8. The height of a trapezoid can be expressed as (4x + 7) while the bases can be expressed as (5x + 6) and (2x-6) and (x + 8). Write and solve an expression to represent the area of the trapezoid. (Hint: A = 0.5h(b₁+b₂)​

The area of the village green is 3552 square meters.

What is area ?

Area is a measure of the size or extent of a two-dimensional surface. It is the amount of space inside a closed figure or shape, and is typically measured in square units such as square meters (m²), square centimeters (cm²), square feet (ft²), square inches (in²), and so on.

To calculate the area of the village green, we can use Heron's formula which is given as:

[tex]Area of triangle = \sqrt{(s(s-a)(s-b)(s-c))}[/tex]

where s is the semi - perimeter (half the perimeter) of the triangle, and a, b, and c are the lengths of its sides.

The semi - perimeter of the triangle can be calculated as:

s = (AB + BC + AC)/2

s = (92 + 75 + 105)/2

s = 136

Using this value of s, we can calculate the area of the triangle as:

Area of triangle = [tex]\sqrt{(136(136-92)(136-75)(136-105))}[/tex]

Area of triangle = [tex]\sqrt{(136(44)(61)(31))}[/tex]

Area of triangle = [tex]\sqrt{(12655264)}[/tex]

Area of triangle = 3552

Therefore, the area of the village green is 3552 square meters.

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The vertices of the dilated image are given.

Dilation is a type of transformation where the figure is enlarged or made smaller such that it preserves the shape but not size.

Every dilated image are similar figures to the original figure.

The point (x, y) with the scale factor of k will dilate to the vertex (kx, ky) if the center of dilation is origin.

1) Given for ΔQRS, Q(-1, 0), R(-1, 2) and S(-2, 1).

Scale factor = 2

Q'(2×-1, 2×0) = Q'(-2, 0)

R'(2×-1, 2×2) = R'(-2, 4)

S'(2×-2, 2×1) = S'(-4, 2)

2) Given for ΔTRK, T(-1, -2), R(1, 0) AND K(0, 1).

Scale factor = 3

T'(-3, -6), R'(3, 0) and K'(0, 3)

3) Given for ΔXYZ, X(-4, 0), Y(-4, 4) and Z(-2, -2).

Scale factor = 1/2

X'(-2, 0), Y'(-2, 2) and Z'(-1, -1)

4) If the center of dilation is not origin, then,

P'(x, y) = O(x, y) + k [P(x, y) - O(x, y)]

where, P'(x, y) is the dilated point, O(x, y) is the center of dilation, k is the scale factor and P(x, y) is the original point.

Given for ΔHAT, H(-1, -1), A(1, 0) and T(-1, 2).

Scale factor = 2

Center of dilation = (-1, 2)

H'(x, y) = (-1, 2) + 2 [(-1, -1) - (-1, 2)]

           = (-1, 2) + 2 (-1+1, -1-2)

           = (-1, 2) + 2 (0, -3)

           = (-1, 2) + (0, -6)

           = (-1, -4)

A'(x, y) = (-1, 2) + 2 [(1, 0) - (-1, 2)]

           = (-1, 2) + 2 (2, -2)

           = (3, -2)

T'(x, y) = (-1, 2) + 2 [(-1, 2) - (-1, 2)]

           = (-1, 2) + 2 (0, 0)

           = (-1, 2)

Hence the vertices of the dilated image are found.

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The polynomial with the given information can be represented as: P(x) = a(x-3)3(x-2)2(x-4). Where a is the leading coefficient. Since the leading coefficient is negative, we can assume that a = -1. Therefore, the polynomial can be written as: P(x) = -(x-3)3(x-2)2(x-4)

This polynomial satisfies all the given conditions:

- The leading coefficient is -1, which is negative.

- The zero of 3 has a multiplicity of 3, as indicated by the exponent of (x-3)3.

- The zero of 2 has a multiplicity of 2, as indicated by the exponent of (x-2)2.

- The zero of 4 has a multiplicity of 1, as indicated by the exponent of (x-4).

Therefore, the polynomial P(x) = -(x-3)3(x-2)2(x-4) is the correct answer.

