivative, we will apply the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), and the constant rule, which states that the integral of a constant is the constant times the variable:

(3 * (x^(1+1))/(1+1)) + (6 * x) + C

Simplifying the expression:

(3x²)/2 + 6x + C

Here, C is the arbitrary constant. So, the evaluated integral of 3x + 6 is:

(3x²)/2 + 6x + C

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The ratios that have a unit rate of 3 include the following:

A. 15/2 cups: 2 1/2 cups

C. 2 cups: 2/3 cups

F. 2 1/2 cups: 5/6 cups

In Mathematics, the unit rate is sometimes referred to as unit ratio and it can be defined as the quantity of material that is equivalent to a single unit of product or quantity.

15/2 cups : 5/2 cups

15/2 ÷ 5/2 : 5/2 ÷ 5/2

15/2 × 2/5  : 1

3 : 1  (True)

1 cup: 1/4 cups

1 × 4 : 1/4 × 1

4 : 1   (False)

2/3 cups: 1 cup

2/3 × 3/2  : 1 × 3/2

1 : 3/2   (False)

3 3/4 cups: 2 cups

(4 × 3 + 3)/4 : 2

15/4 : 2

15/8  : 2/2

15/8 : 1    (False).

2 cups: 2/3 cups

2 × 3/2 : 2/3 × 3/2

3  : 1   (True).

2 1/2 cups: 5/6 cups

(2 × 2 + 1)/2 : 5/6

5/2  : 5/6

5/2 × 6/5 : 5/6 × 6/5

3 : 1  (True).

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Complete Question:

Which ratios have a unit rate of 3? Choose all that apply.

15/2 cups: 2 1/2 cups

1 cup: 1/4 cups

2/3 cups: 1 cup

3 3/4 cups: 2 cups

2 cups: 2/3 cups

2 1/2 cups: 5/6 cups

The height of the replica souvenir is approximately 7.55 inches.

To find the height of the replica souvenir of the Statue of Liberty, we'll use the given scale factor of 1 inch = 20 feet. The original height of the statue is 151 feet. Divide the original height by the scale factor:

151 ft / 20 ft/in = 7.55 inches

The height of the replica souvenir is approximately 7.55 inches.

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The quadratic function in vertex form is y = (8/9)(x - 5)^2 + 7

The vertex form of a quadratic function is given by:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

In this case, we are given that the vertex is (5, 7), so we can write:

y = a(x - 5)^2 + 7

To find the value of a, we can use one of the points on the parabola.

Let's use the point (2, 15):

15 = a(2 - 5)^2 + 7

8 = 9a

a = 8/9

Substituting this value of a into the equation above, we get:

y = (8/9)(x - 5)^2 + 7

Therefore, the quadratic function in vertex form is y = (8/9)(x - 5)^2 + 7

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The values of m and n are

m = 0 and n = 3.125

The graph is attached

The values of m and n are solved using the relationship between miles and kilometers. This type of relationship is a linear proportional relationship. Linear relationship implies the graph will be a straight line graph.

This relationship is that 1 mile equals 0.625 km hence the linear equation is

y = 0.625x

when x = 0, we have that

y = 0.625 * 0

y = 0

when x = 5, we have that

y = 0.625 * 5

y = 3.125

Where x is kilometers and y is miles

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The length that will give a plot with the maximum area allowed is 6 meters.

To find the length that will give a plot with the maximum area, we need to use the formula for the area of a rectangle, which is A = lw, where A is the area, l is the length, and w is the width. In this case, we are given that the area is 45 square meters, and the width is 7.5 meters.

Substituting these values into the formula, we get:

45 = l(7.5)

To solve for l, we divide both sides by 7.5:

l = 45/7.5

Simplifying, we get:

l = 6

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The dimensions of the original square is 8 by 8.

Let x be the original side length of the square. The area of the square is x². When one side is doubled and the adjacent side is diminished by 3, the rectangle's dimensions become 2x and (x-3). The area of the rectangle is (2x)(x-3) = 2x² - 6x.

According to the problem, the area of the rectangle is greater than the area of the square by twice the original side of the square, which is 2x. So we can set up the equation:

2x² - 6x = x² + 2x

Now, solve for x:

2x² - x² = 6x + 2x
x² = 8x
x = 8

So the dimensions of the original square are 8 by 8.

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The statement which correctly describes how to graph the equation shown above include the following: Start with a point at (0, 0). Then go up 1 and 4 to the right.

