You are planning to use a ceramic tile design in your new bathroom. The tiles are blue and white equilateral triangles. You decide to arrange the blue tiles in a hexagonal shapes as shown. If the side of each tile measures 4 centimeters, what will be exact area of each hexagonal shape?

A) 12cm²
B) 24√3 cm²
C) 192 cm²
D) 32√3 cm²​

Answer:

its 60

Step-by-step explanation:

(a) To find the center I and radius R of the circle tangent to each side of a triangle using vectors, we can use the following algorithm:

1. Calculate the midpoints of each side of the triangle.

2. Find the direction vectors of the triangle's sides.

3. Calculate the perpendicular vectors to each side.

4. Find the intersection points of the perpendicular bisectors.

5. Determine the circumcenter by finding the intersection point of the lines passing through the intersection points.

6. Calculate the distance from the circumcenter to any vertex to obtain the radius.

(b) Example: Let A(0, 0), B(4, 0), and C(2, 3) be the vertices of the triangle.

Using the algorithm:

1. Midpoints: M_AB = (2, 0), M_BC = (3, 1.5), M_CA = (1, 1.5).

2. Direction vectors: v_AB = (4, 0), v_BC = (-2, 3), v_CA = (-2, -3).

3. Perpendicular vectors: p_AB = (0, 4), p_BC = (-3, -2), p_CA = (3, -2).

4. Intersection points: I_AB = (2, 4), I_BC = (0, -1), I_CA = (4, -1).

5. Circumcenter I: The intersection point of I_AB, I_BC, and I_CA is I(2, 1).

6. Radius R: The distance from I to any vertex, e.g., IA, is the radius.

(c) Vector equation for parametrization: r(t) = I + R * cos(t) * u + R * sin(t) * v, where t is the parameter, u and v are unit vectors perpendicular to each other and to the plane of the triangle.

(a) Algorithm to find the center and radius of the circle tangent to each side of a triangle using vectors:

1. Calculate the vectors for the sides of the triangle: AB, BC, and CA.

2. Calculate the unit normal vectors for each side. Let's call them nAB, nBC, and nCA. To obtain the unit normal vector for a side, normalize the vector obtained by taking the cross product of the corresponding side vector and the vector perpendicular to it (in 2D, this can be obtained by swapping the x and y coordinates and negating one of them).

3. Calculate the bisectors for each angle of the triangle. To obtain the bisector vector for an angle, add the corresponding normalized side unit vectors.

4. Calculate the intersection point of the bisectors. This can be done by solving the system of linear equations formed by setting the x and y components of the bisector vectors equal to each other.

5. The intersection point obtained is the center of the circle tangent to each side of the triangle.

6. To calculate the radius of the circle, find the distance between the center and any of the triangle vertices.

(b) Example:

Let A = (0, 0), B = (4, 0), C = (2, 3√3) be the vertices of the triangle.

1. Calculate the vectors for the sides: AB = B - A, BC = C - B, CA = A - C.

  AB = (4, 0), BC = (-2, 3√3), CA = (-2, -3√3).

2. Calculate the unit normal vectors for each side:

  nAB = (-0.5, 0.866), nBC = (-0.5, 0.866), nCA = (0.5, -0.866).

3. Calculate the bisector vectors:

  bisector_AB = nAB + nCA = (-0.5, 0.866) + (0.5, -0.866) = (0, 0).

  bisector_BC = nBC + nAB = (-0.5, 0.866) + (-0.5, 0.866) = (-1, 1.732).

  bisector_CA = nCA + nBC = (0.5, -0.866) + (-0.5, 0.866) = (0, 0).

4. Solve the system of linear equations formed by the bisector vectors:

  Since the bisector vectors for AB and CA are zero vectors, any point can be the center of the circle. Let's choose I = (2, 1.155) as the center.

5. Calculate the radius of the circle:

  Calculate the distance between I and any of the vertices, for example, IA:

  IA = √((x_A - x_I)^2 + (y_A - y_I)^2) = √((0 - 2)^2 + (0 - 1.155)^2) ≈ 1.155.

Therefore, the center of the circle I is (2, 1.155), and the radius of the circle R is approximately 1.155.

(c) Vector equation for the parametrization of the circle:

  Let r(t) = I + R * cos(t) * u + R * sin(t) * v, where t is the parameter, and u and v are unit vectors perpendicular to each other and tangent to the circle at I.