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Answer: 50 total stamps = 30 Postcard Stamps + 20 Letter Stamps

Step-by-step explanation:

Postcard Stamp = Ps

Letter Stamp = Ls

Ps = 20c

Ls = 33c

50 total stamps = $12.60

50 total stamps = X Ps + Y Ls

12.60 = 30 Ps + 20 Ls

$12.60 = 30(20c) + 20(33c)

50 total stamps = 30 Postcard Stamps + 20 Letter Stamps

Answer:

You have 30 postcard stamps and 20 letter stamps.

Step-by-step explanation:

To solve this you'll need to set up a system of equations. Let's use P for postcard stamps and L for letter stamps

Remember, since the price is in cents, it'll be 0.2 and 0.33.

The equations can be in any order, this is just the order I chose.

0.2P + 0.33L = 12.60

P + L = 50

Your next step is to cancel out one of the variables to solve for the other. Let's cancel out P and solve for L. (You can switch these if you want, you'll still get the same answsrs.) Remember to multiply by a negative so the variable cancels out.

Here's what your work will look like:

0.2P + 0.33L = 12.60

-0.2(P + L = 50)

Here's what your new equations will look like after distributing:

0.2P + 0.33L = 12.60

-0.2P - 0.2L = -10

Now add these two equations together. When you do so, the P cancels out, and you can now solve for L.

Here's what your new equation will look like after adding the two equations:

0.13L = 2.6

Now, divide both sides by 0.13 to get what L equals. After doing so, you should get L = 20. This means that you have 20 letter stamps.

Your last step is to plug the value of L (which is 20) into either of your original equations to solve for how many postcard stamps you have. Let's use our second equation, P + L = 50. (You can use either original equation and get the same answer, but this one is more simpler to use.)

Here's what your work will look like:

P + 20 = 50

Subtract 20 from both sides:

P = 30. This means you have 30 postcard stamps.

Hope this helps!

[tex](\stackrel{x_1}{-4}~,~\stackrel{y_1}{2})\hspace{10em} \stackrel{slope}{m} ~=~ 3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{ 3}(x-\stackrel{x_1}{(-4)}) \implies y -2= 3 (x +4) \\\\\\ y-2=3x+12\implies {\Large \begin{array}{llll} y=3x+14 \end{array}}[/tex]

Answer: y = 3x + 14

Step-by-step explanation:

The slope-intercept form is y=mx+b

We will plug in -4 for the x, 2 for the y, and 3 for the m, and leave b alone to solve for it.

2 = 3(-4) + b

2 = -12 + b

b = 14

The final equation is y = 3x + 14.

Hope this helps!

Answer:

- 10

Step-by-step explanation:

If y =12 when x is 18

Find x when y =-16

What the relationship between x and y

y = x - 6 and x = y +6

Insert y(-16) into equation above

Y = X - 6

-16= X - 6

Collect like terms

Add 6 to both sides

-10 = X

For the function f(x)=(x-2)^(2)-4, the transformation(s) done to the parent function are a horizontal shift to the right by 2 units and a vertical shift downward by 4 units.

The parent function of the function f(x)=(x-2)^(2)-4 is the quadratic function f(x)=x^(2).

The transformation(s) done to the parent function are a horizontal shift to the right by 2 units and a vertical shift downward by 4 units.

This can be seen by comparing the given function to the parent function. The (x-2) in the given function indicates the horizontal shift to the right, and the -4 at the end of the function indicates the vertical shift downward.

Therefore, the transformation(s) done to the parent function are a horizontal shift to the right by 2 units and a vertical shift downward by 4 units.

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Answer: B. h=192÷ (6⋅4)

Step-by-step explanation:

B. h=192÷ (6⋅4)

This equation can be used to find the height of the prism because the volume of a rectangular prism can be represented by the equation V = l*w*h, where l and w are the length and width, respectively, and h is the height. In this case, given that V=192, the equation h=192÷ (l*w) can be used to find the height, where l=6 and w=4.

The length of the missing side is 15.

What is a theorem?

A statement that can be proven true using well-known mathematical procedures and arguments is known as a theorem. A theorem, in general, is an application of a general principle that joins it to a larger theory. Proof is a procedure used to demonstrate a theorem's correctness.

