What is the area of this trapezoid? 44 in² 64 in² 168 in² 192 in² Trapezoid A B C D with parallel sides D C and A B. Points F and E are between D and C. F E B A form a rectangle with 4 right angles. D F is 2 inches, F E is 14 inches, E C is 2 inches, A B is 14 inches., and E B is 12 inches.

The expression [tex]\(\frac{7x^4 + x + 14}{x + 2}\)[/tex] is [tex]\(7x^3 - 14x^2 + 28x - 55 + \frac{124}{x + 2}\)[/tex].

To find the correct form of the expression [tex]\(\frac{7x^4 + x + 14}{x + 2}\)[/tex] in the form [tex]\(q(x) + \frac{r(x)}{b(x)}\)[/tex], where [tex]\(q(x)\)[/tex] is the quotient and [tex]\(r(x)\)[/tex] is the remainder, we need to perform polynomial long division. Here, [tex]\(b(x) = x + 2\)[/tex].

Step-by-Step Solution:

1. Setup the Division:

  Divide [tex]\(7x^4 + 0x^3 + 0x^2 + x + 14\)[/tex] by [tex]\(x + 2\)[/tex].

2. First Term:

  - Divide the leading term of the dividend by the leading term of the divisor: [tex]\(\frac{7x^4}{x} = 7x^3\)[/tex].

  - Multiply [tex]\(7x^3\)[/tex] by [tex]\(x + 2\)[/tex]: [tex]\(7x^3 \cdot (x + 2) = 7x^4 + 14x^3\)[/tex].

  - Subtract this from the original polynomial:

    [tex]\[ (7x^4 + 0x^3 + 0x^2 + x + 14) - (7x^4 + 14x^3) = -14x^3 + 0x^2 + x + 14. \][/tex]

3. Second Term:

  - Divide the leading term of the new polynomial by the leading term of the divisor: [tex]\(\frac{-14x^3}{x} = -14x^2\)[/tex].

  - Multiply [tex]\(-14x^2\) by \(x + 2\)[/tex]: [tex]\(-14x^2 \cdot (x + 2) = -14x^3 - 28x^2\)[/tex].

  - Subtract this from the current polynomial:

    [tex]\[ (-14x^3 + 0x^2 + x + 14) - (-14x^3 - 28x^2) = 28x^2 + x + 14. \][/tex]

4. Third Term:

  - Divide the leading term of the new polynomial by the leading term of the divisor: [tex]\(\frac{28x^2}{x} = 28x\)[/tex].

  - Multiply [tex]\(28x\)[/tex] by [tex]\(x + 2\)[/tex]: [tex]\(28x \cdot (x + 2) = 28x^2 + 56x\)[/tex].

  - Subtract this from the current polynomial:

    [tex]\[ (28x^2 + x + 14) - (28x^2 + 56x) = -55x + 14. \][/tex]

5. Fourth Term:

  - Divide the leading term of the new polynomial by the leading term of the divisor: [tex]\(\frac{-55x}{x} = -55\)[/tex].

  - Multiply [tex]\(-55\)[/tex] by [tex]\(x + 2\)[/tex]: [tex]\(-55 \cdot (x + 2) = -55x - 110\)[/tex].

  - Subtract this from the current polynomial:

    [tex]\[ (-55x + 14) - (-55x - 110) = 124. \][/tex]

Final Result:

- Quotient [tex]\(q(x) = 7x^3 - 14x^2 + 28x - 55\)[/tex]

- Remainder [tex]\(r(x) = 124\)[/tex]

Thus, the expression [tex]\(\frac{7x^4 + x + 14}{x + 2}\)[/tex] can be written as:

[tex]\[7x^3 - 14x^2 + 28x - 55 + \frac{124}{x + 2}\][/tex]

This is the required form [tex]\(q(x) + \frac{r(x)}{b(x)}\)[/tex].

Final answer:

Shapes can be classified using the distance and slope formulas by calculating side lengths and verifying perpendicular or parallel lines, as demonstrated in identifying a rectangle through side equality and perpendicularity.

Explanation:

You can classify shapes by using the distance and slope formulas to understand their geometry, such as identifying parallelograms or triangles with equal sides. The distance formula, √((x2-x1)² + (y2-y1)²), helps calculate the exact length between two points, allowing one to discern if sides are equal and hence contributing to classifying the shape. The slope formula, (y2-y1) / (x2-x1), helps determine the incline or decline between two points, which is essential for identifying parallel or perpendicular lines, thus assisting in shape classification.