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Answer:

Step-by-step explanation:

B or D

Answer:

Notice that each edge of the cube is 2 yards long, and the height of the pyramid is 2 yards long too.

The slant height refers to the height of a triangle on its face, which forms a right triangle with the height of the pyramid and half side. Using Pythagorean's Theorem, we have

\(s^{2}=1^{2} +2^{2}\\ s=\sqrt{5}\)

Therefore, the slant height is the square root of 5, in yards units. (A)

The surface of the composite figure is the sum of the surface area of both volumes.

\(S_{composite}=S_{cube} +S_{pyramid}\\ S_{composite}=5(2)^{2} +(2)^{2} +\frac{1}{2} (8)(\sqrt{5} )=20+4+4\sqrt{5} \\ S_{composite}\approx 32.9 yd^{2}\)

Therefore, the surface of the composite figure is 32.9 square yards. (B)

The concrete needed will fill the empty space between the cube and the pyramid, so we have to find the difference between their volumes.

\(V_{concrete}=V_{cube} -V_{pyramid}=2^{3}-\frac{1}{3}(2)^{2} (2)=8-\frac{8}{3} \\V_{concrete} \approx 5.33 yd^{3}\)

Therefore, we need 5.33 cubic yards of concrete to make the planter. (C)

Answer:

A. 2.24 yd

B. 53.67 yd²

C. 5.33 yd³

Step-by-step explanation:

Here we have;

Height of the pyramid = 2 yd

Side length of pyramid base = 2 yd

Therefore, the slant height forms a right triangle with the height of the pyramid and half the base hence;

(Slant height)² = (2·yd)² + (2·yd/2)² = 4·(yd)² + (yd)² = 5·(yd)²

∴ Slant height = √5 yards = 2.24 yd

B. The surface area of the cube with one side open is found as follows;

Surface area of container cube = 5 × (2·yd)² = 20 yards²

The surface area of the pyramid = Base area + 1/2 perimeter of base × Slant height

Since the base is open, we have;

The surface area of the pyramid = 1/2 perimeter of base × Slant height

= 1/2 × (4 × (2·yd))×yd·√5 = 4·yd×yd·√5 = 4·√5·(yd)²

Hence total surface area of the planter = Surface area of container cube + surface area of the pyramid

total surface area of the planter = 20·yd² + 4·√5 yd² = 24·√5 yards²

C. The volume of concrete needed to make the planter is the volume of the cube concrete container less the volume of the inverted pyramid

Volume of the cube = 2 × 2 × 2 = 8 yd³

Volume of the inverted pyramid = 1/3 × Base area × Height

Volume of the inverted pyramid = 1/3 × 2 × 2 × 2 = 8/3 yd³

Therefore, volume of the concrete needed = 8 yd³ - 8/3 yd³ = 16/3 yd³ = 5.33 yd³.

Answer:

3. -74

Step-by-step explanation:

\(12 = \frac{2 + z}{-6} \\-72 = 2+ z\\z = -74\)

There are 15 employees in the office and 5 need to be chosen, so there are a total of (15 choose 5) possible ways to select the employees. This can be calculated using the formula for combinations:

(15 choose 5) = 15! / (5! * 10!) = 3003

Now, we need to find the number of ways that the specific group of you, Diana, Kyle, Nancy, and Arthur can be chosen. This is simply (5 choose 5), since we want to choose all 5 members of the group.

(5 choose 5) = 5! / (5! * 0!) = 1

Therefore, the probability of this specific group being chosen is:

(5 choose 5) / (15 choose 5) = 1 / 3003 ≈ 0.0003

So the probability of you, Diana, Kyle, Nancy, and Arthur being chosen is very low, at approximately 0.0003 or 3 in 10,000. This is because there are many other possible groups of 5 employees that could be chosen from the pool of 15, and all of these groups have an equal chance of being selected.

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The estimate of the mean length of the insects Ciara found is 17 millimeters (mm).

To calculate an estimate of the mean length of the insects Ciara found, we need to find the weighted average of the lengths using the given frequencies.

Let's denote the lower limits of the length intervals as L1 = 0, L2 = 10, and L3 = 20.

Similarly, denote the upper limits as U1 = 10, U2 = 20, and U3 = 30.