In Mathematics, the translation a geometric figure or graph to the right simply means adding a digit to the value on the x-coordinate of the pre-image while the translation a geometric figure or graph upward simply means adding a digit to the value on the y-coordinate (y-axis) of the pre-image.

In Mathematics and Geometry, the translation a geometric figure upward simply means adding a digit to the value on the positive y-coordinate (y-axis) of the pre-image;

g(x) = f(x) + N

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

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The artist used a total of $0.33 worth of paint for the project.

The artist purchased 2 liters of paint, which is equivalent to 2,000 milliliters of paint. This amount of paint was used to complete a project, and after the project was finished, there were 350 milliliters of paint left over.

To determine how much paint was used for the project, we subtract the amount of leftover paint from the total amount of paint purchased, which gives us 2,000 - 350 = 1,650 milliliters of paint used for the project.

The cost of the paint is 2 cents per milliliter, which is equivalent to $0.02/100 milliliters or $0.0002 per milliliter. To determine the cost of the paint used for the project, we multiply the amount of paint used by the cost per milliliter.

Therefore, the cost of 1,650 milliliters of paint used for the project can be calculated by multiplying 1,650 milliliters by $0.0002 per milliliter, which gives us $0.33.

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The margin of error at the 99% confidence interval is 1.39%. Interpretation: we can be 99% confident that the true proportion of viewers under 30 in the population falls within the range of 30% ± 1.39%.

To calculate the margin of error at the 99% confidence interval, we can use the formula:

Margin of error = z* × √(p × (1 - p) / n)

where z* is the critical value (2.576 for a 99% confidence interval), p is the sample proportion (0.30), and n is the sample size (5000).

Margin of error = 2.576 × √(0.30 × (1 - 0.30) / 5000) ≈ 0.0139 or 1.39%

The interpretation of this result is that we can be 99% confident that the true proportion of viewers under 30 in the population falls within the range of 30% ± 1.39%. In other words, if we were to conduct the survey multiple times, we would expect the proportion of viewers under 30 to fall within this interval 99 out of 100 times. This information is useful for understanding the level of uncertainty in the survey results and can help guide decision-making based on the findings.

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The amount of money Justin saves in the first month would be 5 times the value of x, where x represents the number of months of gym membership, based on the discounted price provided.

A linear equation is an equation in mathematics that represents a relationship between two variables that is a straight line when graphed on a coordinate plane. It is an equation of the form:

y = mx + b

To calculate the amount of money Justin saves in the first month by joining the gym at the discounted price rather than the regular price, we need to know the discounted price.

The equation given is y = 15x + 20, where y represents the regular price in dollars and x represents the number of months of gym membership. However, we need to know the discounted price, which is not provided in the given information.

Once we have the discounted price, we can substitute it into the equation and calculate the savings. For example, if the discounted price is y = 10x + 20, then we can calculate the savings by subtracting the discounted price from the regular price:

Savings = Regular price - Discounted price

= (15x + 20) - (10x + 20)

= 15x - 10x

= 5x

Hence, the amount of money Justin saves in the first month would be 5 times the value of x, where x represents the number of months of gym membership, based on the discounted price provided.

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The statement "the series converges to 31.25" is true about the given series.

Given series is 25 + 5 + 1 + ....

We can clearly see that given series is infinite geometric series.

First term is a=25

common ratio is r = 5/25

= 1/5

We know that the formula of sum of an infinite geometric series is

S = a / (1 - r)

S = 25 / (1 - 1/5)

S = 25/(4/5)

S = (25*5)/4

S = 125/4

S = 31.5

Therefore, the sum of the infinite geometric series is 31.25.

Since, the sum of the series is a finite number, we can say that the series converges.

Therefore, the statement "the series converges to 31.25" is true about the given series.

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The component of the weight parallel to the inclined plane is approximately 34.72 pounds, and the component perpendicular to the inclined plane is approximately 196.96 pounds.

To analyze the situation, we need to break down the weight of the object into its components parallel and perpendicular to the inclined plane.

Given:

Weight of the object = 200 pounds

Angle of the inclined plane with the horizontal = 10 degrees

First, we find the component of the weight parallel to the inclined plane. This component can be determined using trigonometry:

Component parallel to the inclined plane = Weight * sin(angle)

Component parallel to the inclined plane = 200 pounds * sin(10 degrees)

Component parallel to the inclined plane ≈ 200 pounds * 0.1736

Component parallel to the inclined plane ≈ 34.72 pounds

Next, we find the component of the weight perpendicular to the inclined plane:

Component perpendicular to the inclined plane = Weight * cos(angle)

Component perpendicular to the inclined plane = 200 pounds * cos(10 degrees)

Component perpendicular to the inclined plane ≈ 200 pounds * 0.9848

Component perpendicular to the inclined plane ≈ 196.96 pounds

Therefore, the component of the weight parallel to the inclined plane  and the component perpendicular to the inclined plane is approximately 34.72 pounds and 196.96 pounds respectively.