For example, to classify a quadrilateral as a rectangle, you could calculate the distance between all points to ensure opposite sides are equal and use the slope formula to ensure adjacent sides are perpendicular by checking if the product of their slopes is -1.

To find the stadium's profit margin, subtract its expenses from its revenue and divide by the revenue, then multiply by 100 to get the percentage.

In order to find the stadium's current profit margin, we need to subtract its total expenses from its total revenue and then divide the result by the total revenue. The stadium brings in $16.25 million per year and has football-related expenses of $13.5 million and stadium expenses of $2.7 million per year.

To calculate the profit margin, we subtract the total expenses ($13.5 million + $2.7 million) from the total revenue ($16.25 million): $16.25 million - ($13.5 million + $2.7 million) = $16.25 million - $16.2 million = $0.05 million.

Finally, we divide the profit ($0.05 million) by the total revenue ($16.25 million) and multiply by 100 to get the profit margin as a percentage: ($0.05 million / $16.25 million) * 100 = 0.003076923076923 * 100 = 0.3076923076923%.

The volume of the sand in the top portion of the hourglass is calculated using the volume formula for cones. Given the dripping rate, it takes about 115 seconds for all the sand to drip to the bottom.

To determine how many seconds it will take for all the sand to drip to the bottom, we need to find the volume of sand in the top cone.

The total height of sand in the top portion is 54 millimeters, which means the equivalent of three cones of sand (since 54 mm / 18 mm/cone = 3).

Thus, the total volume of sand is 3 x 1206.37 = 3619.11 cubic millimeters.

Given the sand drips at a rate of 10π cubic millimeters per second, we find the total time by dividing the volume by the rate:

Time = Total Volume / Rate = 3619.11 / 10π ≈ 115.13 seconds.

Therefore, it will take approximately 115 seconds for all the sand to drip to the bottom of the hourglass.

Answer:

[tex]6\frac{2}{3}[/tex] of 18%,  [tex]3\frac{1}{3}[/tex] water

Step-by-step explanation:

The 18% solution added to the water is equal to the 10 liters of 12% solution.

Let's say that the amount of 18% solution is x

18% of x = 0.18x

Let's say that the amount of water is 10-x

Because there is no alcohol in water, it will be 0(10-x), or just 0

0.18x + 0 = 10•0.12

0.18x = 1.2

x = [tex]6\frac{2}{3}[/tex]

10 - [tex]6\frac{2}{3}[/tex] = [tex]3\frac{1}{3}[/tex]

To make a 10 liter solution of 12% alcohol, you need to mix 6.67 liters of an 18% alcohol solution with 3.33 liters of water.

To make a 12% alcohol solution, we need to mix the 18% alcohol solution with water. Let's assume x liters of the 18% alcohol solution and (10 - x) liters of water.

The amount of alcohol in the 18% solution is 0.18x liters, while the amount in the water is 0 liters. In the final solution, the amount of alcohol is 0.12 * 10 = 1.2 liters.

So, we can set up the equation 0.18x + 0 = 1.2 and solve for x. By solving this equation, we find that x = 6.67 liters. Therefore, 6.67 liters of the 18% alcohol solution and 3.33 liters of water should be mixed to make 10 liters of a 12% alcohol solution.

Final answer:

The radius of a sphere with a surface area of 804 cm² is found by using the formula A = 4πr², solving for r, and taking the square root. The correct answer is B. 8cm.

Explanation:

To find the radius of a sphere with a given surface area, we use the formula for the surface area of a sphere, which is A = 4πr². Given that the surface area (A) is 804 cm², we can solve for the radius (r).

Plugging the given surface area into the formula yields:

804 cm² = 4πr²

Next, we divide both sides of the equation by 4π to solve for r²:

r² = 804 cm² / (4π)

To find r, we take the square root of both sides:

r = [tex]\sqrt{(804[/tex]cm² / (4π))

Calculating the right side of the equation gives us the radius:

r ≈ [tex]\sqrt{(804[/tex] cm² / 12.5663706143592)

r ≈ [tex]\sqrt{(64[/tex]

r = 8 cm

Therefore, the correct answer is B. 8cm.

You can expect to have approximately $447 in your account after 1.5 years of continuously compounded interest on your $400 investment.

How much will you have in the account after 1.5 years?