A right triangle's three sides are related in Euclidean geometry by the Pythagorean theorem, also known as Pythagoras' theorem. According to this statement, the areas of the squares on the other two sides add up to the area of the square whose side is the hypotenuse.

The hypotenuse is 17. The length of one leg is 8.

The length other leg is

√(17² - 8²)

= √(289 - 64)

= √(225)

= 15

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Using angle theorems the value of x = 67.5°

Angle theorems are rules that govern the properties of angles.

Given that we have the figure and we desire to find angle x, we proceed as folllows.

Now, we notice that one angle is a right angle and it is equal to 90°.

Also, we notice that all the angles meet at the same point.

So, we have that

x + 3x + 90° = 360° (sum of angles at a point)

Solving for x, we have that

x + 3x + 90° = 360°

4x + 90° = 360°

4x = 360° - 90°

4x = 270°

x = 270°/4

x = 67.5°

So, the value of x = 67.5°

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The number of games in which the player scored more than 25 points is: A. 25%.

In Mathematics, a boxplot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

Based on the information provided about the given box-and-whisker plot (see attachment), the five-number summary for the given data set include the following:

By critically observing the box-and-whisker plots or boxplot (see attachment), we can logically deduce that in 25% of games the player score more than 25 points.

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Answer:

1.7 × .5 = .85

15% smaller than the original

The value of x is 20 and the given two angles are adjacent .

What are parallel lines ?

Parallel lines are lines that are always the same distance apart and never intersect. They have the same slope but different y-intercepts. In other words, they have the same rate of change but different starting points. Parallel lines can be described by linear equations of the form y = mx + b, where m is the slope and b is the y-intercept.

From the given figure

we can say that,

the two angles 7x and x+20 are adjacent

and sum of two angles is 180 degrees.

SO,

7x+x+20 = 180

8x + 20 = 180

8x = 160

x = 160 / 8

x = 20

Therefore, The value of x is 20 and the given two angles are adjacent .

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Answer:

1) To calculate how much it would cost Abi to buy 24 sodas at the vending machines at work, we can multiply the cost of one can ($0.75) by 24 (24 sodas):

Cost at vending machine = $0.75 x 24 = $18

2) To calculate how much Abi can save each day by buying the 24 pack of soda and taking one with her to work, we can subtract the cost of one can at the vending machine ($0.75) from the cost of the 24 pack at Foodtown ($3.25):

Savings each day = $3.25 - $0.75 = $2.50

The inverse of the function f(x) = x² + 3 is, f⁻¹(x) = ± √(3 - x) and it is not a function.

First to be an inverse function that function needs to one to one function, meaning every different preimage must correspond to a different image.

We can obtain the inverse of a function by switching the variables x and y with their respective positions and solving for y in terms of x.

Given, A function f(x) = x² + 3.

Let, y = x² + 3.

x² = 3 - y.

x = ± √(3 - y).

f⁻¹(x) = ± √(3 - x).

The inverse of the function is not a function as two distinct outputs have the same intput.

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Niagara Falls' inclination is roughly 69.8 degrees because Orlando is 178 millimetres due south of the falls, which puts it at a slight angle to Orlando.

Two rays or lines that intersect at the angle's vertex form an angle, a geometric figure in mathematics. Angle sizes range from 0 degrees (a zero angle) to 360 degrees, and they are typically measured in degrees or radians (a full rotation). Mathematical disciplines like geometry, trigonometry, and calculus all use angles as a fundamental concept. They're used to explain how things are positioned in space.

given

The Law of Cosines can be used to calculate Niagara Falls' slope. Let's call the distances from Niagara Falls, Denver, and Orlando, as well as the distance from Denver to Orlando, "a," "b," and "c," respectively. Then:

235mm for a and 178mm for b.

c = 273 mm

For any triangle with side a and angle A across from side a, the following is true in accordance with the Law of Cosines:

A² = 2bc cos - b² + c²

Angle A is the angle at Niagara Falls, and we're trying to find it. The formula above is rearranged to give us:

cos A = (b²+c²-a²)/2bc

If we substitute our current numbers, we get:

cos A is equal to (2 x 178 x 273) / (178 x + 273 - 2352). cos A = 0.3506

Inverse cosine of both sides is calculated, and the result is:

[tex]A = cos^{-1} (0.3506) (0.3506)[/tex]

69.8 degrees A

Niagara Falls' inclination is roughly 69.8 degrees because Orlando is 178 millimetres due south of the falls, which puts it at a slight angle to Orlando.