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The divergence of the given vector field at all points where it's defined is div [tex]F = 4x - 2x × cos(x^2).[/tex]

To find the divergence of the given vector field at all points where it's defined, we will use the following terms:

divergence, vector field, and partial derivatives.

The given vector field is[tex]F = (2x^2 - sin(x^2)) i + 5j - sin(x^2) k.[/tex]

To find the divergence of F (div F), we need to take the partial derivatives of each component with respect to their

respective variables and then sum them up. So, div [tex]F = (∂(2x^2 - sin(x^2))/∂x) + (∂5/∂y) + (∂(-sin(x^2))/∂z)[/tex].

Find the partial derivative of the first component with respect to x:

[tex]∂(2x^2 - sin(x^2))/∂x = 4x - 2x × cos(x^2)[/tex] (applying chain rule).

Find the partial derivative of the second component with respect to y:

∂5/∂y = 0 (since 5 is a constant).

Find the partial derivative of the third component with respect to z:

[tex]∂(-sin(x^2))/∂z = 0[/tex] (since there is no z variable in the component).

Sum up the partial derivatives:

[tex]div F = (4x - 2x × cos(x^2)) + 0 + 0 = 4x - 2x × cos(x^2).[/tex]

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Answer:

hope it helps! :)

Step-by-step explanation:

m < A = 61 bc 90+29=119

180-119=61

ED: 9y-22=5

9y=27

y=3

9(3)-22 =5

ED=5

3x-5=13

3x=18

x=6

[tex]BC^{2}[/tex]+25+=169

[tex]BC^{2}[/tex]=144

BC=12

m<DCE=29

y=3

Answer:

17.2 cm²

Step-by-step explanation:

Alr let me try

The angle is 1.2 you got it right. The rest is in the pics

Answer: A(shaded)=17.15 cm²

Step-by-step explanation:

What you did so far is correct.

Given:

r=5

s=6

Solve for Ф  angle:

s=[tex]\frac{part circle}{wholecircle} 2\pi r[/tex]         This way help you find the portion/percent you want

6=(Ф/360) (2[tex]\pi[/tex]5)       >solve for Ф  Divide by (2[tex]\pi[/tex]5) and multiply by 360

Ф=68.75

Solve for pie/sector

Now that you have the angle,  you can use the same concept for area

Area of sector = [tex]\frac{part circle}{wholecircle} \pi r^{2}[/tex]

Area of sector = [tex]\frac{68.75}{360} \pi 5^{2}[/tex]

Area of sector = 15.0 cm²

Now let find y so we can plug into area of triangle

use tan Ф = opposite/adjacent

tan 68.75 = y/5

y=5 * tan 68.75

y=12.86 cm

Area of triangle = 1/2 b h          b=y=12.86     h =5

Area of triangle = 1/2* 12.86*5

Area of triangle = 32.15 cm²

Now subtract area of sector from triangle

A(shaded)=A(triangle)-A(sector)

A(shaded)=32.15- 15.0

A(shaded)=17.15 cm²

Step-by-step explanation:

(a) [1/2 x 1/4] + [1/2 x 6]

First, we can simplify each term separately:

1/2 x 1/4 = 1/8

1/2 x 6 = 3

Now, we can add these two simplified terms:

1/8 + 3 = 3 1/8

Therefore, [1/2 x 1/4] + [1/2 x 6] simplifies to 3 1/8.

(b) [1/5 x 2/15] - [1/5 x 2/15]

Both terms are the same, so when we subtract them, the result will be zero:

[1/5 x 2/15] - [1/5 x 2/15] = 0

Therefore, [1/5 x 2/15] - [1/5 x 2/15] simplifies to 0.

Using the change of variables u = x and v = x + y, we transform the given region R into a rectangle S, and evaluate the integral as 9 (e^6 - 2e^4 + e^2 - 1).

We need to find a change of variables that maps the region R onto a rectangle in the uv-plane. Let's make the following substitutions

u = x

v = x + y

Then, the region R is transformed into the rectangle S defined by 0 ≤ u ≤ 1 and 0 ≤ v ≤ 2.