Next, we calculate the midpoints of each interval by taking the average of the lower and upper limits.

The midpoints are M1 = (L1 + U1) / 2 = 5, M2 = (L2 + U2) / 2 = 15, and M3 = (L3 + U3) / 2 = 25.

Now, we can calculate the sum of the products of the frequencies and the corresponding midpoints.

This gives us (5 \(\times\) 5) + (6 \(\times\) 15) + (9 \(\times\) 25) = 25 + 90 + 225 = 340.

Next, we calculate the sum of the frequencies, which is 5 + 6 + 9 = 20.

Finally, we divide the sum of the products by the sum of the frequencies to find the weighted average, which is 340 / 20 = 17.

Therefore, the estimate of the mean length of the insects Ciara found is 17 millimeters (mm).

Thus, the mean length of the insects Ciara found is approximately 17 millimeters (mm).

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Answer:

X=8

Step-by-step explanation:

3×X =6×4

3X=24

X= 24÷3

X= 8

The equivalent value of the proportion is ( 3/4 ) = ( 6/8 ) and x = 8

The proportion formula is used to depict if two ratios or fractions are equal. The proportion formula can be given as a: b::c : d = a/b = c/d where a and d are the extreme terms and b and c are the mean terms.

The proportional equation is given as y ∝ x

And , y = kx where k is the proportionality constant

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Given data ,

Let the proportion be represented as A

Now , the value of A is

( 3/4 ) = ( 6/x )

Multiply by x on both sides , we get

( 3/4 )x = 6

Divide by 3 on both sides , we get

( 1/4 )x = 2

Multiply by 4 on both sides , we get

x = 8

Hence , the proportion is ( 3/4 ) = ( 6/8 )

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Answer: Circle 3

Step-by-step explanation: because if you slice it a certain way…

Answer:

A right trapezoid is a trapezoid in which one of the sides is perpendicular to the two bases: In this special case, if you know the length of the perpendicular side, that's the same as the altitude of the trapezoid.

The 90% confidence interval is: (2.51, 3.22)

It is a boundary of values which is eventually to comprise a population value with a certain degree of confidence. It is usually shown as a percentage whereby a population means lies within the upper and lower limit of the provided confidence interval.

We have the following information :

The level of significance is calculated as:

\(\alpha =1-c\\\\\alpha =1-0.90\\\\\alpha =0.10\)

The degrees of freedom for the case is:

df = n - 1

df = 15 - 1

df = 14

The 90% confidence interval is calculated as:

=x(bar) ±\(t_\frac{\alpha }{2}\), df \(\frac{s}{\sqrt{n} }\)

= 2.86 ±\(t_\frac{0.10 }{2}\), 14 \(\frac{0.78}{\sqrt{15} }\)

= 2.86 ± 1.761 × \(\frac{0.78}{\sqrt{15} }\)

= 2.86 ± 0.3547

= (2.51, 3.22)

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On 5th of April he received his raise.

If Steve started his new job on March 1st, then five weeks later would be April 5th. To find the exact day on which he received his raise, we need to know the specific day of the week on which he started his job.

If March 1st was a Monday, for example, then April 5th would also be a Monday, since there are seven days in a week and 35 days (5 weeks) divided by 7 is 5 with no remainder. If he started his job on a different day of the week, then the day on which he received his raise would be different.

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Answer:

Question 5 is positive 8

Step-by-step explanation:

I need to see the number line for question 4

Therefore, x = -1 is the value of the negative number x which satisfies \(|x-3|=|3x|+1\) .  Thus, option B -1 is correct.

To solve the equation  \(|x-3|=|3x|+1,\)  we need to consider two cases:

Case 1: If x-3 is non-negative \((i.e., x > = 3)\)  , then we have  \(|x-3| = x-3\)  , and  \(|3x|+1 = 3x+1.\) Therefore, the equation becomes:

\(x-3 = 3x+1\)

Simplifying this equation, we get:

\(-2x = -4\)

\(x = 2\)

However, x >= 3 in this case, which means that x = 2 does not satisfy the original assumption. Therefore, we reject this solution.