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Complete question:

On a map, Orlando is 178 mm due south of Niagara Falls, Denver is 273 mm from Orlando, and Denver is 235 mm from Niagara Falls. Find the angle at Niagara Falls.

1. Parallelogram: Opposite sides are congruent, Opposite angles are congruent, Consecutive angles are supplementary, Diagonals bisect each other, Diagonals are congruent, Diagonals are perpendicular.

Congruence is a term used in mathematics to describe the relationship between two geometric figures or objects. When two geometric figures are said to be congruent, it means that the two figures have the same size and shape.

2. Rectangle: Two pairs of consecutive congruent sides, but opposite sides are not congruent, Diagonals are perpendicular, Exactly one pair of congruent angles, Diagonals bisect one pair of angles.

3. Rhombus: Opposite sides are congruent, Opposite angles are congruent, Consecutive angles are supplementary, Diagonals bisect each other, Diagonals are congruent, Diagonals are perpendicular.

4. Kite: Opposite sides are congruent, Opposite angles are congruent, Consecutive angles are supplementary, Diagonals bisect each other, Diagonals are perpendicular, Diagonals bisect angles.

5. Square: Opposite sides of a are congruent, Opposite angles of are congruent, Consecutive angles are supplementary, The diagonals of bisect each other, All four sides are congruent.

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The value of g(f(x)) is  3x^(2)+23x+44.

To find g(f(x)), we need to substitute the expression for f(x) into the expression for g(x). This means that wherever we see an "x" in the expression for g(x), we will replace it with the expression for f(x).

g(f(x)) = 3(f(x))^(2)-7(f(x))+4

Now we can substitute the expression for f(x) into the equation:

g(f(x)) = 3(x+5)^(2)-7(x+5)+4

Next, we need to simplify the expression by expanding the squared term and distributing the -7:

g(f(x)) = 3(x^(2)+10x+25)-7x-35+4

g(f(x)) = 3x^(2)+30x+75-7x-35+4

Finally, we can combine like terms to get the polynomial in standard form:

g(f(x)) = 3x^(2)+23x+44

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By 12.57 feet the circumference of outer circle greater than the circumference of the inner​ circle

The circle's boundary is the length of the circle's circumference, sometimes referred to as its perimeter. whereas the size of a circle determines the space it occupies.

The formula for calculating a circle's circumference is written as

Circumference = πD

where D stands for the circle's diameter.

The inner circle is 44 feet in diameter. It means that

22/7 × D = 44

22D = 7 × 44 = 308

D = 308/22

D = 14 ft

The inner circle and the outer circle are separated by 2 feet. This implies that the outer circle's diameter would be

14 + 2 + 2 = 18 ft

Circumference of the outer circle is

22/7 × 18 = 56.57 ft

The distance in feet that the circumference of the outer circle exceeds that of the inner circle is

56.57 - 44 = 12.57 feet

Hence, By 12.57 feet the circumference of outer circle greater than the circumference of the inner​ circle

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The population after 4 years will be 12,155.06. It will take approximately 22.52 years for the population to triple.

The population of a town grows proportionally to its current population. This means that the larger the population, the more it will grow. To determine the population after 4 years, we can use the formula:

P = P0 * (1 + r)^t

Where P is the final population, P0 is the initial population, r is the growth rate, and t is the time in years. Plugging in the values given in the question, we get:

P = 10,000 * (1 + 0.05)^4

P = 10,000 * 1.21550625

P = 12,155.06

Therefore, the population after 4 years will be 12,155.06.

To determine how long it will take the population to triple, we can use the same formula, but set P to 3 * P0 and solve for t:

3 * 10,000 = 10,000 * (1 + 0.05)^t

3 = (1 + 0.05)^t

ln(3) = t * ln(1.05)

t = ln(3) / ln(1.05)

t = 22.52

Therefore, it will take approximately 22.52 years for the population to triple.

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The number of people we would expect to be found within an area of 1400 m by 1.3 km is approximately 25.

The mass of the cube for the object that has the side lengths of 56 cm is 0.043904 g.

1. To find the number of people in an area of 1400 m by 1.3 km, we need to first convert the measurements to the same unit. Since the population density is given in people/km2, we will convert the measurements to kilometers.
1400 m = 1.4 km
Now we can multiply the two measurements to find the area:
1.4 km x 1.3 km = 1.82 km2
Next, we can use the population density to find the number of people in this area:
14 people/km2 x 1.82 km2 = 25.48 people
Therefore, we would expect to find approximately 25 people in an area of 1400 m by 1.3 km.

2. To find the mass of the cube, we first need to find the volume. Since the side lengths are given in centimeters, we will convert them to meters:
56 cm = 0.56 m
Now we can find the volume of the cube by multiplying the side lengths:
0.56 m x 0.56 m x 0.56 m = 0.175616 m3
Next, we can use the density to find the mass of the cube:
0.25 g/m3 x 0.175616 m3 = 0.043904 g
Therefore, the mass of the cube is 0.043904 g.

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The given statement [tex]x = \frac76[/tex] can be proved by using the given equation.

Algebra, a discipline of mathematics where abstract symbols rather than concrete numbers are subjected to arithmetic operations and formal transformations. The idea that there is such a separate branch of mathematics, as well as the term algebra to identify it, came about gradually through time. tries to find solutions to equations that contain one or more unknown quantities. Although early mathematicians would not have been familiar with such a concept, the word "equation" is frequently employed to conveniently represent these operations when discussing the early history of algebra.

As per the given data:

[tex]\frac{2x+6}{5} = 4x -3[/tex]

Simplifying to evaluate the expression:

[tex]\frac{2x+6}{5} = 4x -3[/tex]

[tex]= 2x+6 = 5(4x -3)\\= 2x + 6 = 20x - 15\\= 18x = 21\\= x = \frac{21}{18}\\\\= x = \frac76[/tex]

Hence, the given statement [tex]x = \frac76[/tex] can be proved by using the given equation.

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There is no solution for the equation   -6-(4x)/(x+2)=(8)/(x+2).

To solve for all values of x in the equation -6-(4x)/(x+2)=(8)/(x+2), we need to use the following steps:

Step 1: Multiply both sides of the equation by (x+2) to eliminate the denominators. This gives us:
-6(x+2) - 4x = 8

Step 2: Simplify both sides of the equation by distributing and combining like terms. This gives us:
-6x - 12 - 4x = 8
-10x - 12 = 8

Step 3: Add 12 to both sides of the equation to isolate the variable on one side. This gives us:
-10x = 20

Step 4: Divide both sides of the equation by -10 to solve for x. This gives us:
x = -2

However, we need to check our solution to make sure it is not an extraneous solution. If we plug x = -2 back into the original equation, we get:
-6 - (4(-2))/(-2+2) = (8)/(-2+2)
-6 - (-8)/0 = (8)/0

Since we cannot divide by zero, our solution of x = -2 is an extraneous solution and there are no valid solutions to this equation. Therefore, the answer is no solution.

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Using geometric sequences, 20,480 rabbits are grown during the 12th month.

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the preceding term by a fixed non-zero number called the common ratio.

At the end of the first month, there are 10 x 2 = 20 rabbits.

At the end of the second month, there are 20 x 2 = 40 rabbits.

Similarly, at the end of the third month, there are 40 x 2 = 80 rabbits.

This is a geometric sequence with a common ratio of 2 and a first term of 10. The number of rabbits at the end of the 12th month would be:

10 x 2^12 = 10 x 4096 = 40,960 rabbits

To find the number of rabbits grown during the 12th month, we need to subtract the number of rabbits at the end of the 11th month from the number of rabbits at the end of the 12th month:

[tex]40,960 - (10 x 2^{11}) = 40,960 - 20,480 = 20,480[/tex]

Hence, 20,480 rabbits are grown during the 12th month.

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In order to validate these outliers and their possible influence on any following analysis or modelling, additional research and analysis of the data set may be required.

Data points that differ from the distribution's norm are considered outliers. a figure that falls "outside" of the normal range for the record (either significantly lower or substantially greater). For instance, when the scores are 25, 29, 3, 32, 85, 33, 27, 28, etc., both 3 and 85 are "outliers". Step aside. To spot outliers, look for numbers that are noticeably greater or lower than all other values. Value 4 is an outlier since it is much smaller than all other values. An outlier is a finding in a population sample selected at random that deviates unusually from other values.

There seem to be two possible outliers in the data, according to the scatter diagram you gave.

The data point located at about is the first outlier (60, 30).

The second anomaly is a data point that is situated at around (20, 120). At the upper right corner of the plot, this data point appears to be substantially higher than the other data points.

It's important to keep in mind that identifying outliers from a single scatter plot can be difficult and highly influenced by the surrounding data. In order to validate these outliers and their possible influence on any following analysis or modelling, additional research and analysis of the data set may be required.

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The area of a trapezoidal region is 1/2h(a+b), where a and b are the lengths of the two parallel sides, and h is the height.

The area of a trapezoidal region is given by the formula A = 1/2h(a+b), where a and b are the lengths of the two parallel sides, and h is the height. This formula can be derived by dividing the trapezoid into two right triangles and a rectangle, and finding the area of each of these shapes.

First, let's consider the two right triangles. The height of each of these triangles is h, and the lengths of their bases are (a-b)/2 and (b-a)/2, respectively. The area of each of these triangles is therefore 1/2h(a-b)/2 = 1/4h(a-b).

Next, let's consider the rectangle. The length of this rectangle is b, and the height is h. The area of this rectangle is therefore bh.

The area of the trapezoidal region is the sum of the areas of the two right triangles and the rectangle. This gives us:

A = 1/4h(a-b) + 1/4h(b-a) + bh

Simplifying this equation gives us:

A = 1/2h(a+b)

Therefore, the area of a trapezoidal region is 1/2h(a+b), where a and b are the lengths of the two parallel sides, and h is the height.

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Answer: $12.45.

Without a calculator, we can first round a dinner bill of $83.44 to the closest whole figure and estimate a 15% tip. In this instance, $83.44 can be rounded up to $83.

Then, we can calculate 15% of $83 by using a quick technique. In order to do this, we must first calculate 10% of $83, or $8.30. After that, we can calculate 5% of $83 by subtracting 50% of 10%, or $4.15.

Finally, we can add 10% and 5% together to get an estimate of 15% of $83:

$8.30 (10%) + $4.15 (5%) = $12.45

So, a 15% tip on a dinner bill of $83.44 would be approximately $12.45.

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Answer: A = 1/2(8+12)h

Step-by-step explanation: The formula for area of a trapezoid is 1/2(b1+b2)h so if you just sub in all the variables "a" would be correct

An equation for Area of trapezoid is,

A = (8 + 12) h / 2

We have,

The bases of a trapezoid are 8 centimeters and 12 centimeters, and the height is h centimeters.

We know that,

Area of trapezoid is,

A = (a + b) h / 2

Where, a and b are bases and h is height of trapezoid.

Here,, a = 8 cm,

b = 12 cm

Hence, Area of trapezoid is,

A = (8 + 12) h / 2

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He should use 11.11 pounds of the $0.90 per pound coffee, and 16.89 pounds of the $2.40 per pound coffee for a total cost of 28 pounds.

x + y = 28

0.90x + 2.40y = 38.20

x = 11.11, y = 16.89

The grocer wants to mix 28 pounds of two kinds of coffee, one selling for $0.90 per pound and the other for $2.40 per pound, and sell it for $1.45 per pound. To determine how many pounds of each kind to use in the new mix, we can set up a system of equations. Let x represent the quantity of pounds of coffee priced at $0.90 per pound and y the quantity of pounds of coffee priced at $2.40 per pound. Thus, x + y = 28 (the total number of pounds) and 0.90x + 2.40y = 38.20 (the total cost of the 28 pounds, at $1.45 per pound). Solving this system of equations yields x = 11.11 pounds of the $0.90 per pound coffee and y = 16.89 pounds of the $2.40 per pound coffee. Therefore, the grocer should use 11.11 pounds of the $0.90 per pound coffee, and 16.89 pounds of the $2.40 per pound coffee for a total of 28 pounds.

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