To find the limits of integration in the new variables, we can solve the equations 2[x] + 2y = 2 for x and y in terms of u and v

2[x] + 2y = 2

2u + 2v - 2[x] = 2

[x] = u + v - 1

Since [x] is the greatest integer less than or equal to x, we have

u + v - 1 ≤ x < u + v

Also, since 0 ≤ y ≤ 1, we have

0 ≤ x + y - u ≤ 1

u ≤ x + y < u + 1

u - x ≤ y < 1 + u - x

Now we can evaluate the integral using the new variables

∫∫R 9e^(2x+2y) dA = ∫∫S 9e^(2u+2v) |J| dudv

where J is the Jacobian of the transformation, given by

|J| = det [[∂x/∂u, ∂x/∂v], [∂y/∂u, ∂y/∂v]]

= det [[1, 1], [-1, 1]]

= 2

Therefore, the integral becomes

∫∫S 9e^(2u+2v) |J| dudv = 2 ∫0^1 ∫0^2 9e^(2u+2v) dudv

= 2 ∫0^1 [9e^(2u+2v)/2]_0^2 dv

= 2 ∫0^1 (9/2)(e^(4+2v) - e^(2v)) dv

= 2 (9/2) [(e^6 - e^2)/2 - (e^4 - 1)/2]

= 9 (e^6 - 2e^4 + e^2 - 1)

Therefore, the value of the integral is 9 (e^6 - 2e^4 + e^2 - 1).

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Answer:

66.7%

Step-by-step explanation:

Let people asked bye x.

Then, people considering to change = 29% of x

People considering to become vegetarians = 23% of (29% of x)

                                                                          = 23/100 * 29x/100

                                                                          = 667x/10000

Percentage of people considering to become vegetarians = 667x/10

                                                                                                  = 66.7%

The mean is 35 minutes and the median is 34 minutes.

let's first calculate the mean, median, and mode for Sarah's exercise times: 34, 30, 45, 30, 40, 38, and 28 minutes.

Step 1: Calculate the mean
Add up all the values and divide by the total number of values:
(34 + 30 + 45 + 30 + 40 + 38 + 28) / 7 = 245 / 7 = 35 minutes (rounded)

Step 2: Calculate the median
Arrange the values in ascending order: 28, 30, 30, 34, 38, 40, 45
There are 7 values, so the median is the middle value: 34 minutes

Step 3: Calculate the mode
Determine the value(s) that occur most often: 30 minutes (occurs twice)

Now, should Sarah use the mean or the median to show she exercises for large amounts of time each day? The mean is 35 minutes and the median is 34 minutes. Both values are close and represent the central tendency of the data. However, since the mean is slightly higher than the median, Sarah should use the mean (35 minutes) to show she exercises for a larger amount of time each day.

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Therefore, the linearization predicts that f(2.9, 4.1) is approximately 4.142.

To find the linearization of the function f(x, y) = √x^2 + y^2 at the point (3, 4), we need to find the partial derivatives of f with respect to x and y, evaluate them at (3, 4), and use them to write the equation of the tangent plane to the surface at that point.

First, we have:

∂f/∂x = x/√(x^2 + y^2)

∂f/∂y = y/√(x^2 + y^2)

Evaluating these at (3, 4), we get:

∂f/∂x(3, 4) = 3/5

∂f/∂y(3, 4) = 4/5

So the equation of the tangent plane to the surface at (3, 4) is:

z - f(3, 4) = (∂f/∂x(3, 4))(x - 3) + (∂f/∂y(3, 4))(y - 4)

Plugging in f(3, 4) = 5 and the partial derivatives, we get:

z - 5 = (3/5)(x - 3) + (4/5)(y - 4)

Simplifying, we get:

z = (3/5)x + (4/5)y - 1

This is the linearization of f(x, y) = √x^2 + y^2 at the point (3, 4).

To approximate f(2.9, 4.1), we plug in x = 2.9 and y = 4.1 into the linearization:

z = (3/5)(2.9) + (4/5)(4.1) - 1

z ≈ 4.142

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Based on the information provided, the probability that Ben wins the board game is 30%.

To calculate the probability of Ben winning the board game, let's start by checking the information provided:

Probability for Justin to win: 20% or 0.2

Probability for Cam to win: 50% or 0.5

Now, the total probability is always equivalent to 100% or 0.1. Based on this, let's calculate now the probability that Ben wins the game.

1 - (0.2 + 0.5)  

1 - 0.7 = 0.3

Note: Here is the complete question:

Justin, Cam, and Ben are playing a board game where exactly one player will win. Ben estimates that Justin has a %20 percent chance of winning each game and that Cam has a %50 percent chance of winning each game. What is the probability that Ben will win the board game?

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