Case 2: If x-3 is negative (i.e., x < 3), then we have \(|x-3| = -(x-3) = 3-x,\) and  \(|3x|+1 = -(3x)+1 = 1-3x.\)Therefore, the equation becomes:

\(3-x = 1-3x\)

Simplifying this equation, we get:

\(2x = 2\)

\(x = 1\)

However, \(x < 3\) in this case, which means that \(x = 1\) satisfies the original assumption.

Therefore, x = -1 is the value of the negative number x which satisfies \(|x-3|=|3x|+1\) . option (B) -2 is correct.

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The required solutions of the situation of the function are  x = 3/4  or x = -1/4.

A unique connection where each input only produces one output. A common notation for it is "f(x)," where x denotes the input value. as in f(x) = x/2 ("f of x equals x divided by 2") Since there is only one output ("x/2") for each input ("x"), it is a function: • f(2) = 1.

We have,

f(x)=2x-1

g(x)=3/8x

To find the value of x for f(x) = g(x).

f(x) = g(x)

2x-1 = 3/8x

16x² - 8x = 3

16x² - 8x - 3 = 0

16x² - 12x + 4x - 3 = 0

4x(4x - 3) + (4x - 3) = 0

(4x - 3)(4x + 1) = 0

x = 3/4  or x = -1/4

Thus, required solution are  x = 3/4  or x = -1/4.

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Answer: 1. C=43.96 in  A=153.86 in²

              2. C=56.52 in  A=254.34 in²

Step-by-step explanation:

1.

R=7 in

C=2πR

C=2*3.14*7

C=43.96 in

A=πR²

A=3.14*7²

A=3.14*7*7

A=153,86 in²

2.

R

D=18 in

⇒ R=D/2

⇒ R=18/2

⇒ R=9 in

C=2πR

C=2*3.14*9

C=56.52 in

A=πR²

A=3.14*9²

A=3.14*9*9

A=254.34 in²

The possible amount of time that it could take Wyatt to husk 72 dozen ears of corn is 4 to 6 hours.

What are time and work?

Time is the span of any action or work that occurs or is ongoing. Work is a task or series of activities carried out to accomplish a specific goal.

Given, Wyatt can husk anywhere from  12 to  18 dozen ears of corn per hour.

Thus, to find the time taken to husk 72 dozen ears of corn,

the maximum time = 72/12 =6 hours

And, the minimum time =  72/18 =4 hours

Thus, depending on his speed, Wyatt takes anywhere from  4  to 6 hours to husk  72 dozen ears of corn.

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Answer:

\(26 - 6 = 20\\\)

Suppose there are many other ways to do this, but to get 20, I recommend this equation.

Hope this helps, have a great day! ♣

Answer:

X = 8

Step-by-step explanation:

6 x 8 = 48 - 8 = 40 so 8 is your answer

The expected value of the lottery is $680 dollars which is among the options provided.

Expected value of a lottery refers to the amount that an individual will get on average after multiple trials. It is calculated as a weighted average of possible gains in the lottery with the weights being the probability of each gain.

Assuming that in a lottery you can win 2,000 dollars with a 30% probability, 0 dollars with a 50% probability, and 400 dollars otherwise, the expected value of this lottery is $720 dollars. This is because the probability of winning $2,000 is 30%, the probability of winning 0 dollars is 50%, and the probability of winning $400 is the remaining 20%.

Expected value = 2,000(0.30) + 0(0.50) + 400(0.20)

Expected value = 600 + 0 + 80

Expected value = 680 dollars

So, the expected value of the lottery is $680 dollars which is among the options provided.

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Answer:

1) 1/4<7/12<2/3<5/6

2) 1 1/6<17/12<1 5/8<9/4

Step-by-step explanation:

Answer:

16: 1/4, 2/3, 7/12, 5/6

17: 1 1/6, 17/12, 1 5/8, 9/4

Step-by-step explanation:

Divide each number

16 : 1/4 = .25, 2/3 = .333 , 7/12 = .5888, 5/6 = .8333,

17:  1  + 1/6 = 1.166, 17/12 = 1.41, 1 + 5/8 = 0.625, 9/4 = 2.25

After dividing each fraction, order them from smallest to largest.

I am not sure if this helps, but I tried my best!

Using the center-of-gravity technique, the best site location for the new distribution center in Charlotte is determined to be at coordinates (X, Y) = (27.92, 19.08).

