Answer:
[tex]x^{3}+y^{3}=(a+b)(a^{2}-ab+b^{2})[/tex]
Step-by-step explanation:
The given expression is
[tex]x^{3}+y^{3}[/tex]
This expression represents the sum of two perfect cubes, which is factorize as this product
[tex]x^{3}+y^{3}=(a+b)(a^{2}-ab+b^{2})[/tex]
We can demonstrate this factorization by multiplying the product
[tex](a+b)(a^{2}-ab+b^{2})=a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}=a^{3}+b^{3}[/tex]
Therefore, the answer is
[tex]x^{3}+y^{3}=(a+b)(a^{2}-ab+b^{2})[/tex]
Peter wants to cut a rectangle of size 6x7 into squares with integer sides. What is the smallest number of squares he can get
The smallest number of squares he can get would be 2 × 2 squares.
What is the area of the rectangle?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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Which dot plot has more than one mode?
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.
What is a data set?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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What is the classification for this polynomial?
gh^5+2g^3
Choices:
A. Monomial
B. Binomial
C. Trinomial
which is the correct form of q(x) + r(x)/b(x) for expression 7x^4+x+14/x+2
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].
The perimeter of a fence is 140 feet. The sum of three times the length and two times the width is 180 feet. What are the length and width of the fence?
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:
2l + 2w = 140 (Perimeter equation)3l + 2w = 180 (Given equation)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:
3(70 - w) + 2w = 180210 - 3w + 2w = 180-w = -30w = 30Now that we have the width, we can find the length using the simplified perimeter equation:
l + 30 = 70l = 40Therefore, 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.
Explanation: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.
If vector u = (5, 3) and vector v = (-1, 4), what is the component form of vector u + v?
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:
Sum of x-components: 5 + (-1) = 4Sum of y-components: 3 + 4 = 7Therefore, the component form of vector u + vector v is (4, 7).
A rectangular park is 56 miles wide and 157 miles long. What is the area of the park? Enter your answer in the box as a mixed number in simplest form. plz answer nowPLz!!!
Answer:
1 3/7
Step-by-step explanation:
just finished the k12 test
Use >, <, or = to compare these numbers.
3.08 m ____ 3.8 m
Explain how you can classify shapes, using the distance and slope formula. Provide examples to support your response.
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.
Allan purchased a zero coupon bond four years ago. He paid $3,000 and it has a face value of $3,500. How much money did he earn for holding the bond until maturity
math help again please
A) Inverse Property of Multiplication
B) Commutative Property of Multiplication
C) Associative Property of Addition
D) Commutative Property of Addition
A circular piece of fabric has a radius of 1.2 feet. The fabric sells for $5.40 per square foot. What is the total cost of the circular piece of fabric? A. $6.48 B. $20.35 C. $24.42 D. $40.69
if you purchased a camera in British for 100 pounds what would be the price in U.S dollars if one pound is worth $1.40
$71
$100
$140
$1400
Quadrilateral ABCD is inscribed in this circle.
What is the measure of angle C?
Enter your answer in the box.
°
George and Chin work as landscapers. George charges $90 for a 6-hour job. Chin charges $84 for the same job.The table shows their price structures. An equation representing George’s charges is written in the chart.
Landscaping Cost
George
Chin
George charges one hourly rate for the first three hours and then reduces his rate for additional hours. Chin charges the same initial rate as George for the first two hours and the same reduced rate for additional hours.
mc016-1.jpg
What is the equation for Chin’s charges needed to solve the system and find the cost of the initial and additional hours?
Answer:
C
Step-by-step explanation:
did the test
Answer:
c
Step-by-step explanation:
Write the equation of a line that is perpendicular to y=0.25x-7 and passes through the point (-6,8)
The equation of a line that is perpendicular to y = 0.25x -7 and passing through the point (-6,8) is y = -4x -16.
Explanation: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 equation sin(25o) =9/c can be used to find the length of AB .
What is the length of AB? Round to the nearest tenth.
c = 9/sin(25)
c = 9/0.4226
c = 21.2958
9/sin(25)
9/0.422618261741
21.29
system of equations solve y=2x+1 and 2x-y=3
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.
Will use 3/4 cup if olive oil to make 3 batches of salad dressing. How much oil does Will use for one batch of salad dreesing?
I don’t now this answer please help
Please Find the value of x
(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
Avogadro's number is equal to _____. 6.02 x 1025 6.02 x 1023 6.02 x 1010 6.02 x 1022
Find the radius of a sphere with a surface area of 804 cm^2.
A. 9cm
B. 8cm
C. 64cm
D. 204cm
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.
Find the quotient
A. 7r5
B. 6r1
C. 5r5
D. 5r3
A stadium brings in $16.25 million per year. it pays football-related expenses of $13.5 million and stadium expenses of $2.7 million per year. whatis the stadium's current profit margin
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.
Explanation: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 maximum weight for an elevator is 1600 pounds. You need to move boxes each weighing 40 pounds, and you weigh 145 pounds. Write an inequality that can be used to determine the maximum number of boxes that you can place in the elevator at one time. Assume only you and the boxes are in the elevator.
A pharmacist has an 18% alcohol solution. How much of this solution and how much water must be mixed together to make 10 liters of a 12% alcohol solution?
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.
Explanation: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.
The circumference of any circle equal to__ times the diabetes.
A. About 3.07
B. About 3.41
C. Exactly 3
D. About 3.14
Can someone please help me I really would appreciate it please.
1. What happens to shadows at sunrise and sunset? How are these shadows different from those cast at noon?
2. Study the information in the table below. What relationship do you see between a planet's speed of rotation and its length of day?
Diameter of planet Speed of Rotation Length of the Day
Planet A 3,000 miles 29 miles/second 24 hours
Planet B 2,990 miles 60 miles/second 14 hours
Planet C 3,020 miles 120 miles/second 8 hours