To find out how much you'll have in your account after 1.5 years, we can follow these steps:

1. Formula: Use the formula for continuous compounding:

[tex]Final\ amount = Initial\ amount * (1 + annual\ interest\ rate)^{time}\[/tex]

2. Plug in the values: We know:

Initial amount = $400

Annual interest rate = 7.6% (remember to convert it to a decimal by dividing by 100, so 7.6% becomes 0.076)

Time = 1.5 years

3. Calculation: Substitute the values and calculate:

Final amount = $400 * (1 + 0.076)^{1.5}

[tex]Final\ amount = $400 * (1 + 0.076)^{1.5} = 446.5[/tex]

4. Round the answer: Round the final amount to the nearest dollar:

Final amount  = 447

The ordered pairs are (0, 1), and (2, -1) which are solutions to the inequality 2x+3y>-1 option first and fifth are correct.

It is defined as the expression in mathematics in which both sides are not equal they have mathematical signs either less than or greater than known as inequality.

We have an inequality:

2x+3y>-1

The above inequality is a straight line and represents a region.

After plotting the inequality on the graph.

The points are in the region of the inequality:

These points satisfy the inequality.

Thus, the ordered pairs are (0, 1), and (2, -1) which are solutions to the inequality 2x+3y>-1 option first and fifth are correct.

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Final answer:

The perimeter problem is solved using two simultaneous equations derived from the given information about the perimeter and sum of sides. The solution reveals that the width (w) of the fence is 30 feet and the length (l) is 40 feet.

Explanation:

We are given that the perimeter of a fence is 140 feet. The perimeter formula for a rectangle is P = 2l + 2w, where l is the length and w is the width of the rectangle.

According to the problem, we also know that 3 times the length plus 2 times the width equals 180 feet (3l + 2w = 180).

Let's assign variables: Let l represent the length and w represent the width of the fence. We can now set up two equations:

We can solve these equations simultaneously to find the values for l and w.

First, simplify the perimeter equation by dividing everything by 2: l + w = 70. Now substitute l from the simplified equation into the given equation to find w:

Now that we have the width, we can find the length using the simplified perimeter equation:

Therefore, the width (w) is 30 feet, and the length (l) is 40 feet.

The length of the fence is 40 feet and the width is 30 feet.

To solve this problem, we can set up a system of equations. Let's denote the length of the fence as 'l' and the width as 'w'. From the given information, we have the following equations:

2l + 2w = 140   (equation 1)

3l + 2w = 180   (equation 2)

We can solve this system of equations by eliminating a variable. Multiply equation 1 by 3 and equation 2 by 2 to eliminate 'w':

6l + 6w = 420   (equation 3)

6l + 4w = 360   (equation 4)

Subtract equation 4 from equation 3 to eliminate 'l':

6w - 4w = 420 - 360

2w = 60

w = 30

Substitute the value of 'w' into equation 1 to solve for 'l':

2l + 2(30) = 140

2l + 60 = 140

2l = 80

l = 40

Therefore, the length of the fence is 40 feet and the width is 30 feet.

Answer:

c on edge

Step-by-step explanation:

took the exam

Table 4 represents the linear function that has slope of 5 and a y-intercept of 20.

Further explanation:

The linear equation with slope [tex]m[/tex] and intercept [tex]c[/tex] is given as follows.

[tex]\boxed{y = mx + c}[/tex]

Given:

The y-intercept is [tex]\left( { - 9, - 3} \right).[/tex]

The slope of the linear function is [tex]- 6.[/tex]

Explanation:

The linear function can be expressed as follows,

[tex]y = 5x + 20[/tex]

In table 1,

Substitute [tex]-60[/tex] for [tex]x[/tex] and 8 for [tex]y[/tex] in linear function to check whether the point satisfy the linear function.

[tex]\begin{aligned}8&= 5\left({ - 60} \right) + 20\\8&= - 300 + 20\\8&= - 280\\\end{aligned}[/tex]

Table 1 is not correct.

In table 2,

Substitute 20 for [tex]x[/tex] and [tex]0[/tex] for [tex]y[/tex] in linear function to check whether the point satisfy the linear function

[tex]\begin{aligned}0&= 5\left( {20} \right) + 20\\0&= 100 + 20\\&0\ne120\\\end{aligned}[/tex]

Table 2 is not correct.

In table 3,

Substitute [tex]-20[/tex] for [tex]x[/tex] and 0 for [tex]y[/tex] in linear function to check whether the point satisfy the linear function

[tex]\begin{aligned}- 20&= 5\left( 0 \right) + 20\\- 20 &\ne 20\\\end{aligned}[/tex]

Table 3 is not correct.

In table 4,

Substitute [tex]-4[/tex] for [tex]x[/tex] and 0 for [tex]y[/tex] in linear function to check whether the point satisfy the linear function

[tex]\begin{aligned}0&= 5\left( { - 4} \right) + 20\\0&= - 20 + 20\\0&= 0\\\end{aligned}[/tex]

Table 4 is correct.

Table 4 represents the linear function that has slope of 5 and a y-intercept of 20.

Learn more:

1. Learn more about the equation in the intercept form https://brainly.com/question/1473992.

2. Learn more about binomial and trinomials https://brainly.com/question/1394854.

3. Learn more about range and domain of the function https://brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Linear equation

Keywords: linear function,  numbers, slope 5, y-intercept of 20, slope intercept, inequality, equation, linear inequality, shaded region, y-intercept, graph, representation.

For this case we have the following system of equations:

[tex]y = 2x + 1\\2x-y = 3[/tex]

We cleared "y" of the second equation:

[tex]y = 2x-3[/tex]

Now we equate the equations:

[tex]2x+1=2x-3\\1 = -3[/tex]

Since 1 is not equal to -3 then the system has no solution.

This can also be observed, knowing that the lines are parallel since they have the same slope, [tex]m = 2.[/tex]

Answer:

The system has no solution.

The equation of a line that is perpendicular to y = 0.25x -7 and passing through the point (-6,8) is y = -4x -16.

In the mathematical realm, a line that is perpendicular to another has a slope which is the negative reciprocal of the original line's slope. The given equation, y = 0.25x - 7, has a slope of 0.25. Thus, the slope of the perpendicular line would be the negative reciprocal, which is -1/0.25 = -4. Now, a line's equation can be written in the form y = mx + b, where m is the slope and b is the y-intercept. Although we now know our perpendicular line's slope, we still need the y-intercept. To find it, we use the formula b = y - mx, substituting in our known slope (-4) and given point (-6,8). After calculation, b = 8 - (-4*-6) = -16. Therefore, the equation of a line that is perpendicular to y = 0.25x - 7 and passes through the point (-6,8) is y = -4x -16.

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The smallest number of squares he can get would be 2 × 2 squares.

The area of the rectangle is the product of the length and width of a given rectangle.

The area of the rectangle = length × Width

We are given that Peter wants to cut a rectangle of size 6x7 into squares with integer sides.

The area of rectangle = 6 x7 = 42

Peter can cut 5 squares, by making;

4 × 4,

3 × 3 and 2 × 2 squares.

The smallest number of squares is 2 × 2 squares.

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Answer:

C

Step-by-step explanation:

did the test

Answer:

c

Step-by-step explanation:

Answer:

The dot plot representing the calico crayfish has more than one mode.

Step-by-step explanation:

We know that mode of the data is the value corresponding to the highest frequencies.

or we may say the mode of a data set is the number that occurs most frequently.

Clearly from the dot plot we could see that the data representing the calico crayfish have two quantities with four dots ( one is when number of calico crayfish are 5 and the other when number of calico crayfish are 8)

Also by making a frequency table we may check it as:

Number of calico crayfish         frequency

      1                                                    1

      2                                                   0

      3                                                    1

      4                                                    3          

     5                                                    4

      6                                                     1    

      7                                                     2

      8                                                     4  

      9                                                     0  

     10                                                     2

The dot plot shows calico crayfish has more than one mode. In the data set each column of a table represents a specific variable and each row represents a specific record of the data sets.

A data set is a set of information corresponds to one or more database tables in the case of tabular data,

From the table of the calico crayfish, it is observed that the data no 5 on the x-axis the frequency is 4. As well as on data no 8 the frequency is 4.

The dot plot shows calico crayfish has more than one mode. because it shows the same frequency at two other variables;

Hence the dots plot shows calico crayfish has more than one mode.

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(x+8)*8 = (9+7)*9

(x+8) *8 = 16*9

(x+8) *8 = 144

x+8 = 144/8

x+8 = 18

x = 18-8

x = 10

c = 9/sin(25)

c = 9/0.4226

c = 21.2958

9/sin(25)

9/0.422618261741

21.29

So, the component form of vector u + v is (4, 7).

Final answer:

The component form of the sum of vector u = (5, 3) and vector v = (-1, 4) is found by adding corresponding components, yielding the sum vector u + v = (4, 7).

Explanation:

To find the component form of the sum of two vectors, vector u, and vector v, you simply add the corresponding components of each vector. The given vectors are vector u = (5, 3) and vector v = (-1, 4). By adding the x-components (5 and -1) and the y-components (3 and 4), we get the sum of the vectors.

The component form of vector u + vector v is calculated as follows:

Therefore, the component form of vector u + vector v is (4, 7).