The center-of-gravity technique is used to find the optimal location for a facility based on the distribution of demand. In this case, we will calculate the weighted average of the coordinates (X, Y) of the three sites, with the weights being the number of weekly packages flowing to each site.

To calculate the X-coordinate of the center of gravity, we use the formula:

Xc = (X1 * W1 + X2 * W2 + X3 * W3) / (W1 + W2 + W3)

Similarly, for the Y-coordinate:

Yc = (Y1 * W1 + Y2 * W2 + Y3 * W3) / (W1 + W2 + W3)

Substituting the given values:

Xc = (16 * 26000 + 35 * 12000 + 40 * 10000) / (26000 + 12000 + 10000) ≈ 27.92

Yc = (28 * 26000 + 10 * 12000 + 18 * 10000) / (26000 + 12000 + 10000) ≈ 19.08

Therefore, the best site location for the new distribution center in Charlotte is approximately at coordinates (X, Y) = (27.92, 19.08) based on the center-of-gravity technique.

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Answer:

There is one solution, x = 2 - 3y

Step-by-step explanation:

Nonnegativity conditions impose b. upper bounds on the decision variables in an optimization problem.  The correct answer is b. Upper bounds on the decision variables.

They ensure that the variables cannot take negative values and are typically used when the variables represent quantities that cannot be negative, such as quantities of goods or resources.

By setting an upper bound of zero or a positive value, the nonnegativity condition restricts the feasible region of the optimization problem to only include nonnegative values for the decision variables.

This is a common constraint in many optimization models to reflect real-world limitations or practical considerations.

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Answer:

- ⅓

Step-by-step explanation:

p + q = p - ⅓

q = (p - p) - ⅓

q = - ⅓

The length of the rectangular plot of land is 168 ft.

To find the length of the rectangular plot of land, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In the given diagram, we can see that the plot of land forms a right triangle, where the length of one leg is 874 ft and the length of the diagonal is 890 ft. Therefore, the length of the rectangular plot can be found as follows:

\(h^2 = a^2 +b^2\\b^2 = h^2 - a^2\\b = \sqrt{h^2-a^2} \\b= \sqrt{(890)^2- (874)^2}\\ b= \sqrt{792100- 763876}\\ b= \sqrt{28224}\\ b= 168\)

So, the length of the rectangular plot of land is 168 ft.

The lengths of the legs of a right triangle are related to the lengths of the sides of a rectangle in the following way:

If we draw a rectangle with sides of length "a" and "b", then the diagonal of the rectangle (which is the hypotenuse of the right triangle formed by the sides of the rectangle) will have a length equal to the square root of (\(a^2 + b^2\)).

Conversely, if we have a right triangle with legs of length "a" and "b", then we can form a rectangle by making the legs of the triangle the sides of the rectangle. The length of one side of the rectangle will be "a" and the length of the other side will be "b".

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The probability of randomly selecting a red six or a black king from a 52-card deck is 1/13.

To find the probability of selecting a red six or a black king from a 52-card deck, we need to determine the number of favorable outcomes (red six or black king) and divide it by the total number of possible outcomes (52 cards).

There are 2 red sixes (hearts and diamonds) and 2 black kings (spades and clubs) in a deck.

Since we want to select either a red six or a black king, we can add these numbers together to get a total of 4 favorable outcomes.

Since there are 52 cards in a deck, the total number of possible outcomes is 52.

Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes: Probability = Number of favorable outcomes / Total number of possible outcomes Probability = 4 / 52 Probability = 1 / 13

Therefore, the probability of randomly selecting a red six or a black king from a 52-card deck is 1/13.

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The empirical probability of an event is calculated by dividing the number of times the event occurred by the total number of trials or observations. In this case, Joan rolled the number 2 seven times out of a total of 37 rolls.

To find the empirical probability of rolling a 2, we divide the number of times Joan rolled a 2 (7) by the total number of rolls (37):

Empirical probability of rolling a 2 = Number of times 2 occurred / Total number of rolls = 7 / 37 ≈ 0.189 Therefore, the empirical probability that Joan rolls a 2 is approximately 0.189 or 18.9%.

It's important to note that empirical probability is based on observed data and can vary from the true or theoretical probability. As more trials are conducted, the empirical probability tends to converge towards the true probability.

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Answer:

try 45 if wrong sorry

Step-by-step explanation: